3C. Graphs versus function tables
3D. Notes
A table of the function “twenty-five minus
the square”, or “twenty-five with minus
on the square” for each item of the
argument However, the characteristics of the
function can be seen more easily in a graph of
the function, produced by plotting each column
of the table as follows: starting at
an arbitrary point on a sheet of squared
paper measure a distance Graphs can be produced
quickly and accurately by the computer as follows: Plots of polynomial functions may show
further interesting characteristics. For
example, study the plot of the following function,
use Remarks: The result of To anyone familiar with the functions
sine and cosine of trigonometry, it may be
evident that the functions Moreover, the
cosine and sine of a given angle are
sometimes defined as the coordinates of a
point on the unit circle of radius 1, located
at the given angle (that is, the arc length
along the circumference). It should therefore
come as no surprise that the plot of
The expression In addition to providing an overall view of
a function, its graph shows the local
behaviour, the slopes of the individual
segments reflecting its local rates of growth
and decay. Moreover, the graph provides
a direct view of the area under it, a result
of considerable significance.
This will be illustrated by a
plot of a semi-circle. The function The function The last result is the approximate area
of the complete circle, and is therefore an
approximation to the constant Tables of functions that apply pairwise
to their arguments can, however, provide
similar insights. Moreover, they can provide
other information (such as the rate of
change of the rate of change) not readily
grasped from a graph.
To this end we will define a pairwise
operator Pairwise relations such as Any reader puzzled by certain notations
(such as the double use of
4B. Notes
Polynomials are important for a number of reasons:
* Because of the wide choice of coefficients available,
polynomials can be defined
to approximate most functions of practical interest.
* As already illustrated for sums, products,
and composition of polynomials, they
are closed under a number of important functions,
in the sense that the resulting
function is again a polynomial. These include: In most of these cases, the Taylor operator can be used
to obtain the coefficients
of the resulting polynomial.
With increasing use of computer experimentation,
it becomes important to learn to
use the available tools. In particular:
5B. Truncated power series
5C. Notes
On the other hand, the power series:
As illustrated by the last column, these
truncated power series are approximations
to the trigonometric sine function (on radian
arguments). Moreover, the Taylor
operator However, it was possible to
confine that discussion to rather
elementary ideas, whereas a meaningful
discussion of the uses of power series
would quickly lead to more advanced and
less familiar mathematical notions
outside the experience of many readers.
The same is true of many topics (such as the
derivative, symbolic logic, sets, and
permutations), and we will leave the reader
to observe the importance of topics as
they are exploited in later work. In other
words, some faith is expected of the
reader – a belief that topics will be introduced
only if they are both important and
interesting.
6B. Derivatives of polynomials
6C. Taylor coefficients
6D. Notes
Considered as a sample of points on a
continuous graph of the function (using an
infinite number of points), these sloping
lines are secants (cutting lines) to the
continuous curve, and the slope at a point
is the tangent (touching line) to the curve, which may
be approximated by a secant with a small interval.
The expression Multiplication of
the sums gives the square of Similar calculations for the
product This
is all embodied in the calculation Note that the zero value of the graph of the
derivative occurs at the argument
value for which the original function reaches
its low point, that is, where its graph
is horizontal.
The phrase derivative of In this chapter we have skirted the issue
by confining attention to polynomial
functions, for which the limit of the secant
slope is easily obtained. We will,
however, extend these results to the many
important functions that can be
approximated by the power series (themselves
polynomials) that were discussed in
Chapter 5.
Can we be certain that the derivative of
a polynomial approximation to a
function
7B. The name "exponential"
The simplest case is where the rate of
growth is equal to the size -- this is described
by the exponential function, denoted
here by Since a polynomial may be found that can
approximate almost any function, it
should be possible to find one that approximates
the exponential. Consider the
following: In the dyadic use of the symbol
8B. Harmonics
8C. Decay
If Moreover, the acceleration is caused by the
“restoring force” exerted by the
spring, which is proportional to its extension
as measured by the position In the simplest case
9B. Equations
9C. The method of false position
9D. Newton’s method
9E. Roots of Polynomials
9F. Logarithms
The need for inverse functions arises frequently.
For example, the heat produced
by an electric heater may be given by the
function The function The element of Although Newton’s method converged rather rapidly for
the function Nevertheless, the function The logarithm is discussed further in Chapter 20.
10B. Word formation
10C. Parsing
10D. Conventional mathematical Notation
Formal grammar
becomes particularly important in
writing, where the reader has no recourse to
interjected questions, but must extract
meaning from the text alone.
Similar considerations apply in mathematics.
Although the Hogben remarks cited
in our preface emphasize the importance of
language and grammar, we have
introduced the grammar of our language J
only informally, allowing us to
concentrate on the mathematical notions
it has been used to convey.
This informal approach has been made
tolerable by limiting the writing required of
a student in Exercises to imitation
of expressions already used, and by providing
guidance through conversation with the
computer, a native speaker of the
language. This chapter will address the
formal grammar of J, and will also
comment on the more loosely structured
grammar of conventional mathematical
notation.
In a written English sentence this
process is so simple as to go unnoticed, although
the separation by spaces is somewhat
complicated by things such as hyphens and
apostrophes. In oral communication it
is also unnoticed, not because the breaking
of a continuous stream of sound into words
is simple, but because (except in the
case of a foreign language) it is so deeply ingrained.
In the case of J the word formation requires some attention,
although it is simple
enough to have been thus far treated informally by example.
The computer
provides a word-formation function that may be applied to a
sentence enclosed in
quotes. Thus: Alternative spellings for the national use
characters (which differ from
country to country) are discussed under Alphabet ( Numbers are denoted by digits, the
underbar (for negative signs and for
infinity and minus infinity -- when
used alone or in pairs), the period (used
for decimal points and necessarily
preceded by one or more digits), the
letter A numeric list or vector is denoted by
a list of numbers separated by
spaces. A list of ASCII characters is
denoted by the list enclosed in single
quotes, a pair of adjacent single quotes
signifying the quote itself:
Names (used for pronouns and other surrogates,
and assigned referents by
the copula, as in A primitive
or primary may be denoted by a
single graphic (such as A sentence is evaluated by executing
its phrases in a sequence determined by the
parsing rules of the language. For
example, in the sentence Moreover, the left argument of an
adverb or conjunction is the entire verb
phrase that precedes it. Thus, in the
phrase One important consequence of these rules is that
in an unparenthesized expression
the right argument of any verb is the result of
the entire phrase to its right. The
sentence Parsing proceeds by moving successive elements
(or their values except in the case
of proverbs and names immediately to the left
of a copula) from the tail end of a
queue (initially the original sentence prefixed by
a left marker §) to the top of a
stack, and eventually executing some eligible
portion of the stack and replacing it
by the result of the execution. For example,
if Parsing can be observed using the trace
conjunction The classes of
the first four elements of the stack are
compared with the first four columns of
the table, and the first row that agrees in
all four columns is selected. The bold
italic elements in the row are then subjected
to the action shown in the final column, and are replaced by its result. If no row is
satisfied, the next element is transferred from the queue:
It includes a discussion of the
learnability of mathematical notation as
compared with other specialized notations
and with native languages, as well as the
importance of such learnability.
SOME HISTORICAL VIEWS ON NOTATION
Mathematicians have often noted the power
and importance of notation. In Part IV
of his two-volume A History of Mathematical
Notations [3], Florian Cajori offers
the following examples: Nothing in the history of mathematics is to
me so surprising or impressive
as the power it has gained by its notation or language ...
But in Napier’s time, when there was practically
no notation, his discovery
or invention [of logarithms] was
accomplished by mind alone, without any
aid from symbols. J.W.L. Glaisher
Some symbols, like an, ..., log n,
that were used originally for only positive
integral values of n stimulated
intellectual experimentation when n is
fractional, negative, or complex, which
led to vital extensions of ideas. F. Cajori
The quantity of meaning compressed into
small space by algebraic signs, is
another circumstance that facilitates
the reasonings we are accustomed to
carry on by their aid. Charles Babbage
Symbols which initially appear to have
no meaning whatever, acquire
gradually, after subjection to what
might be called experimenting, a lucid
and precise significance. E. Mach In Sections 740 ff, Cajori laments the
lack of standardization of mathematical
notation in the following excerpts:
§740 Uniformity of mathematical notations
has been a dream of many
mathematicians ...
§741 The admonition of history is clearly
that the chance, haphazard procedure of
the past will not lead to uniformity.
§746 In this book we have advanced many
other instances of “muddling along”
through decades, without endeavor on the
part of mathematicians to get together
and agree on a common sign language.
§748 In the light of the teaching of history
it is clear that new forces must be
brought into action in order to safeguard the
future against the play of blind
chance. ... We believe that this new agency
will be organization and co-operation.
To be sure, the experience of the past in
this direction is not altogether reassuring.
However, in William Oughtred: A Great
Seventeenth-Century Teacher of
Mathematics [4], Cajori touches on a
quite different agency that bears upon the
development and teaching of mathematics --
the mathematical instrument. On
page 88 we have:
PROGRAMMING LANGUAGES
Early computers provided a small set of operations
such as addition and
multiplication, and were controlled by programs
specifying sequences of these
operations. For example, on the Harvard Mark IV [5], the program: Programs were later developed to translate from
languages more congenial to
mathematicians into equivalent machine language
programs for subsequent
execution. For example, in BASIC, a language
largely derived from FORTRAN
(Formula Translator) [6], one might write: Moreover, many have moved to
Computer Science, unwittingly following the advice of
Newton, as expressed in
the following excerpt from page 95 of Cajori [4]: The proliferation of programming languages
shows no more uniformity than
mathematics. Nevertheless, programming
languages do bring a different
perspective. Not only do they provide
precisely defined grammars, but they
address a much wider range of applications
than do any one of the notations
treated by Cajori. Moreover, they sometimes
adopt notions from advanced
mathematics and apply them fruitfully at elementary levels.
For example, the essential notion of Heaviside’s
operators [7] (introduced in the
treatment of differential equations) is the
application of an entity (called an
operator) which, applied to a function,
produces a related function; a notion more
familiar in the form f ' for the derivative of f.
In APL [8], an operator (denoted
by /) is used to provide reduction over a vector
argument. The sum over a vector
is obtained as follows: ECONOMY OF SYMBOLS
Far from introducing further symbols in
the manner of Oughtred’s list (of some
150) shown in §181 of Cajori [3], programming
languages are largely confined to
a standard alphabet (ASCII) whose special characters
are limited to the following
familiar list: TRAINS Some writers, such as March and Wolf
in their Calculus [10], denote the
function that is the sum of functions INFLECTION A symbol formed by appending a dot
or colon to a given function
symbol will be said to be inflected, and such
inflection is used to assign related
symbols to related functions, providing names
that therefore have some mnemonic
value. For example, in his Laws of Thought [9],
George Boole used The analogy between
times and and was helpful, and
the conflict with the normal use of the symbols
was of no concern within the
confines of logic. In a wider context it is a
concern, and J uses A UNIFORM NOTATION
Cajori recognizes the importance of
individual invention, but warns against
individual efforts to impose a uniform system: §742 This confusion is not due to the
absence of individual efforts to introduce
order. Many an enthusiast has proposed
a system of notation for some particular
branch of mathematics, with the hope,
perhaps, that contemporary and succeeding
workers in the same field would hasten
to adopt his symbols and hold the
originator in grateful remembrance. Oughtred
in the seventeenth century used over
one hundred and fifty signs for elementary
mathematics -- algebra, geometry, and
trigonometry. However,
such a system can be used as a basis for the
discussion of mathematical notation: the
precision provided (or enforced) by
programming languages and their execution
can identify lacunas, ambiguities, and
other areas of potential confusion in
conventional notation. Discussion here will focus
on such matters, and may suggest remedies.
Discussion will also center on a single
programming language (J), for the following reasons:
* As indicated in my A Personal View of APL [12],
it was designed “as a simple,
precise, executable notation for the teaching of
a wide range of subjects”.
* It has received some use in mathematical exposition.
* Its simple grammar is defined by a ten-by-four
table in terms of six parts of
speech.
* It uses vectors, matrices, and higher-dimensional
arrays, based on the unifying
concept of rank or order in Tensor Calculus [13].
* In addition to the bond operator already discussed,
it embraces some fifteen or
more operators (such as dual, Taylor, and
Hypergeometric) of interest in
mathematics.
GRAMMAR
The evaluation, interpretation, or execution
of the mathematical expression (or
sentence) In natural language, the word-formation is
largely unnoticed, except in listening to
an unfamiliar language. In programming
languages, the word-formation and
parsing are commonly lumped together under
the term syntax.
In J, the word-
formation may be made explicit by applying
the word-formation function ( Although programming languages adopt the
term grammar from English, most use
terms for the parts of speech adopted from
mathematics. J, on the other hand, uses
terms from English, avoiding certain ambiguities
(such as the use of operator both
as a synonym for function in elementary math, and
as an operator in the sense of Heaviside),
and providing additional distinctions. The
following fragment exemplifies all of the
parts of speech: The choice of English terms is based on the
analogy between the “action words”
verb and function (which derives from the
Latin fungio, to perform). Moreover,
adverb describes the action of an operator
in applying to a verb to produce a verb,
and conjunction reflects the use of the
copulative conjunction and to produce a
verb such as "run and hide". The word
proverb (in the sense used here) is
pronounced with a long o.
The term variable merits further comment.
The expressions However, the use of the term
variable for the pronoun SYNONYMS, ANONYMS, AND SUBSTITUTION
A synonym of a word is a word with a similar,
but not necessarily equivalent,
meaning. Synonyms that occur in mathematical
notation can be misleading if they
are carelessly interpreted.
For example,
division may be expressed as a%b or
a/b. Are they equivalent? If so,
are a%b%c and and a/b/c and a%b/c and a/b%c
equivalent? Or do they differ in order of execution?
It may be said that “one would never write
such forms”, but if not, why not? Are
there any rules that forbid it, and how is
the beginning student to know? Notice
that a/ 3/4 would probably be accepted, if
only because the form 3/4 no longer
denotes division: it is the commonly-accepted
form of the constant three-quarters.
Synonyms for multiplication raise further questions. For example, are the following
expressions all executed in the same
order: a%b*c and a%bc and a/b*c and 3/4c?
The term anonym (a nameless person) will
be used for any nameless entity: in the
product abc, the matrix product M N (or MN),
and the power xn, the functions
applied are unnamed. Although this anonymity
normally goes unremarked, it has
consequences. As remarked in the section on
Programming Languages, the
notation Moreover, the strict use
of the anonymous form abc forbids the
use of multi-character mnemonic names
such as age and area in elementary algebra,
giving the unfortunate impression that
algebra is about letters rather than about the use of names.
Multi-character names that end in digits do get
used: a2 is interpreted as a single
name rather than as the product of a and 2, although 2a
is interpreted as two times a.
An important general notion is the validity of
substituting equals for equals, and it
is particularly valuable in mathematics. Nevertheless,
conventional mathematical
notation does not always permit it. For example, using
the anonymous form xy: In APL and J there is no heirarchy, the
need for parentheses in polynomials (as in
many other expressions) being obviated
by the use of vectors, as in The expression The math result is the first element
less the sum of the rest (i.e., Since Heirarchy introduces ambiguities, particularly for
synonyms and anonyms, which
may or may not be accorded the same positions:
are a/bc and a/b*c equivalent?
CONSTANTS
Mathematics reserves various symbols for constants,
such as e (for Euler’s
number), and the Greek pi. Programming languages
use “scientific” notation of the form
Notations such as NEUTRAL OR IDENTITY ELEMENT
In Section 2.6 of Concrete Mathematics [2],
Graham, Knuth and Patashnik state
that “... a product of no factors is conventionally
taken to be 1 (just as a sum of no
terms is conventionally 0).”
These choices are not
mere conventions, and may be
based more firmly on the neutral or identity
element of a function, that is, a value
Using The indicated identities hold as well
for the case EMPTY CASES IN DEFINITION
In a definition of the form x(x-1)...(x-m+1)
(As in Eq 2.43 of Concrete
Mathematics [2] together with the note: m factors
for m not equal to 0), the notation does not
show explicitly what happens in the case of m=0.
In vector notation, this definition
(of the falling factorial function)
becomes Because the factors in the expression change in
steps like the stope in a mine, these
functions are collectively called stopes, and the
fit operator Furthermore, DUALS
It is a familiar observation that almost every
stated task t is preceded by some
preparation p, and followed by restoration
(the inverse of p). This is sometimes
expressed as performing “t under p”, as
in “surgery under anesthetic”. In
mathematics, the terms dual and duality
are sometimes used, as in “or is the dual
of and under (or with respect to) negation”,
and “the duality of and and or”. The
use of under is ubiquitous in mathematics
and related disciplines, but often cloaked
in other terms, such as similarity in the
product S-1 A S, where S is a matrix of
eigenvectors that maps to “natural” coordinates.
We will express “t under p” as
This last example uses the function LEARNABILITY OF LANGUAGE
The learnability of a discipline depends
strongly on the learnability of the language
used. In teaching mathematics we should
therefore consider the learnability of the
language used, and compare it with that of
specialized languages (such as musical
notation), and of our native tongue. For example,
a child can quickly learn to
transcribe the following tune from musical notation
to a piano, perhaps by merely
watching it done by an adult: Further refinements such as the indication of time,
sharps, flats, and phrasing may
be added and learned with equal facility. Two matters
warrant further comment:
* A less graphic presentation of the notes (such as
the sequence e c a c e c)
would be more difficult for a beginner to learn.
Moreover, the phenomenal rate
of transcription by a pianist could hardly be
achieved using parallel lists of
named notes. Admittedly, this speed is due in
part to the musical coherence of
the piece transcribed, just as our speed in reading
a meaningful English
sentence is much greater than in reading gibberish.
* The ease of learning musical notation claimed in
this example is due in part to
the tool used; in this case the piano. The linear
arrangement of the piano keys
mirrors the vertical movement over the musical
staff, and proves much easier
to understand than the multiple strings (and
absence of frets) on a violin.
With the computer and programming languages,
mathematics has newly-acquired
tools, and its notation should be reviewed
in the light of them. The computer may,
in effect, be used as a patient, precise,
and knowledgeable “native speaker” of
mathematical notation.
In The Language Instinct [14], Pinker has
treated the phenomenal learnability of
natural languages as an inborn instinct in
children, and in The Symbolic Species
[15], Deacon has presented the co-evolution
of language and the human brain as
an explanation of this miracle of an inborn
facility. In Deacon’s view, language has
developed under evolutionary pressure to have,
at least for elementary purposes, a
uniform structure that is easily learned,
particularly by infants. For example, the
pattern in the sentences: applies uniformly to other nouns and verbs.
Moreover, these patterns extend to
simple compound sentences and tenses: Of course a child never hears a phrase such
as “holded the doll”, but uniformly
applies the pattern used in “dropped”,
finally adopting the anomalous forms of
strong verbs such as hold. According to
Pinker, this adoption occurs at a time
quite independent of badgering by adults
(No, dear, you held the doll).
However, in contrast to our native tongue,
mathematical notation is neither
uniform nor easy to learn. Consider the following
examples, expressed first in
English: A review of earlier sections will suggest other
examples of non-uniform notation
that further increase the difficulties for a beginner.
Experienced mathematicians
may well accept such non-uniformities as obvious or
natural.
Moreover, they will be properly concerned
about the continued readability
of older literature if any change were to be made.
However, this is not a new
problem, and there are known ways in which such
change may be addressed.
Are such non-uniformities essential to mathematics,
or are they simply relics of the
“...chance, haphazard procedure of the past...” cited
by Cajori, and therefore
replacable? To illustrate the possibilities, we will
repeat the foregoing examples
together with equivalent expressions in two distinctly
different programming
languages: A PROGRAM FOR MATHEMATICAL NOTATION
We cannot expect our mathematical notation to be
as accessible as our native
language. In particular, it can never enjoy the
ever-present opportunity to
experiment and to provide immediate and gratifying
results that is enjoyed by our
native language, at least not for very young children.
However, we might make
mathematical notation more accessible by:
* Complementing existing notation by notation possessing
a simple and uniform
grammar and using only familiar and readily available symbols.
* Making this complementary notation executable on a
computer, so that a
student might easily obtain gratifying results that
encourage exploration.
* Providing material to guide exploration in a
restrained manner that does not
stifle the thrill of independent discovery.
EXPLORATION
We now present examples of using the computer
in exploring a few functions and
operators, as well as in exploring the grammar
of the notation used. We invite
mathematicians to ponder the question of how
they might be presented in
conventional notation.
Exp 1 From the functions: Observe the results and attempt to identify
and name the functions used; but do not
spend too much time, since the next exploration
provides better tools.
Exp 2 Enter Exp 3 Define and experiment with functions
such as Exp 4 Repeat explorations 2 and 3 using the
following symmetric lists, and
comment on any interesting characteristics
of the resulting tables: Exp 6 Use the results of the following
expressions (and similar experiments) to
state in English (and in conventional math notation)
the relations among the tables
generated: Exp 7 Enter expressions of the following forms,
and then describe in English (and
in mathematical notation) the effects of Exp 8 Examine the production of tables of Stirling
numbers by the expression Exp 9 To explore the rhematic (word-formation) rules
of J, enter expressions
such as:
Exp 10 Enter the expressions Tracing may be switched off by
entering The parsing table from Section C illuminates
the annotations provided by the trace.
Exp 11 For further simple explorations use
Exercises from Chapter 1 of Iverson
[16].
SUMMARY
In this section we have attempted to:
* Use the strict grammar and precise
interpretation of programming languages to
illuminate some of the vagaries of a mathematical
notation that has, as De
Morgan said, “...grown up without much looking to,
at the dictates of
convenience ...”
* Indicate the potentially benign effects on the
learnability of mathematical
notation that can be provided by simple uniform
notation combined with the
opportunity for precise and independent exploration
using computers.
* Sketch the possibilities of adopting some
complementary and non-conflicting
notation executable on computers.
11B. Informal proofs
11C. Formal proofs
11D. Inductive proofs
11E. Recursive definition
11F. Guessing games
Although proofs are an important (and many
might say the essential) part of
mathematics, we will spend little time on them in this book.
In introducing his book Proofs and
Refutations: The Logic of Mathematical
Discovery [17], Imre Lakatos makes the following point:
The main point of the present book is to
exploit new tools for the exploration of
relations and patterns, that can be used
by both mathematicians and laymen to find
those guesses that are amenable to, and
worthy of, proof. We recommend the
reading of Lakatos at any point: his book
is highly entertaining, instructive, and
readable by any layman with the patience
to look up the meanings of a small
number of words such as polyhedron, polygon, and convex.
The following quotes from Lakatos reflect
his view of the importance of guessing: I have had my results for a long time,
but I do not yet know how I am to arrive at
them.
Gauss
If only I had the theorems! Then I should
find the proofs easily enough.
Riemann
I hope that now all of you see that proofs,
even though they may not prove, certainly
do help to improve our conjecture.
Lakatos
On the other hand those who, because
of the usual deductive presentation of
mathematics, come to believe that the
path of discovery is from axioms and/or
definitions to proofs and theorems, may completely forget about the possibility and
importance of naive guessing.
Lakatos We cite two further comments on
the effects of excessive formalism in the
teaching of mathematics: Plato’s exaltation of mathematics as
an august and mysterious ritual had its roots in
dark superstitions which troubled, and
fanciful puerilities which entranced, people
who were living through the childhood of
civilization, when even the cleverest people
could not clearly distinguish the difference
between saying that 13 is a ‘prime’
number and saying that 13 is an unlucky
number. His influence on education has
spread a veil of mystery over mathematics
and helped to preserve the queer
freemasonry of the Pythagorean brotherhoods,
whose members were put to death for
revealing secrets now printed in school books.
It reflects no discredit on anyone if this
veil of mystery makes the subject distasteful.
Plato’s great achievement was to invent a
religion which satisfies the emotional needs
of people who are out of harmony with
their social environment, and just too
intelligent or too individualistic to seek
sanctuary in the cruder forms of animism.
Hogben
… the author has the notion that mathematical
formulas have their “secret life,”
behind their Golem-like appearance. To bring
out the “secret life” of mathematical
relations by an occasional narrative digression
does not appear to him a profanation of
the sacred rituals of formal analysis but
merely an attempt to a more integrated way of
understanding The reader who has to struggle
through a maze of “lemmas,”
“corollaries,” and “theorems,” can easily
get lost in formalistic details, to the
detriment of the essential elements of
the results obtained. By keeping his mind on the
principal points he gains in depth,
although he may lose in details. The loss is not
serious, however, since any reader
equipped with the elementary tools of algebra and
calculus can easily interpolate the
missing details. It is a well-known
experience that
the only truly enjoyable and profitable
way of studying mathematics is the method of
“filling in details” by one’s own efforts.
Lanczos[18]
However, it is important to remember
that in addition to the mathematics that is
important to the intelligent layman,
there is another mathematics, as expressed by
G.H. Hardy in his A Mathematician’s Apology [19]: What about other steps, such as the
equivalence of For the layman it may be best
to accept informal proofs, but harbor a
healthy suspicion that obvious ideas are sometimes wrong.
In a further quotation from Lakatos:
… You are interested only in
proofs which ‘prove’ what
they have set out to prove. I am interested
in proofs even if they do not
accomplish their intended task. Columbus
did not reach India but he
discovered something quite interesting. Suppose that by assuming that If, independently of this assumption, we are able
to show further that equality does
hold for some specific value (such as A recursive definition therefore depends
upon three functions, such as
A recursively defined function provides
a convenient base for an hypothesis for use
in an inductive proof. For example a
function We will
use this last relation in an
inductive proof that the sum of the first The quotes from Lakatos in Section 11A
emphasize the importance of guessing at
relationships, and the use of proofs
to test and refine the resulting conjectures. We
will now discuss techniques for guessing,
emphasizing the use of the computer.
However, games are seductive; the reader
should perhaps limit the time spent on
this section, but plan to return to it again and again.
EACHBOX and BOX
The list function
DIAGONAL
It is sometimes necessary to apply a function
to each diagonal of a table. For
example:
DIFFERENCES
If
The letters in a word can be sorted into
alphabetical order by using the function The last result indicates that there
are only six anagrams of the three-letter word,
with the indices The last results illustrate that anagrams
of lists of numbers may be made, and that
if these lists are indices (such as In making tables of anagrams by hand
as required in the first list of Exercises, it
was probably easy to do the first two
by simply jotting down the anagrams
unsystematically. However, for larger
tables it becomes difficult to avoid
repetitions and to ensure that the table
is complete, especially if one does not know
how many there are in all.
The number of anagrams of an A comparison of Exercises 3 and 4 indicates
that it is easier to write the table for a
word whose letters are in some systematic order,
preferably alphabetic. The result
of If This suggests a method for making permutations
of the next higher order: replicate
and modify the table of permutations of
order
13B. Or, and, and not
13C. Lists and sets
13D. Classification
The list of positive numbers that occur
in a list Otherwise one might
confuse the question of the ordering of
the defining list
14B. Commutativity
14C. Associativity
14D. Symmetry
14E. Distributivity
14F. Distributivity of dyadic functions
14G. Parity
For example, Section 11C showed
that Certain important identities are expressed
as general properties of functions,
referred to as commutativity, associativity,
symmetry, distributivity, and parity.
The expressions A function that yields the same result for
any association imposed by parentheses is
said to be associative. For example, A function that applies to any permutation
of a list to yield the same result is said
to be symmetric. For example: Since double and halve each distribute
over addition, and since they may be
expressed as the equivalent
functions However, The situation is more accurately stated
by saying that the monadic functions
As we will show in Chapter 19, we can use complex numbers
to similarly express the sine and cosine as odd and even
parts of a function closely related to the exponential.
15B. Dot product as a linear vector function
15C. Matrix inverse
Comment on the matrices
that represent them and their inverses.
16B. Products of polynomials
16C. Multiplication of decimal numbers
16D. Other bases
16E. Remainder, Divisibility, and Integer part
16F. Notes
ax0+bx1+cx2+dx3
ax3+bx2+cx1+dx0
Descending order is commonly used in high school,
but ascending order is more
suitable in advanced math, because in
truncated power series a “largest exponent”
may not be clearly identified.
To compare these schemes we will define
two functions called The final example suggests that a descending
polynomial with right argument It follows that the elements of An equivalent list of single digits can,
however, be obtained by “carrying” all but
the final digit to the next higher element,
as illustrated in the following sequence: Not only is this
combined multiplication-and-carry
more difficult to execute accurately, but the
resulting record is almost impossible to check
for possible errors except by
repeating the entire process – a repetition that
invites repetition of the same errors.
The remainder function is denoted by
If the remainder The number The integer quotient can also be obtained by
applying the integer part function
III II; IIIoII; III IIo; IIIoIIo
...If our base is b, we need only (b-1)
other signs
e.g. if b=10 the other signs we need are nine in all.
We can then express any nameable number however great
without enlarging our stock in trade.
...Its invention liberated the human intellect from the prison
bars of the counting frame. The new script was a
complete model of the mechanical process one performs
with it. With a sign for the empty
column, 'carrying over' on slate, paper or parchment is
just as easy as carrying over on the abacus.
...In mediaeval Europe, the name for such rules
was algorithms, a corruption of the name of a
thirteenth century mathematician, spelled Al
Khwarismi or Alkarismi.
17B. Designing an algorithm
17C. Explicit definition
A recipe may call for some thing (such as Hollandaise
sauce) which is itself specified by a recipe; in mathematics or
programming such a thing is more commonly called
a function or component function. For
example, in the program the program for the function
EXAMPLE 1 Define a function for the decaying sine curve.
We begin by assigning values to a list argument In Section 11E, the agenda operator
18B. Area under a graph as a function
18C. Polynomial approximations
We will first make a
similar graph, using a finer grid (with a spacing 0.05 between points): Consider
the point
The rate of change
In other words, the area under the graph of a function
is the anti-derivative of the function. Since this area
can be viewed as the aggregation or integration of
the component areas, it is also called the integral
of the function.
This situation is analogous to the equation-solving of
Chapter 9, which gives the inverse of a function for some
chosen point, but not the inverse function itself.
A practical solution to the anti-derivative of a function The expression The function
19B. Complex numbers
19C. Division
19D. The Exponential Family
Such negative numbers were once regarded as curious
absurdities, but are found to serve consistently and usefully
under addition, subtraction, and multiplication.
Similarly, the introduction of division as the inverse of
multiplication leads to the consistent and useful notion
of fractional numbers.
Because the square of any number (positive or negative)
is non-negative, the introduction of the square root as the inverse
of the square leads to a further extension when applied to
a negative number. These new numbers were (and still are) called
imaginary. For example:
The function
20B. Mathematics: from the Birth of Numbers
20C. Concrete mathematics
20D. Computer Resources
20E. Conclusion
These facts make it possible to present to the layman
a simple view of calculus as the
study of the rate of change of a function,
and to use it to provide insight into matters such
as the sine and cosine functions (so useful in
trigonometry and the study of mechanical and
electrical vibrations), and into the exponential
and its inverse the logarithm (so useful in
growth processes and matters such as the familiar
musical scale). These matters have now been addressed, but
there remain many practical and interesting topics (such
as probability and statistics) that are well within the
grasp and interest of a layman. Rather than pursue these, we will conclude
by suggesting sources for further study.
Although there are many excellent elementary treatments of
math (including some high-school texts), they are all couched
in conventional notation, which we have (except for the
discussion in Chapter 10) largely ignored. This may make
their study difficult.
However, the use of a foreign (that is, unfamiliar)
language also offers advantage: the effort of translation
often forces one to "look behind the words" to gain a clearer
understanding of their import.
We will discuss just two texts, in
addition to resources available on the computer.
The first text (Gullberg, Mathematics: From the Birth
of Numbers [20]) is serious, entertaining, and extensive,
and provides a wealth of interesting historical detail. As
stated in a foreword by Peter Hilton (an emeritus professor of
mathematics):
This is not to deny the usefulness of mathematics;
this very usefulness, however, tends to conceal and
disguise the cultural aspect of mathematics.
...
Further, the study of mathematics starts with the teaching
of arithmetic, a horrible, wretched subject, far removed from
real mathematics, but perceived to be useful. As a result,
vast numbers of intelligent people become 'mathematics avoiders'
even though they have never met mathematics. Their desire to
avoid the tedium of elementary arithmetic, with its boring,
unappetizing algorithms and pointless drill-calculations, is
perfectly natural and healthy.
To those intelligent people, it must seem absurd to liken
mathematics to music as an art to be savored and enjoyed even
in one's leisure time. Yet that is how it should appear and could
appear if it were playing its proper role in our (otherwise) civilized
society. Just as an appreciation of music is a hallmark of the
educated person, so should be an appreciation of mathematics. The
author of this book is a splendid example of an educated person bearing
this hallmark.
The second (Graham, Knuth and Patashnik, Concrete
Mathematics [2]) is presented as "A foundation for Computer
Science". As stated in the preface:
This text is recommended not only because of its clarity and
coverage, but also because of my Concrete Math Companion [21],
which may prove helpful in the matter of notation.
It provides an extensive commentary on the Graham text,
expressed in the executable notation used here.
CONTINUED FRACTIONS On page 127, Gullberg says:
N = a0 + 1
-----------------------
a1 + 1
------------------
a2 + 1
-------------
a3 + 1
--------
a4 + ...
or, in a kind of accepted mathematical "shorthand",
N = [a0, a1, a2, a3 ...]
The fibonacci numbers (treated by Gullberg
on page 287) may also be obtained as continued fractions.
For example: On page 143, Gullberg shows examples of continued fraction
approximations to irrational numbers such as the square root
of six:
Multiplying all entries of any row multiplies the determinant
by the same factor.
The area of a triangle is one-half the
absolute value of the determinant of
the matrix obtained by bordering the three-by-two
table of its coordinates by a column of ones. Similarly, the tetrahedron specified by a four-by-three
table of points in three dimensions may be bordered by ones
and subjected to the determinant to yield six times ( The following practicable
systems of logarithms are used:
lg x The common logarithm of x
(to base 10) (also called log x ...)
ln x The natural logarighm of x (to base e)
lb x The binary logarithm of x (to base 2)
loga x The logarithm of x to the base a
(also called alog x, especially
in older texts)
1. Zero is a number. 2. Every natural number or
zero, a, has an immediate successor a + 1. GENERATING FUNCTIONS and TAYLOR SERIES In Section 7.2
GKP states: G(z) = g0 + g1z + g2z2
+ ... (7.12) and we say that G(z), or G for short, is
the generating function for the sequence
<g0,g1,g2, ...>, which we also call
<gn>. The coefficient gn
of zn in G(z) is sometimes denoted [zn]G(z).
For example, the
functions for the exponential, sine, and the product of the sine and the
exponential may be generated as follows: a1 + a2 +...+ an
where each ak is a number that
has been defined somehow.
...the number of ways to partition a set of n things into k
nonempty subsets In Section 5F, CMC introduces the falling factorial
function The required transformation matrices are given by:
These topics are presented as labs in the sense that,
at any point in the presentation, you may enter your
own J expressions to experiment with the ideas presented. Or
you may use the facilities described in Section 4B to
capture and revise expressions already used in the presentation
or in your own experiments.
For example, choosing "Fractals, Visualization, & J" leads
to a function which, applied repeatedly to the one-element
list
DEMONSTRATIONS The demos include a game (called Pousse), a table of
distances between major cities, and numerous graphical displays
of objects in two and three dimensions. By choosing the
Definition option in some displays you may study, and even
modify, the programs that produce them.
DICTIONARY DEFINITIONS Further information about J may
be obtained from the Help menu as described in Section 4B, but
definitions may also be obtained more directly by pressing
key F1 to display the vocabulary, and then clicking on
the desired item.
For example, selecting the item
A definition may also be displayed by placing the cursor immediately
to the left of a symbol on the screen and pressing F1 while
holding down the CONTROL key.
The view which we shall explore is that mathematics is the
language of size, shape and order and that it is an essential
part of the equipment of an intelligent citizen to
understand this language. If the rules of mathematics are
the rules of grammar, there is no stupidity involved when
we fail to see that a mathematical truth is obvious. The rules
of ordinary grammar are not obvious. They have to be learned.
They are not eternal truths. They are conveniences without
whose aid truths about the sorts of things in the
world cannot be communicated from one person to another.
.......
Before the assembled court, Euler accosted him with the following
pronouncement, which was uttered with due gravity: ....... Like many of us, Diderot
had stagefright when confronted with a sentence in size language.
He left the court abruptly amid the titters of the assembly,
confined himself to his chambers, demanded a safe conduct, and
promptly returned to France. This discussion may be found by first clicking on Chapter 20 in
the list at the beginning of the text, then clicking on Section
20C, and then moving back up the page to the last part of Section 20B.
The function denoted by Compare your definitions with those given below.
In his imaginative and persuasive Vision in Elementary
Mathematics [22], W.W. Sawyer uses similar (hand-drawn)
pictures to introduce elementary concepts in arithmetic and
algebra, clarifying the ideas by "translating" from pictures
to notation and back.
Sawyer's pictures do not even require the somewhat abstract
notion of the numeral Addition is shown as:
For further justification of the choice of Click on any one of the Chapters listed at the beginning of the text, and
then click on the chapter heading to return to the list of chapters.
Click on any one of the Sections listed at the beginning of
the selected chapter, and then click on the Section heading to return to the chapter.
Again select a chapter and a section within it. Then click on the
label beginning with S that appears to the right of the section heading.
Click on the heading of the supplemental section to return to
the corresponding main section.
Click on the Help menu to display the items for which information is
available. Then click on Foreign Conjunction, and then on item 9 to see the
case used in defining Finally press the Escape key (Esc) to remove the Help display.
A deinition may be displayed directly without displaying the Vocabulary: simply
place the cursor just to the left of the symbol, and press F1 while holding down the
control key (Ctrl).
Experiment with their joint use, employing the shape function
Examine a plot of the sine function ( Look up the definition of Finally, plot
Test this assertion for various values of
Experiment with the growth function Then use other initial brackets to approximate the other two roots, and apply
Experiment with other polynomial functions. Note that a polynomial will have
one less root than the number of elements in its coefficient, but that some of them
will be complex in the sense defined in Chapter 19.
The method of false position will find only the real (that is, non-complex)
roots. The result of We will use the function Experiment with multiplying other pairs of complex numbers.
In particular, verify that the product of The (monadic) function Make a table of the sum of squares (3A. Plotting
S3A.
x=: i:5 can be prepared as follows:
f=: 25 with - on *:
x=: i:5
x,:f x
_5 _4 _3 _2 _1 0 1 2 3 4 5
0 9 16 21 24 25 24 21 16 9 0
Study of such a function table can reveal
much about the behaviour of the
function: for example, from 0 at the
argument _5,
it grows at an ever-decreasing
rate to a high point at 0 25, and then declines
at an ever-increasing rate to 5 0.
x to the right (or
left, if negative), and a distance f x
upward (or downward, if negative), and mark
the resulting point. Then join adjacent points
by "lines", and points to arguments by "sticks".
load 'plot'
PLOT=: 'line,stick' with plot
PLOT x;f x
A plot can be removed from the screen
(to permit further computation) by pressing
the key labelled Esc.
However, anyone unfamiliar with such
graphs should perhaps make a few by hand,
using simple functions such as the cube
(f=: ^ with 3), negation (f=: -), and
the identity (f=: ]).
x,:fs x or
x,.fs x to make its table,
and compare your observations with the
remarks given below:
fs=: 0 1 0 _1r6 0 1r120 0 _1r5040 with p.
x=: 1r3*i:10
PLOT x;fs x
fs appears to be an
oscillating function that begins at about
(-3+1r7),0; drops at a decreasing
rate to a low of _1; rises
through 0 0 to a high of 1; and again
drops to about (3+1r7),0.
PLOT x;f x is said to be a
plot of f x “against” or “versus” the
argument x. A function can also be plotted
against another function, as illustrated
in the following exercise.
fc=: 1 0 _1r2 0 1r24 0 _1r720 0 1r40320 with p.
set 5
x,.fc x
PLOT x;fc x
(fc x),.(fs x)
PLOT (fc x);(fs x)
fs and fc are
approximations of them. fs against fc in
the preceding exercise
produced an approximate circle.
d,:e
produces a two-rowed table from the
lists d and e, and
expressions of the form c,d,:e
form multi-rowed tables from further lists.
Moreover, PLOT x;c,d,:e plots each of the
lists against x in a single graph,
providing visual comparison of the functions
represented by the lists.
PLOT x;(fs x),:fc x)
PLOT x;(fs x),:fc x-3r2)3B. Local behaviour and area
S3B.
cir=: 0 with o. gives the square
root of one minus the square
of its argument. Its plot for the
argument a=: 1r5*i:5 is therefore a rough
approximation to a semi-circle with a radius of one unit. Thus:
cir=: 0 with o.
a=: 1r5 * i:5
a
_1 _4r5 _3r5 _2r5 _1r5 0 1r5 2r5 3r5 4r5 1
PLOT a;cir a
sum=: +/ sums a list, and
sum cir a therefore sums the
altitudes of the graph of the
semi-circle, thus giving an approximation to its area
(except that the sum must be multiplied
by 1r5, the common width of the
trapezoids that form the area). Thus:
sum=: +/
1r5 * sum cir a
1.51852
2*1r5 * sum cir a
3.03705
pi.
Better approximations are provided by a larger
number of points:
b=: 1r1000 * i:1000
2 * +/1r1000 * cir b
3.14156
3C. Graphs versus function tables
S3C.
Insights provided by the graph of the
function cir that are not provided so
directly by its function table are
based upon the pairwise views of adjacent points;
the slopes of the line segments between
them reflect the rate of change of the
function, and the trapezoids defined by
them provide a basis for the area.
pw, a sum function, a commuted
difference function, and an average or
mean function. Thus:
pw=: 1 : '2 with (u.\)'
sum=: +/
dif=: -~/
mean=: sum % #
y=: cir a
y
0 0.6 0.8 0.917 0.98 1 0.98 0.917 0.8 0.6 0
mean y
0.69
mean pw y NB. Average heights of the trapezoids
0.3 0.7 0.86 0.95 0.99 0.99 0.95 0.86 0.7 0.3
dif pw y
0.6 0.2 0.12 0.063 0.02 _0.02 _0.063 _0.12 _0.2 _0.6
We will further illustrate the pairwise
operator on a function for the Fibonacci
numbers, using a definition discussed
under “generating functions” in [2]:
fib=: (0 1 with p. % 1 _1 _1 with p.)t.
x=: 1 2 3 4 5 6 7 8 9 10 11 12 13
fib x
1 1 2 3 5 8 13 21 34 55 89 144 233
sum pw fib x
2 3 5 8 13 21 34 55 89 144 233 377
sum pw sum pw fib x
5 8 13 21 34 55 89 144 233 377 610
dif pw fib x
0 1 1 2 3 5 8 13 21 34 55 89
gm=: %/pw fib x NB. Pairwise ratios
set 4
gm NB. Appproximations to golden mean
1 0.5 0.6667 0.6 0.625 0.6154 0.619 0.6176 0.6182 0.618 0.6181 0.618
gm * 1 + gm
2 0.75 1.111 0.96 1.016 0.9941 1.002 0.9991 1 0.9999 1 1
dif pw f x are
easily seen in a graph of f, but
repeated use (as in dif pw dif pw f x) are
not. They can, however,
provide interesting results when
applied to familiar functions. For example:
sqr=: ^ with 2
sqr x
1 4 9 16 25 36 49 64 81 100 121 144 169
dif pw sqr x
3 5 7 9 11 13 15 17 19 21 23 25
dif pw dif pw sqr x
2 2 2 2 2 2 2 2 2 2 2
dif pw dif pw dif pw sqr x
0 0 0 0 0 0 0 0 0 0
dp=: dif pw
dp sqr x
3 5 7 9 11 13 15 17 19 21 23 25
dp dp sqr x
2 2 2 2 2 2 2 2 2 2 2
cube=: ^ with 3
cube x
1 8 27 64 125 216 343 512 729 1000 1331 1728 2197
dp cube x
7 19 37 61 91 127 169 217 271 331 397 469
dp dp cube x
12 18 24 30 36 42 48 54 60 66 72
dp dp dp cube x
6 6 6 6 6 6 6 6 6 6
p2=: 2 with ^
and the use of the pair-wise quotients %/ pw
and %~/ pw on it.3D. Notes
S3D.
Chapters 2 and 3 introduced two important
facilities for the study of functions, the
function table, and graphs. Chapter 1
introduced one particularly important
function, the polynomial -- important
because it can approximate, and therefore be
used to study, a wide range of functions.
The next chapter examines it further.
* for both
signum and multiplication in Section 1D) may
wish to turn now to the two-page
discussion of trains, inflections, ambivalence,
and bonding in Section 10D.
4A. Bonding
4A. Bonding
S4A.
The polynomial function introduced in Chapter 1
is a function of two arguments,
the first of which is commonly called the
coefficients of the function. Using the
example from Chapter 1:
x=: 4
2 3 4 p. x
78
c=: 2 3 4
c p. x
78
g=: c with p.
g x
78
The expression g=: c with p. illustrates the
fact that the function p. can be
bonded with a list of coefficients to define a
specific polynomial function g. Consider
the following examples:
c2=: 1 2 1
c3=: 1 3 3 1
c4=: 1 4 6 4 1
y=: 0 1 2 3 4 5
f2=: c2 with p.
f2 y
1 4 9 16 25 36
f3=: c3 with p.
f3 y
1 8 27 64 125 216
(f2 y) * (f3 y)
1 32 243 1024 3125 7776
g=: f2 * f3
g y
1 32 243 1024 3125 7776
The Taylor operator t. applies to a polynomial
such as f3 to produce a function
which, applied in turn to an integer i, gives
coefficient i of the polynomial. For
example:
f3 t. 2
3
f3 t. 0 1 2 3 4 5 6
1 3 3 1 0 0 0
(f2 * f3) t. 0 1 2 3 4 5 6
1 5 10 10 5 1 0
The result of f2 f3 y is said to be the result
of applying f2 to the result of f3. The
corresponding function is denoted by f2 on f3.
For example:
f3 y
1 8 27 64 125 216
f2 1 8 27 64 125 216
4 81 784 4225 15876 47089
f2 f3 y
4 81 784 4225 15876 47089
h=: f2 on f3
h y
4 81 784 4225 15876 47089
h t. 0 1 2 3 4 5 6 7 8
4 12 21 22 15 6 1 0 0
g1=: f2 * f2
g2=: f2 * f2 * f2
g3=: f3 - f2
g4=: f3 on g
SUM c with p. + d with p.
DIFFERENCE c with p. - d with p.
PRODUCT c with p. * d with p.
COMPOSITION (c with p.) on (d with p.)
SLOPE or rate of increase over an interval
DERIVATIVE or limit of the slope over small intervals
AGGREGATE or area under the graph of a function
INTEGRAL or limit of the area for small intervals
4B. Notes
S4B.
This brief treatment of polynomials
is enough to provide a basis for the treatment
of the important topics of Power Series,
Slope and Derivative, Growth and Decay,
and Vibrations in the next four chapters.
We therefore defer further discussion of
the polynomial to Chapter 16: Polynomials
and Number Systems. That chapter
may, however, be attempted at any point.
5A. Introduction
5A. Introduction
S5A.
Elements of a list may be identified and selected
by its indices, beginning at zero.
For example:
a=: 1 2 3 4 5 6 ^ 3
a
1 8 27 64 125 216
0 { a
1
1 { a
8
5 { a
216
6 { a
|index error
| 6 {a
A polynomial whose coefficients may be expressed
as a function of their indices is
called a power series. For example:
g=: ! with 3
g i. 8
1 3 3 1 0 0 0 0
(g i.8) with p. 0 1 2 3 4 5 6
1 8 27 64 125 216 343
g 0 gives the number of distinct ways that zero
things can be chosen from three
things; g 1 gives the number of ways that one thing
can be chosen, and so on, to
the case g 4 which shows that four things can be
chosen from three in no ways.
The resulting coefficients are those used in c3
in Section 4A; h=: ! with 4
gives those used in c4, and so on.
The coefficients:
0 1 0 _1r6 0 1r120 0 _1r5040
1 0 _1r2 0 1r24 0 _1r720 0 1r40320
used in Section 3A can also be expressed as
power series. Both lists are reciprocal
factorials (such as 1r24 and 1r120) multiplied
by _1 or 0 or 1. The power
series function for the first is given by:
ps=: % on ! * 2 with | * _1: ^ 3: = 4: | ]
ps 0 1 2 3 4 5 6 7 8 9 10r1
0 1 0 _1r6 0 1r120 0 _1r5040 0 1r362880 0
pc for the
second list of coefficients given above.5B. Truncated power series
S5B.
For the power series g=: ! with 3, the coefficients
following the first four are
all zero, and the truncated series g i.4 therefore
defines a polynomial that is
equivalent to the longer list
produced by g i.8. Thus: g i.4
1 3 3 1
(g i.4) with p. x=: 0 1 2 3 4 5 6
1 8 27 64 125 216 343
g i.8
1 3 3 1 0 0 0 0
(g i.8) with p. x
1 8 27 64 125 216 343
ps=: % on ! * 2 with | * _1: ^ 3: = 4: | ]
never “terminates” with all zeros. However, the
reciprocal factorial factor (% on !) ensures
that successive terms diminish rapidly
in magnitude, and a short series
may therefore provide a good approximation. For example:
] c12=: ps i. 12r1
0 1 0 _1r6 0 1r120 0 _1r5040 0 1r362880 0 _1r39916800
] c10=: ps i. 10r1
0 1 0 _1r6 0 1r120 0 _1r5040 0 1r362880
set 9
dec (c12 with p. ,c10 with p.) 2
0.909296136 0.909347443
However, the definition of the polynomial
in Section 1I shows that successive
coefficients are multiplied by successive
powers of the argument x. For large
arguments, the growth of this factor may be
fast enough to overpower the
decrease in the coefficient. For example:
dec (c12 with p. ,c10 with p.) 5
_1.1336173 0.08963018
For small arguments (particularly for those
less than one in magnitude), reasonably
short power series of the type that includes
a reciprocal factorial factor provide
good approximations. For example:
dec ((ps i.20) with p.,(ps i.18) with p.) 5
_0.958933165 _0.958776369
y=: 1r5 * i:5
y
_1 _4r5 _3r5 _2r5 _1r5 0 1r5 2r5 3r5 4r5 1
sin=: 1 with o.
((ps i.10) with p.,.(ps i.8) with p.,.sin) y
_0.841471010 _0.841468254 _0.841470985
_0.717356093 _0.717355723 _0.717356091
_0.564642473 _0.564642446 _0.564642473
_0.389418342 _0.389418342 _0.389418342
_0.198669331 _0.198669331 _0.198669331
0 0 0
0.198669331 0.198669331 0.198669331
0.389418342 0.389418342 0.389418342
0.564642473 0.564642446 0.564642473
0.717356093 0.717355723 0.717356091
0.841471010 0.841468254 0.841470985
t. can be used to produce the power
series for the sine as follows:
sin t. i. 12r1
0 1 0 _1r6 0 1r120 0 _1r5040 0 1r362880 0 _1r39916800
sign sin t. i. 12r1
0 1 0 _1 0 1 0 _1 0 1 0 _1
^
q=: (^t.i),.(1 with o.t.i),.(2 with o.t.i=: i.12x)
q,.*q
5C. Notes
S5C.
Power series are of great importance in math,
and it is tempting to digress in a
discussion of reasons for this importance,
much as was done for polynomials in
Section 4A.
6A. Approximation to the derivative (rate of change)
6A. Approximation to the derivative (rate of change)
S6A.
As remarked in Section 3B, the slopes of
the segments of a graph show the
average rate of change of a function between
adjacent points. Plotting more points at lesser
intervals gives a better approximation. For example:
c=: 4 _3 _2 1
f=: c with p.
x1=: i.4
PLOT x1;f x1
x2=: 1r10*i.31
PLOT x2;f x2
x3=: 1r100 * i.301
PLOT x3;f x3
((f x+r)-(f x))%r gives the
slope of the graph of f
between arguments x and x+r as the ratio of
the rise (f x+r)-(f x) to the
run r. For example:
x=: 2
r=: 1r10
((f x+r)-(f x)) % r
1.41
x=: 0 1 2 3 4 5 6
r=: 1r1000
((f x+r)-(f x)) % r
_3.002 _3.999 1.004 12.007 29.01 52.013 81.016
r=: 1r10000
((f x+r)-(f x)) % r
_3.0002 _3.9999 1.0004 12.001 29.001 52.001 81.002
For these small values of the run, the slopes
appear to be “approaching a limiting
value” given approximately by the run of one
ten-thousandth. This limiting
value is the derivative of the function f,
that is, the slope of the tangent.
The value r=: 0 might seem appropriate, but this
only gives the meaningless
division of a zero rise by a zero run. Thus:
r=: 0
((f x+r)-(f x)) % r
0 0 0 0 0 0 0
The desired result is given by the derivative
operator d., with f d.1 giving the
(first) derivative of f and f d.2 giving the
second derivative (that is, the
derivative of the derivative), and so on. For example:
f d.1 x
_3 _4 1 12 29 52 81
f d.2 x
_4 2 8 14 20 26 32
f d.1 2 3 4 5 x
_3 _4 6 0 0
_4 2 6 0 0
1 8 6 0 0
12 14 6 0 0
29 20 6 0 0
52 26 6 0 0
81 32 6 0 0
Moreover, the application of the Taylor
operator to the resulting derivatives show
them to be terminating power series, that
is, ordinary polynomials:
d=: f d.1 t. i.7
d
_3 _4 3 0 0 0 0
f d.2 t. i.7
_4 6 0 0 0 0 0
d with p. x
_3 _4 1 12 29 52 81
d with p. d.1 x
_4 2 8 14 20 26 32
f d.2 x
_4 2 8 14 20 26 32
The coefficients d of the first derivative
polynomial must bear some relation to the
coefficients c of the original polynomial f.
We will explore this relation by
examining their ratios, as seen in their divide table:
d % table c
+--+----------------+
| | 4 _3 _2 1|
+--+----------------+
|_3|_3r4 1 3r2 _3|
|_4| _1 4r3 2 _4|
| 3| 3r4 _1 _3r2 3|
| 0| 0 0 0 0|
| 0| 0 0 0 0|
| 0| 0 0 0 0|
| 0| 0 0 0 0|
+--+----------------+
The diagonal of successive integers 1 2 3 suggests
that d may be obtained from c by
multiplying by successive integers, and
rotating the result one place to the left:
c
4 _3 _2 1
#c NB. number of elements in c
4
i.#c
0 1 2 3
c * i.#c
0 _3 _4 3
1 |. c * i.#c
_3 _4 3 0
d
_3 _4 3 0 0 0 0
We will first test this relation on another
polynomial function and then, in the
following section, examine the question of
why the relation holds in general:
c2=: ! with 5 i.6
c2
1 5 10 10 5 1
f2=: c2 with p.
d2=: f2 d.1 t. i.5
d2
5 20 30 20 5
1 |. c2 * i.#c2
5 20 30 20 5 0
6B. Derivatives of polynomials
S6B.
The polynomial f=: 4 1 3 2 with p. is a sum of
four monomials. Thus:
f=: 4 1 3 2 with p.
f0=: 4 with * on (^ with 0)
f1=: 1 with * on (^ with 1)
f2=: 3 with * on (^ with 2)
f3=: 2 with * on (^ with 3)
(f0,f1,f2,:f3) x
4 4 4 4 4 4 4
0 1 2 3 4 5 6
0 3 12 27 48 75 108
0 2 16 54 128 250 432
(f0+f1+f2+f3) x
4 10 30 76 160 294 490
f x
4 10 30 76 160 294 490
Any slope of a function that is a sum of functions
equals the sum of the
corresponding slopes of the component functions,
and the derivative of a sum of
functions is therefore the sum of the derivatives
of the corresponding functions.
For example:
(f0 d.1 , f1 d.1 , f2 d.1 ,: f3 d.1) x
0 0 0 0 0 0 0
1 1 1 1 1 1 1
0 6 12 18 24 30 36
0 6 24 54 96 150 216
(f0 d.1 + f1 d.1 + f2 d.1 + f3 d.1) x
1 13 37 73 121 181 253
f d.1 x
1 13 37 73 121 181 253
Similar remarks apply to the multiplication
that occurs in the monomials. For
example, in f2=: 3 with * on (^ with 2), the
multiplication by three
of the square function (^ with 2) multiplies
each of its slopes by three, and
therefore multiplies its derivative by three.
It remains to determine the derivative of power
functions such as the square. If
f=: ^ with 2, then the rise (f x+r)-(f x) is
given by the square of
x+r (that is, (x+r)*(x+r)) minus the square
of x. x plus 2*x*r plus
the square of r, and subtraction
of the square of x then leaves a rise of 2*x*r
plus the square of r. The slope of
the square function (for the run r) is then
given by dividing this sum by r to obtain
(2*x)+r. The derivative is then given by the
case for r=: 0, that is, 2*x (or,
equivalently, 2*x^1).
(x+r)*(x+r)*(x+r)gives 3*x^2 for
the derivative of the cube ^ with 3, and,
in general, gives n*x^(n-1)for the
derivative of ^ with n.
The contribution of a monomial cn * x ^ n
to the derivative polynomial is
therefore the monomial n * cn * x ^ (n-1),
which therefore appears as a
coefficient n * cn displaced one place to
the left in the list of coefficients. d=: 1: |. c * i.#c
given in the
preceding section for the coefficients d
of the derivative polynomial.
These results will be summarized by defining
a function der which, applied to a
list of coefficients of a polynomial, gives
the list of coefficients of the derivative
polynomial. We will illustrate its use on the
polynomial graphed in Section A, and
will graph it together with the secant slopes
of the function so that they can be
compared:
der=: 1: |. ] * i. on #
c=: 4 _3 _2 1
d=: der c
d
_3 _4 3 0
x2=: 1r10*i.31
PLOT x2 ; (c with p. ,: d with p.) x2
f correctly suggests that it
is a function derived from f, but it is only one among many (such
as the inverse) also derived from f. The phrase
slope of f would be more informative, and could be
distinguished from the associated secant slope of f.
6C. Taylor coefficients
S6C.
It is also interesting to compare the Taylor
coefficients of the derivative of the
polynomial d with p. with the result of the
function der:
d with p. d.1 t. i.8
_4 6 0 0 0 0 0 0
der d
_4 6 0 0
As a foretaste of the growth and oscillating
functions of the next two chapters, we
will also show Taylor series for the exponential and
sine functions:
exp=: ^
exp t. i=: i.10r1
1 1 1r2 1r6 1r24 1r120 1r720 1r5040 1r40320 1r362880
exp d.1 t. i
1 1 1r2 1r6 1r24 1r120 1r720 1r5040 1r40320 1r362880
exp d.2 t. i
1 1 1r2 1r6 1r24 1r120 1r720 1r5040 1r40320 1r362880
sin=: 1 with o.
]s=: sin t. i.12r1
0 1 0 _1r6 0 1r120 0 _1r5040 0 1r362880 0 _1r39916800
der s
1 0 _1r2 0 1r24 0 _1r720 0 1r40320 0 _1r3628800 0
sin d.1 t. i.11r1
1 0 _1r2 0 1r24 0 _1r720 0 1r40320 0 _1r3628800
der der s
0 _1 0 1r6 0 _1r120 0 1r5040 0 _1r362880 0 0
der der der s
_1 0 1r2 0 _1r24 0 1r720 0 _1r40320 0 0 0
der der der der s
0 1 0 _1r6 0 1r120 0 _1r5040 0 0 0 0
s
0 1 0 _1r6 0 1r120 0 _1r5040 0 1r362880 0 _1r39916800
6D. Notes
S6D.
For an arbitrary function f (such as any one
of the functions of trigonometry), the
matter of the limiting value of the secant
slope ((f x+r)-(f x)) % r as r
“approaches” zero (discussed in Section 6A)
raises difficult questions that are
properly answered only by lengthy analysis
of the notion of limits. This is what
makes Calculus such a forbidding subject.
f is a good approximation to the
derivative of f ? Yes, but only for
functions that are “uniformly continuous”.
This is true of a wide range of functions
of practical interest, including all of
those to be treated in subsequent chapters.
7A. Growth polynomials
7A. Growth polynomials
S7A.
It is a common observation that growing
things (such as young plants and animals,
young commercial companies, and a colony
of bacteria) change not at a constant
rate, but at a rate roughly proportional to present size.
^ . In a graph of this function this
relation may be seen approximately in
the slopes of the secants. Thus:
x=: 1r2*i.11
x
0 1r2 1 3r2 2 5r2 3 7r2 4 9r2 5
set 4
^ x
1 1.649 2.718 4.482 7.389 12.18 20.09 33.12 54.6 90.02 148.4
PLOT x;^x
c=: 1 1 1r2 1r6 1r24 1r120 1r720 1r5040
der=: 1:|.]*i.@:# NB. Gives coeffs of derivative
d=: der c
d
1 1 1r2 1r6 1r24 1r120 1r720 0
dec c with p. x
1 1.649 2.718 4.481 7.381 12.13 19.85 32.23 51.81 82.22 128.6
dec d with p. x
1 1.649 2.718 4.478 7.356 12.01 19.41 30.95 48.56 74.81 113.1
The two polynomials differ only in the
term 1r5040*x^7, and are therefore
nearly equal for reasonably small values of x. Moreover,
the pattern required of
further coefficients is clear: coefficient k
must be 1%!k. Thus:
! i. 12r1
1 1 2 6 24 120 720 5040 40320 362880 3628800 39916800
a=: 1 % ! i. 12
b=: der a
(a with p. ,. b with p. ,. ^) x
1 1 1
1.649 1.649 1.649
2.718 2.718 2.718
4.482 4.482 4.482
7.389 7.389 7.389
12.18 12.18 12.18
20.08 20.08 20.09
33.11 33.08 33.12
54.55 54.44 54.6
89.8 89.42 90.02
147.6 146.4 148.4
^(x+3)
and (^x) * (^3), and comment on the
results.
(^x) * (^-x)
^(x+-x)
set 2
^-x
% ^ x
decay=: ^ on -
decay x
decay t. i. 10
7B. The name Exponential
S7B.
^, the
expression x^e
is said to denote the base x
to the exponent e, and the
function x with ^
may therefore be said to be an
exponential function. For example:
e=: 1r10*i.11
e
0 1r10 1r5 3r10 2r5 1r2 3r5 7r10 4r5 9r10 1
3^e
1 1.1 1.2 1.4 1.6 1.7 1.9 2.2 2.4 2.7 3
3 with ^ e
1 1.1 1.2 1.4 1.6 1.7 1.9 2.2 2.4 2.7 3
^e
1 1.1 1.2 1.3 1.5 1.6 1.8 2 2.2 2.5 2.7
2 with ^ e
1 1.1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.9 2
The exponential function ^ appears to lie
between the functions 2 with ^ and
3 with ^. Thus:
set 7
(^,.2.5 with ^,.2.75 with ^,.2.71 with
^,.2.718 with ^,.1x1 with ^) e
1 1 1 1 1 1
1.105171 1.095958 1.106454 1.104834 1.105159 1.105171
1.221403 1.201124 1.22424 1.220658 1.221377 1.221403
1.349859 1.316382 1.354565 1.348624 1.349817 1.349859
1.491825 1.4427 1.498763 1.490005 1.491763 1.491825
1.648721 1.581139 1.658312 1.646208 1.648636 1.648721
1.822119 1.732862 1.834846 1.818786 1.822005 1.822119
2.013753 1.899144 2.030172 2.009456 2.013607 2.013753
2.225541 2.081383 2.246292 2.220115 2.225356 2.225541
2.459603 2.281109 2.485418 2.452858 2.459374 2.459603
2.718282 2.5 2.75 2.71 2.718 2.718282
The base that gives the exponential exactly is
called Euler’s number, and is
denoted in math by e and in J by 1x1 .
8A. Introduction
8A. Introduction
S8A.
Vibrations are commonly seen in
the motions of a clock pendulum, of a
piano string, and of a weight (such as a plumb bob suspended on a
rubber band or steel spring). If the
bob is pulled straight down a certain distance from
its rest position and released, it
visibly oscillates above and below its rest position
until it is brought to rest by friction
in the air and in the suspension.
b is a function that gives the position of the bob
as a function of time, then its
speed or velocity is given by b d.1 (the rate of change
of position), and its
acceleration is given by the
second derivative b d.1 d.1
(the rate of change of
its velocity).
b of the
bob. In other words, the second
derivative b d.1
is proportional to -b.
-b is equal
to b d.1 d.1, and
we seek a polynomial with
this property. Consider the coefficients of the
function fs introduced in Section 3A:
c=: 0 1 0 _1r6 0 1r120 0 _1r5040
der c
1 0 _1r2 0 1r24 0 _1r720 0
der der c
0 _1 0 1r6 0 _1r120 0 0
-der der c
0 1 0 _1r6 0 1r120 0 0
The required pattern is clear: it is the same
as for the exponential of Chapter 7,
except that alternate coefficients are zero,
and that the other coefficients alternate
in sign.
The function with this property is called the
sine, commonly abbreviated to sin:
sin=: 1 with o.
sin t. i.12
0 1 0 _1r6 0 1r120 0 _1r5040 0 1r362880 0
_1r39916800
sin t. 20 21 22 23
0 1r51090942171709440000 0 _1r25852016738884976640000
8B. Harmonics
S8B.
The function sin on (2:*]) applies the sin to the
double of its argument, and therefore produces a vibration
(or oscillation)
of twice the frequency. It is called an harmonic of
the oscillation sin. This may be seen in a combined plot
of the two functions: x=: 1r10 * i.65
plot x;(sin x) ,: (sin on (2:*]) x)
2 in the function sin on (2:*])
by other integers to produce other harmonics, and plot the
results.cos and some of its harmonics.
cos against its harmonic
cos on (2:*]).8C. Decay
S8C.
Because of "friction" of some sort, vibrations
commonly decay, usually at an approximately exponential rate
determined by a function of the form decay=: ^ on - on (] % [).
sin * 6 with decay .
6, and plot them together with their non-decaying
counterparts.
9A. Inverse
9A. Inverse
S9A.
The predecessor function (<:) introduced
in 1D is said to be the inverse of the
successor function (>:) because it “undoes”
its work. For example:
>:1 2 3 4 5
2 3 4 5 6
<:>:1 2 3 4 5
1 2 3 4 5
Conversely, >: is also the inverse
of <:,
and we may simply say that the two are
inverses.
h=: (* with 4) on sqr,
where sqr=: *: is the square function. In
order to determine the voltage
required to produce a desired heat, we
need the inverse v=: sqrt on (% with 4),
where sqrt=: %: is the square root function. Thus:
h=: (* with 4) on sqr
sqr=: *:
v=: sqrt on (% with 4)
sqrt=: %:
h 0 1 2 3 4 5 6 7
0 4 16 36 64 100 144 196
v h 0 1 2 3 4 5 6 7
0 1 2 3 4 5 6 7
v 0 1 2 3 4 5 6 7
0 0.5 0.707107 0.866025 1 1.11803 1.22474 1.32288
h v 0 1 2 3 4 5 6 7
0 1 2 3 4 5 6 7
For many (but not all) functions, the
inverse operator INV=: ^:_1 produces the
inverse function. For example:
INV=: ^:_1
<:INV 0 1 2 3 4 5 6 7
1 2 3 4 5 6 7 8
cube=: ^ with 3
cuberoot=: cube INV
cube 0 1 2 3 4 5 6 7
0 1 8 27 64 125 216 343
cuberoot cube 0 1 2 3 4 5 6 7
0 1 2 3 4 5 6 7
^ with 4 INV 0 1 2 3 4 5 6 7
0 1 1.18921 1.31607 1.41421 1.49535 1.56508 1.62658
^ with 4 ^ with 4 INV 0 1 2 3 4 5 6 7
0 1 2 3 4 5 6 7
f=: ^ with 3 + ^ with 4
f 0 1 2 3 4 5 6 7
0 2 24 108 320 750 1512 2744
f INV 0 1 2 3 4 5 6 7
|domain error
| f INV 0 1 2 3 4 5 6 7
9B. Equations
S9B.
However, for a specific argument,
say y=: 20, we could find the value of the
inverse of f by guessing the result x, and
applying f to it to guide us to an
improved guess, by either increasing or
decreasing x. For example:
y=: 20
x=: 2
f x
24
x=: x-1r2
set 5
f x
8.4375
x=: x+1r4
dec f x
14.738
dec f x=: x+1r8
18.951
dec f x=: x+1r16
21.365
dec f x=: x-1r32
20.131
dec f x=: x-1r64
19.535
dec f x=: x+1r128
19.831
dec f x=: x+1r256
19.981
dec f x=: x+1r512
20.056
dec x
1.9043
This problem is commonly stated as the
equation (f x)=y, implying that one is
to find a value of x such that the relation
is true. Moreover, the problem of solving
equations is commonly introduced with little
or no clue as to why the matter is important.
g=: - with y on f differs
from f in subtracting y from the
result, and a solution of (g x)=0 is therefore
a solution of (f x)=y. A
solution of (g x)=0 is said
to be a root of the function g.
9C. The method of false position
S9C.
If one knows two values a and b
such that g a and g b
differ in sign, then a
root of g must lie somewhere between them; the
list ab=: a,b is said to bracket
the root. Evaluation of f at its
midpoint m (that is, m=: mean ab) will either
yield zero (in which case m is a root of g),
or it will differ in sign from one of the two results of g ab.
ab that differs can be used
with m to form a tighter bracket, whose
mean will give a better approximation to the
root of g. This process may be
repeated to obtain as close an approximation
as desired. For example:
g=: - with y on f
g ab=: 1,2
_18 4
mean=: +/ % #
]m=: mean ab
1.5
g m
_11.563
]ab=: m,1{ab
1.5 2
g ab
_11.563 4
In general, a bracket of a root of a function g
may be tightened by the function tighten,
defined and illustrated below:
tighten=: mean,(sign on{. on g = sign on g on mean) { ]
tighten ab=: 1,2
1.5 2
tighten tighten tighten ab
1.875 2
g tighten tighten tighten ab
_1.0486 4
The function tighten may be paraphrased
as follows: Compare the sign of
the first element ({.) of the result of g
for equality with the sign of g on the
mean, and use the result (0 or 1) as an
index to select from the argument; finally,
catenate the selection with the mean.
The operator ^: can be used to apply a
function repeatedly, as illustrated below:
tighten^:3 ab
1.875 2
z=: tighten^:0 1 2 3 4 5 6 7 8 9 10 11 12 ab
z;g z
+---------------+-------------------------+
| 1 2| _18 4|
| 1.5 2| _11.5625 4|
| 1.75 2| _5.26172 4|
| 1.875 2| _1.04858 4|
| 1.9375 1.875| 1.36501 _1.04858|
|1.90625 1.875| 0.131333 _1.04858|
|1.89063 1.90625| _0.465246 0.131333|
|1.89844 1.90625| _0.168624 0.131333|
|1.90234 1.90625| _0.0190637 0.131333|
| 1.9043 1.90234| 0.0560301 _0.0190637|
|1.90332 1.90234| 0.018457 _0.0190637|
|1.90283 1.90332|_0.000309859 0.018457|
|1.90308 1.90283| 0.00907195 _0.000309859|
+---------------+-------------------------+
9D. Newton’s method
S9D.
If x=: 2 is an appoximation to the root
of g, then
(as suggested by the small
“triangle” with apex at the point x,g x in the
accompanying graph) a good
correction is provided by dividing g x by the
derivative g d.1 x that gives the
slope of the tangent to the
graph at the point x,g x:
PLOT x;g x=: 1r10*i.22
x=: 2
(g , g d.1 , g%g d.1) x
4 44 0.0909091
x=: x - (g%g d.1) x
g x
0.24124
x=: x - (g%g d.1) x
g x
0.00106658
newton=: ] - g % g d.1
newton 2
1.90909
g newton 2
0.24124
newton^:(i.4) 2
2 1.90909 1.90287 1.90284
g newton^:(i.4) 2
4 0.24124 0.00106658 2.1139e_8
g newton ^:(i.3 5) 25
406230 128520 40655 12855.8 4061.08
1279.01 399.129 121.057 33.5929 7.07075
0.658183 0.00775433 1.11692e_6 2.13163e_14 3.55271e_15
g, its
convergence is not guaranteed in general, especially
for initial guesses far from the root.
9E. Roots of Polynomials
S9E.
The (possibly multiple) roots of a polynomial
c with p. may, in principle, be obtained by the
methods of sections 9C and 9D, but it may be difficult
to find suitable brackets or initial guesses. However,
the function roots=: > on {: on p. may
be applied to the coefficients c to
obtain the roots. For example: roots=: > on {: on p.
c=: _1 0 1
r=: roots c
r
1 _1
c p. r
0 0
d=: 6 1 _4 1
s=: roots d
s
3 2 _1
d p. s
3.55271e_15 2.66454e_15 0
Because of the limited precision of the computations leading
to the roots s, the application of the polynomial
to them yields tiny results, but not exact zeros. The following
function (the product of the sign function with the magnitude)
serves to make such results more readable by "zeroing" tiny
results: zero=: **|
zero d p. s
0 0 0
A graph of the polynomial 1 0 1 with p.
(one plus the square) indicates that it has no roots,
because the graph never drops to zero:
e=: 1 0 1
x=: i:4
PLOT x;e p. x
roots applies
as follows: roots e
0j1 0j_1
e p. roots e
0 0
The root 0j1 is an example of an imaginary
number, to be discussed in Chapter 19. Its square is negative
one: 0j1*0j1
_1
9F. Logarithms
S9F.
The logarithm (^.) is the function inverse to
the exponential:
^ ^:_1
^.
^. ^:_1
^
]x=: i.8
0 1 2 3 4 5 6 7
^x
1 2.71828 7.38906 20.0855 54.5982 148.413 403.429 1096.63
^.^x
0 1 2 3 4 5 6 7
^.x
__ 0 0.693147 1.09861 1.38629 1.60944 1.79176 1.94591
^^.x
0 1 2 3 4 5 6 7
As discussed in Section 7B, the exponential ^
is equal to e with ^, where e=: 1x1
is Euler's number. Similarly, e with ^. is
inverse to e with ^, as illustrated below:
e=: 1x1
e
2.71828
e with ^ x
1 2.71828 7.38906 20.0855 54.5982 148.413 403.429 1096.63
e with ^. e with ^ x
0 1 2 3 4 5 6 7
10 with ^ x
1 10 100 1000 10000 100000 1e6 1e7
10 with ^. 10 with ^ x
0 1 2 3 4 5 6 7
10 with ^. 1 10 100 1000 10000 100000
0 1 2 3 4 5
The function 10 with ^. is the base-10
logarithm, more commonly used in elementary
math than the natural log ^. .
10A. Introduction
10A. Introduction
S10A.
Despite its importance, the grammar of our native
language is not acquired by
formal instruction, but by imitation and casual
guidance in conversation. Formal
grammar becomes important only at a later stage,
in addressing more complex and
more precise uses of language.
10B. Word formation
S10B.
Grammar is commonly taken to concern
parsing the words of a sentence into
phrases, and determining the order in
which the phrases are to be interpreted.
However, it is worth noting that the
comprehension of a sentence must begin by
first breaking it into its component words.
;: '12+27'
+--+-+--+
|12|+|27|
+--+-+--+
;: '12+.27'
+--+--+--+
|12|+.|27|
+--+--+--+
;: '12+0.27'
+--+-+----+
|12|+|0.27|
+--+-+----+
;: '+/1 2 3 4'
+-+-+-------+
|+|/|1 2 3 4|
+-+-+-------+
The formal rules are stated in the J Dictionary as follows:
The alphabet is standard ASCII, comprising
digits, letters (of the English
alphabet), the underline (used in names and numbers),
the (single) quote,
and others (which include the space) to be referred to as graphics.
It would clearly be confusing if it were possible to assign the
name a.).
e (as in 2.4e3 to
signify 2400
in exponential form), and the letter
j to separate the real and imaginary
parts of a complex number, as in
3e4j_0.56. Also see the discussion of Constants.
'can''t' is the five-character abbreviation
of the six-character word
'cannot'. The ace a: denotes
the boxed empty list <$0 .
prices=: 4.5 12) begin with a
letter and may
continue with letters, underlines, and digits.
A name that ends with an
underline or that contains two consecutive
underlines is a locative, as
discussed in Section II.I.
+ for
plus) or by a graphic modified by one or more
following inflections (a period or colon), as
in +. and +: for or and nor.
A primary may also be
an inflected name, as in e. and o.
for membership and pi times. A
primary cannot be assigned a referent. + to multiplication, as in +=: * .
10C. Parsing
S10C.
The complete parsing rules are stated
in the J dictionary as follows. The leading
five-point summary will suffice for most purposes:
10%3+2, the phrase
3+2 is evaluated first to obtain a
result that is then used to divide 10. In
summary:,"2-a is
equivalent to (,"2)-a, not to ,"(2-a).
+/ . */b, the rightmost
adverb /
applies to the verb derived from the
phrase +/ . *, not to the verb * .
3*p%q^|r-5 can therefore be read from
left to right: the overall result
is 3 times the result of the remaining phrase,
which is the quotient of p and the
part following the %, and so on.
a=: 1 2 3, then b=: +/2*a
would be parsed and executed as follows:
§ b =: + / 2 * a
§ b =: + / 2 * 1 2 3
§ b =: + / 2 * 1 2 3
§ b =: + / 2 * 1 2 3
§ b =: + / 2 * 1 2 3
§ b =: + / 2 4 6
§ b =: + / 2 4 6
§ b =: + / 2 4 6
§ b =: 12
§ b =: 12
§ 12
§ 12
The foregoing illustrates two points: 1) Execution of
the phrase 2 * 1 2 3 is
deferred until the next element (the /)
is transferred; had it been a conjunction, the
2 would have been its argument, and the
monad * would have applied to 1 2 3;
and 2) Whereas the value of the name a
moves to the stack, the name b (because it
precedes a copula) moves unchanged, and
the pronoun b is assigned the value 12.
13!:16 (q.v.).
The executions in the stack are confined
to the first four elements only, and
eligibility for execution is determined
only by the class of each element (noun,
verb, etc., an unassigned name being
treated as a verb), as prescribed in the
following parse table.
EDGE VERB NOUN ANY 0 Monad
EDGE+AVN VERB VERB NOUN 1 Monad
EDGE+AVN NOUN VERB NOUN 2 Dyad
EDGE+AVN VERB+NOUN ADV ANY 3 Adverb
EDGE+AVN VERB+NOUN CONJ VERB+NOUN 4 Conj
EDGE+AVN VERB VERB VERB 5 Trident
EDGE CAVN CAVN CAVN 6 Trident
EDGE CAVN CAVN ANY 7 Bident
NAME+NOUN ASGN CAVN ANY 8 Is
LPAR CAVN RPAR ANY 9 Paren
Legend: AVN denotes ADV+VERB+NOUN
CAVN denotes CONJ+ADV+VERB+NOUN
EDGE denotes MARK+ASGN+LPAR
10D. Conventional mathematical notation
S10D.
This section is a discussion of mathematical
notation from the vantage point of
programming languages, with emphasis on
ideas from programming that might be
adopted without conflict in mathematical
notation.
By relieving the brain of all unnecesary work,
a good notation sets it free
to concentrate on more advanced problems, and
in effect increases the
mental power of the race... By the aid of
symbolism we can make transitions
in reasonings almost mechanically by
the eye... A. N. Whitehead
On the other hand, Cajori comments
on the paucity of publications concerning the
general question of notation, and
quotes De Morgan (“a close student of the
history of mathematics”) as follows:
Mathematical notation, like language,
has grown up without much looking
to, at the dictates of convenience and
with the sanction of the majority.
In that preface William Forster quotes the
reply of Oughtred to the
question of how he (Oughtred) had for so
many years concealed his
invention of the slide rule from himself
(Forster) whom he had taught so
many other things. The reply was:
On page 93 Cajori continues with:
That the true way of Art is not by Instruments, but by
Demonstration: and that it is a preposterous course of vulgar
Teachers, to begin with Instruments, and not
with the Sciences, and
so in-stead of Artists, to make their Scholers only doers of tricks,
and as it were Iuglers ... That the vse of instruments is indeed
excellent, if a man be an Artist: but contemptible being set and
opposed to Art. And lastly, that he meant to commend to me, the
skill of Instruments, but first he would haue me well instructed in
the Sciences.
It may be claimed that there is a middle ground which more nearly
represents the ideal procedure in teaching. Shall
the slide rule be placed in
the student’s hands at the time when he is engaged
in the mastery of
principles? Shall there be an alternate study of
the theory of logarithms and
of the slide rule -- on the idea of one hand washing
the other -- ...
Writing in the early 1900’s, Cajori could not have
foreseen the advent of that
universal mathematical instrument, the modern
computer, nor its spawning of the
development and study of the notations of programming languages.
00 0080 10
00 0081 11
10 0082 11
loaded the number contained in register 80 into the
accumulator, added register
81 to it, and stored the resulting sum in register 82.
Let a=3
Let b=5
Let c=(a+b)*(a-b)
Although a few symbols from programming languages
(such as FORTRAN’s * for
multiplication) have been widely adopted in math,
the mathematical notation used
in exposition has been largely unaffected. On the
other hand, many mathematicians
have been forced to learn programming languages,
or to work with programmers
who translate their mathematical expressions.
“I doubt not but that there shall be in convenient time,
brought to light
many precepts which may tend to ye perfecting of
Navigation, and the help
and safety of such whose Vocations doe inforce
them to commit their lives
and estates in the vast Ocean to the providence
of God.” Thus farr that
very good and judicious man Mr. Oughtred. I will
add, that if instead of
sending the Observations of Seamen to able
Mathematicians at Land, the
Land would send able Mathematicians to Sea,
it would signify much more
to the improvemt of Navigation and safety of
Mens lives and estates on
that element.
The need to make translation manageable has
led to the development of languages
with simple and precisely defined grammars,
that is, rules for word-formation and
parsing. Although mathematical notation
undoubtedly possesses parsing rules, they
are rather loose, sometimes contradictory,
and seldom clearly stated. It is perhaps
significant that Cajori’s History makes
no mention of the matter of grammar.
v<-2 7 1 8 2 8
+/v
28
The operator applies to any function, such
as * (times), >. (max),
or <. (min), and
may therefore obviate the use of many symbols
such as the capital Greek sigma and pi for summation and
product:
(*/v),(>./v),(<./v)
1792 8 1
Because of their application to a broad range
of topics, their strict grammar, and
their strict interpretation, programming
languages can provide new insights into
mathematical notation. We begin with the matter of symbols.
= < > _ + * - % ^ $ ~ | . : , ;
# ! / \ [ ] { } " ' ` @ & ?
This economy is achieved in part by adopting from
mathematics the use of
reserved words such as sin, cos, e, and log -- reserved
from use as names for
variables. Further economies are implicit in notions
that have been used in
mathematics but not systematically exploited. We will
illustrate them by their use
in J, a language derived from APL.
f and g
by f+g. This notion can be exploited
more generally, including constructs such
as f+g*h for f added to the product of
g and h. For example,
using % for divide,
and # for number of elements:
mean=: +/ % # NB. Arithmetic mean or average
mean 1 2 3 4 5
3
com=: ] - +/ % # NB. ] is the identity function
com 1 2 3 4 5 NB. Center on mean
_2 _1 0 1 2
Non-mathematical functions can be used in a train.
For example, the function ;
(which boxes each argument and catenates the boxed
results) can be used to
display results for convenient comparison:
(];mean;com;] - +/ % #) 1 2 3 4 5
+---------+-+-----------+-----------+
|1 2 3 4 5|3|_2 _1 0 1 2|_2 _1 0 1 2|
+---------+-+-----------+-----------+
1 and 0 to
denote true and false, and used the symbols for
times and plus to denote the
logical functions and and or.
*. for and and +.
for or. Similarly, = is used as a truth-function
(like < and >), whereas =: is used
as a copula (to assign a referent to a name), and <.
is used for lesser-of (that is,
minimum). Thus:
p=: 0 0 1 1 [ q=: 0 1 0 1
(p *. q);(p +. q);(p + q);(p < q);(p <. q)
+-------+-------+-------+-------+-------+
|0 0 0 1|0 1 1 1|0 1 1 2|0 1 0 0|0 0 0 1|
+-------+-------+-------+-------+-------+
AMBIVALENCE In the expression a-b the symbol -
denotes subtraction (a function of two arguments),
whereas in -b it
denotes negation, a different (albeit
related) function of one argument. This
ambivalence (a “binding power” of either
two or one) introduces no confusion in
conventional notation, and could be
exploited for other functions:
a=: 2
b=: 5
(a-b),(-b) NB. Subtraction and Negation (0-b)
_3 _5
(a%b),(%b) NB. Division and Reciprocal (1%b)
0.4 0.2
(a^b),(^b) NB. Power and Exponential (e^b)
32 148.413
(a^.b),(^.b) NB. Base-a log, Natural log (e^.b)
2.32193 1.60944
BONDS In their Combinatory Logic [11],
Curry and Feys developed a set of
combinators, which we will treat as operators
in the sense of Heaviside. One of
these bonds a function of two arguments
(such as ^) to one of its arguments (such
as 3) to produce a function of one argument
(the cube), an operation that has
come to be called Currying in honour of its author.
For example:
cube=: ^ with 3
log10=: 10 with ^.
(cube ; log10) 10 50 100 200
+-------------------+-------------------+
|1000 125000 1e6 8e6|1 1.69897 2 2.30103|
+-------------------+-------------------+
Conclusion. Taken together, trains,
inflection, ambivalence, and bonds provide
highly mnemonic representations for
a host of functions. Their advantages can be
fully appreciated only by careful
comparison with the alternatives now used in
mathematics.
§727 Invention of Symbols. -- Whenever the
source is known, it is found to have
been individualistic -- the conscious suggestion of one mind.
Comments about the hope of grateful remembrance
appear unkind; perhaps Oughtred
invented symbols to clarify his own thinking,
and as a tool for teaching.
Nevertheless, Cajori’s main point is sound --
any attempt to impose an individual
system is vain and hopeless. log(a+b)*(a-b) is carried out by
identifying the primitive operations
(such as addition) and then performing
them in an appropriate order. The process
begins with word-formation (identification
of the individual words: log, left
parenthesis, a, +, b, etc.), and continues
with parsing the sequence of words
according to a formal or informal grammar
based upon the parts of speech
(variables, functions, punctuation, etc.).
;:) to a
quoted character string:
;:'log(a+b)*(a-b)'
+---+-+-+-+-+-+-+-+-+-+-+-+
|log|(|a|+|b|)|*|(|a|-|b|)|
+---+-+-+-+-+-+-+-+-+-+-+-+
sqr=: ^ with 2
area=: sqr 4
+/\4#3
3 6 9 12
PARTS OF SPEECH
MATH TERMS ENGLISH (J) TERMS
Constants 2 3 4 Nouns
Variable area Pronoun
Functions ^ # Verbs
Function sqr Proverb
Operators / \ Adverbs
Operator with Conjunction
( ) Punctuation
=: Copula
2+2 and 2*2 and
2^2 yield identical results, only for the
specific argument 2, but 2*2
and sqr 2
are identical for any argument. Such an
identity is usually expressed by using a
variable argument in the form x*x and sqr x .
pi in the
expression pi=: 3.14159 is jarring, since
there is no intent to allow its value
to vary. Although the term pronoun is not
commonly used in mathematics, Hogben
uses it (p 432) to refer to the symbol e
used for Euler’s number.
+/v can replace the Greek sigma for summation and, more
generally, f/ can replace other such
symbols, but only for functions that
are not anonymous.
x=: 3 [ y=: 4
(x*y),(3*4),(xy),(34) NB. xy is NOT a J expression
12 12 12 34
HIERARCHY AND ORDER OF EXECUTION
The polynomial expression 3x+4x2+2x4 evaluates
correctly only under the
convention that the power function is
executed before multiplication, and
multiplication before addition. This
heirarchy is adopted throughout mathematics,
and in almost all programming languages.
+/3 4 2*x^1 2 4,
and in +/c*x^e.
1-2-3-4-5 may be interpreted
differently according to how it is
parenthesized: the following interpretations
are used in mathematics and in J:
(((1-2)-3)-4)-5 NB. Math
1-(2-(3-(4-5))) NB. J
1-(2+3+4+5)), and
the J result is the alternating sum (1+3+5)-(2+4).
f/v inserts the
function f between elements of v,
the J expression -/v gives the alternating
sum, a commonly-needed result that is expressed in
math as the sum over
(Greek sigma) (-1)i v.
Similarly,
the alternating product is expressed as %/v.
3e6 to denote 3*10^6. Producing no conflict,
this has also been widely adopted in
mathematics. 3j4 (a complex
number) and 1r2 (a rational)
and 1r2p1 could be similarly adopted for many
convenient constants, as
illustrated by the following examples from J:
2p1 1r2p1 1r4p_1 NB. Multiples of powers of pi
6.28319 1.5708 0.0795775
3x2 2b101 NB. 3 times e sqared,101 in base 2
22.1672 5
2ad30 NB. Complex number, magnitude 2 at 30 deg.
1.73205j1
n (when it exists) such
that n f x yields x
for any argument x. For example, 0 is
the neutral of +, and 1 is the neutral
of * .
k{.v
and k}.v to take and drop the first k elements
of a vector v, we may write identities
concerning sums and products over partitions of a vector:
v=: 2 3 5 7 11
k=: 3
(k{.v);(k}.v);((+/k{.v) + (+/k}.v));(+/v)
+-----+----+--+--+
|2 3 5|7 11|28|28|
+-----+----+--+--+
(*/v);((*/k{.v) * (*/k}.v))
+----+----+
|2310|2310|
+----+----+
k=: 0 (for
which k{.v is an
empty vector), but only because the
results of +/i.0 and */i.0
are defined to
be the neutrals of + and *.
The neutral
of minimum (<./i.0) is infinity, in
effect defining it.
*/x-i.m,
correctly including the case
of the empty vector that occurs when m is zero.
Moreover, */x+s*i.m covers a
host of cases; in particular, the
values _1 and 0
and 1 for s give the falling
factorial, the ordinary power function, and the
rising factorial. Non-integer values
of s are equally useful in the finite calculus.
!. provides them in
expressions of the form ^!.s.
p.!.s
provides the corresponding
polynomial functions (in which the power function ^
used in the ordinary
polynomial is replaced by the stope function ^!.s).
Compare the use of ^!.s
with the notations discussed in the introductory
Note on Notation in [2].
t under p, using the “under”
operator under=: &. . Thus:
under=: &.
0 0 1 1 +. under -. 0 1 0 1 NB. Or under not is and
0 0 0 1
3 + under ^. 4 NB. Times as add under natural log
12
3 – under (10 with ^.) 4 NB. Division using base-10 log
0.75
(<2 1 3) + under p. (<1 0 4) NB. Add polynomials
+-+------------------+
|2|3.68614 1 0.813859|
+-+------------------+
p. to map
the polynomial roots given by the
arguments to corresponding coefficients,
which are then added and mapped back
to give the equivalent multiplier 2
and roots 3.68614 1 0.813859.
--------------------------------
O O
--------------------------------
O O O
--------------------------------
O
--------------------------------
--------------------------------
Hold the doll
Drop the ball
Drop the ball and hold the doll
I dropped the ball and holded the doll
(base 10) logarithm of 3 log10 3
negation of 3 -3
factorial 3 3! (Order of noun and verb reversed)
(# of combs of) 3 above 5 in large parentheses
3 to the power 5 3 superscript 5
3 beside 5 (3,5) (Parens mandatory for vectors)
this is 3 Let t=3 (Single-letter names only)
that is 5 Let u=5
this plus that t+u
(is) this lessthan that t<u Usually treated as assertions,
(is) this equalto that t=u rather than as more usable truth-
functions that return results
English Math J C
logarithm of 3 log10 3 10^.3 log10(3.0)
negation of 3 3 -3 -3
factorial 3 3! !3 fact(3)
(# of combs of) 3 above 5 in parens 3!5 outof(3,5)
3 to the power 5 3 superscript 5 3^5 pow(3.0,5.0)
3 beside 5 (3,5) 3,5
this is 3 Let t=3 this=: 3 this=3
that is 5 Let u=5 that=: 5 that=5
this plus that t+u this+that this+that
(is) this equalto that t=u this=that this==that
+ - * ^ ! +. *. ^. < = > >:
select a
few for exploration by entering expressions such as:
16 + 36
52
16 +. 36
4
x=: 0 1 2 3 4 5 and
y=: x and x +/ y to produce an
addition table. Then enter expressions for further
tables to identify each of the
functions listed in Exp 1. Which of the functions
appear to be commutative?
f=: +/*-/ and
g=: </+.=/
]y=: x=: 0 1 2 3 4 5 6 - 3
_3 _2 _1 0 1 2 3
Exp 5 Evaluate the familiar sum over (denoted by capital
Greek sigma) cixi
at x=0,
assuming that (as is often stated) zero to the
power zero is undefined.
2 3 5 7;11 13
x (-/ ; -~/) y
x (</ ; =/ ; g ; <:/) y
~ ,
and state what part of speech it is in J:
x (-/ ; -~/) y
x (!/ ; !~/) y
x (*./ ; *.~/) y
S1=: %.S2=: |: (^/%.^!._1/)~ i.10r1
(%. N is matrix inverse, and M %. N is “matrix divide”)
;: 'TABLE=: x (*./ ; *.~/) y'
trace=: 13 !: 16
and 9 trace 1 to
invoke the tracing of subsequent expressions, that
is, to show their detailed
parsing and execution. Then enter TABLE=: x (*./ ; *.~/) y
and other expressions from earlier explorations.
9 trace 0 .
11A. Introduction
11A. Introduction
S11A.
Its modest aim is to elaborate the
point that informal, quasi-empirical,
mathematics does not grow through a
monotonous increase of the number of
indubitably established theorems but
through the incessant improvement of
guesses [Italics added] by speculation
and criticism, by the logic of proofs and
refutations.
Just send me the theorems, then I shall find the proofs.
Chrysippus
There are then two mathematics. There is
the real mathematics of the real
mathematicians, and there is what I will
call ‘trivial’ mathematics, for want of a better
word. The trivial mathematics may be
justified by arguments which would appeal to
Hogben, or other writers of his school,
but there is no such defence for the real
mathematics, which must be justified as
an art if it can be justified at all. There is
nothing in the least paradoxical or unusual
in this view, which is that held commonly
by mathematicians.
11B. Informal proofs
S11B.
We will illustrate a proof by first guessing
at a relation, using the function odd
from Section 2E:
odd=: 1:+2:*i.
odd 6
1 3 5 7 9 11
+/odd 6
36
(+/ odd 1),(+/odd 2),(+/odd 3),(+/odd 4),(+/odd 5)
1 4 9 16 25
It appears that the sum of the first n odd integers
is n*n, a relation we will
demonstrate by using a few facts illustrated by the following:
|. odd 6
11 9 7 5 3 1
(+/odd 6),(+/|.odd 6)
36 36
(odd 6)+(|.odd 6)
12 12 12 12 12 12
6 # 2*6 NB. Six copies of twice six
12 12 12 12 12 12
half=: % with 2
half 6 # 2*6
6 6 6 6 6 6
+/half 6 # 2*6
36
+/6#6
36
6*6
36
We now write a sequence of (obviously?) equivalent sentences:
n=: 6
+/odd n
36
+/|.odd n
36
half (+/odd n)+(+/|.odd n)
36
half +/(odd n)+(|.odd n)
36
half +/6 # 2*6
36
+/6 # 6
36
n*n
36
Because a value (6) was assigned to the
argument n, each of the foregoing
sentences were executable, and were
executed to give results whose equality
provided some assurance that an error
had not been made. We will, however, write
an informal proof as a similar sequence,
omitting the results that would have been
produced by their execution. Thus:
+/odd n
+/|.odd n
half (+/odd n)+(+/|.odd n)
half +/(odd n)+(|.odd n)
half +/6 # 2*6
+/6 # 6
n*n
11C. Formal proofs
S11C.
In a formal proof it is necessary to state
clearly the justification for each
equivalence, that is, the reason why each
statement is believed to be equivalent to
the one that precedes it. For example, we
will begin to convert the foregoing
informal proof to a formal proof as follows:
+/odd n
+/|.odd n
half (+/odd n)+(+/|.odd n)
half +/(odd n)+(|.odd n)
half +/6 # 2*6
+/6 # 6 NB. Half of sum is sum of halves
n*n NB. Def of multiplication (in 1G)
A stickler for logic might ask why he
should believe that half of a sum equals the
sum of the halves of the list summed.
A convincing answer is not as easy as might
be supposed, leading to an appreciation
of the following statement from Lakatos:
TEACHER: This decomposition of the
conjecture suggested by the proof
opens new vistas for testing. The
decomposition deploys the conjecture on
a wider front, so that our criticism has more targets.
+/odd n and +/|.odd n?
It may seem obvious, but what is special
about +/ that would not apply for
another function such as %/ ? A mathematician
might say (and offer proofs that)
functions such as +/
and */ and gcd/ and >./ are
symmetric, meaning that
they give the same result when applied to
any permutation (re-ordering) of their
arguments.
TEACHER: I agree with you that the conjecture
has received a severe
criticism by Alpha’s counterexample.
But it is untrue that the proof has
"completely misfired". If, for the time
being, you agree to my earlier
proposal to use the word ‘proof’ for a
‘thought experiment’ which leads to
a decomposition of the original conjecture
into ‘subconjectures’, instead of
using it in the sense of a ‘guarantee of
certain truth’, you need not draw
this conclusion.
+/even n, and write an informal proof of
the equivalence.
+/i.n
+/
on i. t. i. 411D. Inductive proofs
S11D.
+/ on odd n
were equal to n*n for a certain
value of n, we were able to use this
assumption to prove that +/ on odd n+1
must therefore equal (n+1)*(n+1), we
could then conclude that equality must
hold for all succeeding values, n+2
and n+3, and so on.
n=: 0),
we may further conclude that the
relation is true for all integers greater
than or equal to that value. For example:
+/ on odd n+1
(+/odd n)+(2*n)+1 NB. From definition of odd
(n*n)+(2*n)+1 NB. Assumed "induction hypothesis"
(n+1)*(n+1) NB. Simple algebra
11E. Recursive definition
S11E.
A function that is defined in terms of
itself is said to be recursively defined. For
example, a factorial function fac might
be defined by saying that fac n is
n * fac n-1. However, it is necessary to
further define its value for some
specific argument; for example, to
define fac 0 as 1.
]*fac on <: to perform the recursion,
the constant function 1: for the
specific argument 0, and the “conditional
function” = with 0 to choose
between them. Thus:
fac=: (]*fac on <:) ` 1: @. (= with 0)
fac 0
1
fac 4
24
!4
24
In the foregoing, the tie operator ` forms
a gerund from the two functions to
which it is applied, and the agenda
operator @.selects one of them according to the
result of the conditional function = with 0.
sodd for the sum of the odd
integers might be recursively defined as follows:
sodd=: (sodd on <: + 2 with * - 1:)`0: @. (= with 0)
(sodd 0),(sodd 1),(sodd 2),(sodd 3),(sodd 4)
0 1 4 9 16
The first and “main” function of the three
used in this recursive definition may be
interpreted as follows: sodd n is equal to
(sodd n-1) + (2*n) – 1 or,
equivalently, that sodd n+1
equals (sodd n) + (2*n+1) – 1.
n odd numbers
is n*n, using the inductive hypothesis
of Section D as follows:
sodd n+1
(sodd n) + (2*n+1) – 1 NB. Recursive def of sodd
(n*n) + (2*n+1) – 1 NB. Induction hypothesis
(n+1)*(n+1) NB. Simple algebra
n=: 6
d=: 0 1r6 _1r2 1r3
+/*:i.n
55
d p. n
55
11F. Guessing games
S11F.
i. applies to an argument
such as 3 4 to produce a table, but
we may also wish to apply it to each of the
arguments 3 and 4 to produce separate
lists. Thus:
a=: 3 4
i. a
0 1 2 3
4 5 6 7
8 9 10 11
i.3
0 1 2
i.4
0 1 2 3
eachbox=: &.>
i. eachbox a
+-----+-------+
|0 1 2|0 1 2 3|
+-----+-------+
PREFIX
In summing the list 2 7 1 8, we may also wish
to obtain the “subtotals” or
“partial sums” by applying sumation over each
prefix of the list. Thus:
pref=: \
sum=: +/
a=: 2 7 1 8
sum a
18
sum pref a
2 9 10 18
*/pref a
2 14 14 112
<./pref a
2 2 1 1
>./pref a
2 7 7 8
diag=: /.
c=: 1 2 1
d=: 1 3 3 1
t=: c */d
t
1 3 3 1
2 6 6 2
1 3 3 1
sum diag t
1 5 10 10 5 1
f is a function and g is a guess at
an equivalent function, then f-g gives the
needed correction. For example:
x=: 0 1 2 3 4 5 6 7r1
each=:"0
f=: +/ on i. each NB. Sums of integers
f x
0 0 1 3 6 10 15 21
g=: (^ with 2 % 2:) each
g x
0 1r2 2 9r2 8 25r2 18 49r2
(f-g) x
0 _1r2 _1 _3r2 _2 _5r2 _3 _7r2
corr=: (- % 2:) each
corr x
0 _1r2 _1 _3r2 _2 _5r2 _3 _7r2
gc=: g+corr NB. Corrected guess
(f=gc) x
1 1 1 1 1 1 1 1
RATIOS
The ratio f%g may sometimes provide a better
guide to correction. Consider, for
example, the following sums of powers:
f1=: +/ on (^&1) on i. each
f2=: +/ on (^&2) on i. each
f3=: +/ on (^&3) on i. each
f4=: +/ on (^&4) on i. each
f5=: +/ on (^&5) on i. each
(f1,f2,f3,f4,:f5) x
0 0 1 3 6 10 15 21
0 0 1 5 14 30 55 91
0 0 1 9 36 100 225 441
0 0 1 17 98 354 979 2275
0 0 1 33 276 1300 4425 12201
For f1 the equivalent function g1 is,
of course, the function gc derived above,
and we will use the ratio f3%g1 to look
for a definition of a function g3
corresponding to f3. Thus:
g1=: gc
(f3%g1) x
0 0 1 3 6 10 15 21
g1 x
0 0 1 3 6 10 15 21
It therefore appears that g3 may be
expressed as the product of g1 with itself:
g3=: g1*g1
(f3=g3) x
1 1 1 1 1 1 1 1
TAYLOR COEFFICIENTS
For functions such as g1 and g3 that
are expressed directly in terms of powers,
quotients, products, and sums, the Taylor
operator may be applied to give the
coefficients of an equivalent polynomial. For example:
x=: 0 1 2 3 4 5 6 7r1
c1=: g1 t. x
c1
0 _1r2 1r2 0 0 0 0 0
c1&p. x
0 0 1 3 6 10 15 21
f1 x
0 0 1 3 6 10 15 21
c3=: g3 t. x
c3
0 0 1r4 _1r2 1r4 0 0 0
(f3=c3&p.) x
1 1 1 1 1 1 1 1
POLYNOMIAL APPROXIMATIONS
The Taylor operator does not apply to
functions such as the square root, and we
use instead an expression that gives
coefficients that approximate the function.
Thus:
f=: %:
f x
0 1 1.4142 1.7321 2 2.2361 2.4495 2.6458
(f x)*(f x) NB. To verify that f is square root
0 1 2 3 4 5 6 7
c=: (f x) %. (x ^/ i.#x)
c p. x
0 1 1.4142 1.7321 2 2.2361 2.4495 2.6458
It may be noted that the polynomial appeared
to give not just an approximation,
but the exact result. However, the result
may be far from correct for arguments
other than the x used in computing the
coefficients, especially so for arguments
outside the range of the elements of x.
For example:
c p. x+1r2
0.64666 1.2301 1.5796 1.8717 2.1204 2.3468 2.5436 2.8157
f x+1r2
0.70711 1.2247 1.5811 1.8708 2.1213 2.3452 2.5495 2.7386
c p. 9
5.9577
f 9
3
12A. Anagrams
12A. Anagrams
S12A.
Although we have discussed the importance
of relations and analysis (proofs) in
math, our exclusive use of numeric arguments
has probably given the impression
that math is exclusively about numbers.
To redress the balance, we will now treat
some relations concerning English words,
beginning with sorting and anagrams.
sort=: /:~ (which applies equally to numeric arguments).
For example:
sort=: /:~
w0=: 'REVEL'
sort w0
EELRV
w1=: 'LEVER'
sort w1
EELRV
w0=w1
0 1 1 1 0
(sort w0)=(sort w1)
1 1 1 1 1
As shown in the last results the words are not equal,
but they are similar, in the
sense that they are anagrams, or permutations of
one another, containing the same
letters but in a different order. Although
it is not an English word, we will also call
the word 'EELVR' an anagram.
w2=: 'AH' (including itself).
w3=: 'ART' .
w4a=: 'SPOT' .
w4b=: 'ABCD' .
To check your work, use the anagram function A. as
illustrated below:
0 A. w2=: 'AH'
AH
1 A. w2
HA
0 1 A. w2
AH
HA
0 1 2 3 4 5 A. w3=: 'ART'
ART
ATR
RAT
RTA
TAR
TRA
6 A. w3
|index error
| 6 A.w3
i.6. Similar tests
will show that there are twenty-four anagrams
of four-letter words, leading to the following tables:
(i.24) A. w4a=: 'SPOT'
SPOT
SPTO
SOPT
SOTP
STPO
STOP
PSOT
PSTO
POST
POTS
PTSO
PTOS
OSPT
OSTP
OPST
OPTS
OTSP
OTPS
TSPO
TSOP
TPSO
TPOS
TOSP
TOPS
trans=: |: NB. Function to transpose a table
trans (i.24) A. w4a=: 'SPOT'
SSSSSSPPPPPPOOOOOOTTTTTT
PPOOTTSSOOTTSSPPTTSSPPOO
OTPTPOOTSTSOPTSTSPPOSOSP
TOTPOPTOTSOSTPTSPSOPOSPS
trans (i.24) A. w4b=: 'ABCD'
AAAAAABBBBBBCCCCCCDDDDDD
BBCCDDAACCDDAABBDDAABBCC
CDBDBCCDADACBDADABBCACAB
DCDBCBDCDACADBDABACBCABA
trans (i.24) A. i.4
0 0 0 0 0 0 1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 3 3
1 1 2 2 3 3 0 0 2 2 3 3 0 0 1 1 3 3 0 0 1 1 2 2
2 3 1 3 1 2 2 3 0 3 0 2 1 3 0 3 0 1 1 2 0 2 0 1
3 2 3 1 2 1 3 2 3 0 2 0 3 1 3 0 1 0 2 1 2 0 1 0
(trans (i.24) A. i.4) { 'arst'
aaaaaarrrrrrsssssstttttt
rrssttaassttaarrttaarrss
strtrsstatasrtatarrsasar
tstrsrtstasatrtarasrsara
trans (i.24) A. 'arst'
aaaaaarrrrrrsssssstttttt
rrssttaassttaarrttaarrss
strtrsstatasrtatarrsasar
tstrsrtstasatrtarasrsara
i.4) they
may be used to index a word to
produce its anagrams.
anag=: trans on (i. on ! on # A. ]) on
various lists of numbers and letters.n-letter
word can be obtained by a simple
argument: the leading letter can be
chosen in n ways, so that
the total is n times
the number of anagrams of an (n-1)-letter
word,
leading to the product n times n-1
times n-2, etc. -- in other
words, !n. Thus:
#w4a
4
!#w4a
24
(i.24) A. 'ABCD' suggests a systematic
approach: choose the first letter
of the word as the initial letter, and
append to it the table for the remaining letters
organized by the same principle; then
choose the second letter as the initial, and so
on.
i.4 3 2 and (i.4 3 2) A. 'ABCD'
and <"2(i.4 3 2) A. 'ABCD' . Study the
results, and try similar expressions for
shorter and longer words.p is any one of the six permutations
of 0 1 2, then 3,p is a permutation of
order four. However, if p is prefixed by
any of the other possible beginnings of a
permutation of order four (2 or
1 or 0)
the result is not a permutation, but it can
be made so by incrementing each element
of p that is greater than or equal to the
prefix. Thus:
P=: (i.6) A. 0 1 2
P;(3,.P);(P>:2);(P+P>:2);(2,.P+P>:2);(1,.P+P>:1);(0,.P+P>:0)
+-----+-------+-----+-----+-------+-------+-------+
|0 1 2|3 0 1 2|0 0 1|0 1 3|2 0 1 3|1 0 2 3|0 1 2 3|
|0 2 1|3 0 2 1|0 1 0|0 3 1|2 0 3 1|1 0 3 2|0 1 3 2|
|1 0 2|3 1 0 2|0 0 1|1 0 3|2 1 0 3|1 2 0 3|0 2 1 3|
|1 2 0|3 1 2 0|0 1 0|1 3 0|2 1 3 0|1 2 3 0|0 2 3 1|
|2 0 1|3 2 0 1|1 0 0|3 0 1|2 3 0 1|1 3 0 2|0 3 1 2|
|2 1 0|3 2 1 0|1 0 0|3 1 0|2 3 1 0|1 3 2 0|0 3 2 1|
+-----+-------+-----+-----+-------+-------+-------+
n, and prefix each by an element of
i.n+1. The scheme can also be used as
the basis for a recursive definition of a
function for making tables of permutations.
13A. Introduction13A. Introduction
S13A.
Boolean or symbolic logic concerns propositions,
a proposition being simply a
function that produces one of only two
possible results. These two results are
commonly referred to as true and false,
and (following George Boole, the father of
symbolic logic) will be denoted by the
numbers 1 and 0. For example:
f1=: <&0
f2=: >&0
x=: i:3
x
_3 _2 _1 0 1 2 3
f1 x
1 1 1 0 0 0 0
f2 x
0 0 0 0 1 1 1
The proposition f1 is true (that is, gives the
result 1)
for any negative number,
and is said to define the set of negative
numbers; f2 defines the set of positive
numbers.
y may be obtained by using the
result f2 y to copy from y each element
of y for which f2 y is true. For
example:
y=: 3 1 4 1 5 9 - 3
y
0 _2 1 _2 2 6
f2 y
0 0 1 0 1 1
(f2 y) # y
1 2 6
13B. Or, and, and not
S13B.
The function or=: +. may be applied to two
propositions to yield a proposition
that is true if either of the propositions is true. For example:
or=: +.
g=: f1 or f2
g x
1 1 1 0 1 1 1
Similarly for and=: *., the proposition
proposition h=: f1 and f2 is true
only if both f1 and f2 are true. Since
a number cannot be both positive and
negative, h can never be true, and is
said to define an empty set. Thus:
and=: *.
h=: f1 and f2
h x
0 0 0 0 0 0 0
(h x) # x
(g x) # x
_3 _2 _1 1 2 3
The (monadic) function not=: -. negates a
boolean result. For example:
not=: -.
(f1,.(not on f1),.g,.(not on g),.h,.(not on h)) x
1 0 1 0 0 1
1 0 1 0 0 1
1 0 1 0 0 1
0 1 0 1 0 1
0 1 1 0 0 1
0 1 1 0 0 1
0 1 1 0 0 1
13C. Lists and sets
S13C.
The set of English vowels is defined by the
proposition vow, developed as
follows:
s=: 'do you love me'
'aeiou' =/ s
0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 1 0 0 1
0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 1 0 0 1 0 0 0 1 0 0 0 0 0
0 0 0 0 0 1 0 0 0 0 0 0 0 0
or/'aeiou' =/ s
0 1 0 0 1 1 0 0 1 0 1 0 0 1
vow=: or/ on ('aeiou'&(=/))
vow s
0 1 0 0 1 1 0 0 1 0 1 0 0 1
(vow s)#s
oouoee
(not vow s)#s
d y lv m
alphabet=: 'abcdefghijklmnopqrstuvwxyz '
(vow alphabet)#alphabet
aeiou
It is important to understand that the
set of English vowels is defined by the
function vow, not by
the list 'aeiou'
(nor by the same list that resulted from
applying the function to the universe
of the entire alphabet).
'aeiou' or repetitions
of its elements (neither of which affect
the definition of the function) with the
question of whether a set is ordered (which it is not).
13D. Classification
S13D.
A single proposition can be used to classify
its arguments according to whether they
belong to the set it defines, and several
propositions can be used to provide
several classifications. Consider, for
example, a list of transactions that are
recorded as a list of account numbers
(an) and a
corresponding list of credits (cr)
to the accounts, as well as the list of
all possible account numbers (all). Thus:
an=: 1010 1040 1030 1030 1060 1010 1040 1040 1050
cr=: 131 755 458 532 218 47 678 679 934
all=: 1010 1020 1030 1040 1050 1060
c=: all =/ an
c
1 0 0 0 0 1 0 0 0
0 0 0 0 0 0 0 0 0
0 0 1 1 0 0 0 0 0
0 1 0 0 0 0 1 1 0
0 0 0 0 0 0 0 0 1
0 0 0 0 1 0 0 0 0
eachrow=: "1
cr +/ on * eachrow c
178 0 990 2112 934 218
14A. Introduction
14A. Introduction
S14A.
A pair of expressions (such as x*y and y*x)
that give identical results for all
possible values of the arguments are said to
form an identity. Identities are
important because they allow a function to
be expressed in different ways, each of
which may possess particular advantages,
such as ease of evaluation or insight
into the behaviour of the function.
+/ odd n (the sum of the first n odd
numbers) is identical to the more
easily computed product n*n. Similarly, the sum
of the squares of the first n positive
integers (+/(1+i.n)^2) is identical to the
polynomial 0 1r6 1r2 1r3 p. n.
14B. Commutativity
S14B.
The commute operator ~ interchanges the
arguments of the function to which it is
applied. For example:
3 -~ 5
2
5 – 3
2
from=: -~
3 from 5
2
into=: %~
5 into 3
0.6
If f~ is identical to f,
then the function f is said to be commutative.
For example,
+ and * are commutative,
but - and
% are not.
<, =, [(left), ],
<.
(min), >., +:
(greatest common divisor), and *: (least common multiple).
f table i:4
for both commutative and non-commutative
functions, and observe the symmetry
of the former.14C. Associativity
S14C.
3-(4-5) and (3-4)-5 apply the
same function to the same
arguments, but yield different results because
the parentheses associate them
differently; subtracting 5 from 4 and then
subtracting the result from 3 to yield 4
in the first case, and subtracting 4 from 3
and then subtracting 5 from the result to
yield _6 in the last.
3+(4+5)
and (3+4)+5 both yield 12.
14D. Symmetry
S14D.
+/2 3 5 7
17
+/3 5 2 7
17
-/2 3 5 7
_3
-/3 5 2 7
_7
If f is both associative and commutative,
then f/ is symmetric. For example:
all=: (i.!3) A. 2 3 5r1
all
2 3 5
2 5 3
3 2 5
3 5 2
5 2 3
5 3 2
lcm=: *.
eachrow=: "1
lcm/ eachrow all
30 30 30 30 30 30
%/ eachrow all
10r3 6r5 15r2 6r5 15r2 10r3
14E. Distributivity
S14E.
If f(x g y) is equivalent to (f x)g(f y),
we say that the (monadic)
function f distributes over the (dyadic)
function g. The two parts of this identity
may also be stated as f on g and (f on [) g (f on ]).
For example, +: (double) and -: (halve)
each distribute over both addition and
subtraction, facts that may be tested in the following manner:
3 (+:on+) 5
16
3 (+:on[ + +:on]) 5
16
The operator over defined below can be
used in the expression f over g to
test whether f distributes over g. Thus:
over=: 2 : '(u. on v.) = ((u. on [) v. (u. on ]))'each
(+:over - table ,. -: over + table ,. +: over * table) i:3
+--+----------------+--+----------------+--+----------------+
| |_3 _2 _1 0 1 2 3| |_3 _2 _1 0 1 2 3| |_3 _2 _1 0 1 2 3|
+--+----------------+--+----------------+--+----------------+
|_3| 1 1 1 1 1 1 1|_3| 1 1 1 1 1 1 1|_3| 0 0 0 1 0 0 0|
|_2| 1 1 1 1 1 1 1|_2| 1 1 1 1 1 1 1|_2| 0 0 0 1 0 0 0|
|_1| 1 1 1 1 1 1 1|_1| 1 1 1 1 1 1 1|_1| 0 0 0 1 0 0 0|
| 0| 1 1 1 1 1 1 1| 0| 1 1 1 1 1 1 1| 0| 1 1 1 1 1 1 1|
| 1| 1 1 1 1 1 1 1| 1| 1 1 1 1 1 1 1| 1| 0 0 0 1 0 0 0|
| 2| 1 1 1 1 1 1 1| 2| 1 1 1 1 1 1 1| 2| 0 0 0 1 0 0 0|
| 3| 1 1 1 1 1 1 1| 3| 1 1 1 1 1 1 1| 3| 0 0 0 1 0 0 0|
+--+----------------+--+----------------+--+----------------+
over to test whether +: distributes
over gcd and lcm.d=: * with 2 and h=: % with 2 it
appears that numbers other
than 2 would do as well, that is, any multiple or
fraction function would distribute
over addition (and subtraction). Thus:
(((* with 2.5)over + table),.((% with 4)over + table))i:3
+--+----------------+--+----------------+
| |_3 _2 _1 0 1 2 3| |_3 _2 _1 0 1 2 3|
+--+----------------+--+----------------+
|_3| 1 1 1 1 1 1 1|_3| 1 1 1 1 1 1 1|
|_2| 1 1 1 1 1 1 1|_2| 1 1 1 1 1 1 1|
|_1| 1 1 1 1 1 1 1|_1| 1 1 1 1 1 1 1|
| 0| 1 1 1 1 1 1 1| 0| 1 1 1 1 1 1 1|
| 1| 1 1 1 1 1 1 1| 1| 1 1 1 1 1 1 1|
| 2| 1 1 1 1 1 1 1| 2| 1 1 1 1 1 1 1|
| 3| 1 1 1 1 1 1 1| 3| 1 1 1 1 1 1 1|
+--+----------------+--+----------------+
Since * is commutative, 2.5 with* would
serve as well as *with 2.5 in
the foregoing, but would 4 with% give
the same result as %with 4 ?
14F. Distributivity of dyadic functions
S14F.
Although we have discussed the distribution of
monadic functions over dyadic
functions, it is more common in mathematics to
discuss the distribution of dyadic
functions over dyadic functions. Thus,
because a*(b+c) is equivalent to
(a*b)+(a*c), it is said that multiplication
distributes over addition. Similarly
(a+b)*c is equivalent to (a*c)+(b*c), and
again multiplication distributes
over addition.
(a+b)%c is equivalent to (a%c)+(b%c)
and we might conclude
that division also distributes over addition.
On the other hand, a%(b+c) is not
equivalent to (a%b)+(a%c), and it is
sometimes said that division “distributes
to the left” over addition.
x with * and * with x and % with x
distribute over addition, but that
x with % does not.
Finally, the idea of a monadic function
distributing over addition leads to the important notion of
linear functions, treated in Chapter 15.
14G. Parity
S14G.
A graph of the square function appears to be reflected
(as in a mirror) in the vertical line through the origin.
This is so because the square of -x equals the
square of x for every x. Thus:
sqr=: ^ with 2
x=: i:1j10
x
_1 _0.8 _0.6 _0.4 _0.2 0 0.2 0.4 0.6 0.8 1
sqr x
1 0.64 0.36 0.16 0.04 0 0.04 0.16 0.36 0.64 1
PLOT x;sqr x
Similarly, a graph of the cube is reflected in the origin,
because the cube of -x equals the negative
of the cube of x:
cube=: ^ with 3
PLOT x;cube x
(cube -x)=(-cube x)
1 1 1 1 1 1 1 1 1 1 1
(sqr x)=(sqr x)
1 1 1 1 1 1 1 1 1 1 1
Simple algebra shows that for any function f
, the function ef=: f + f with - is even, and
that the function of=: f - f with - is odd.
For example: f=: ^ NB. The exponential (which is neither even nor odd)
ef=: f+f with -
of=: f-f with -
PLOT x;(f,ef,:of) x
The functions hef=: ef % 2: and
hof=: of % 2: are called the even
and odd parts of f because they are,
respectively, even and odd, and because they sum to f:
hef=: ef % 2:
hof=: of % 2:
(f=hef+hof) x
1 1 1 1 1 1 1 1 1 1 1
PLOT x;(hef,hof,:hef+hof) x
The odd and even parts of a function are often functions
of interest in their own right. For example, hof
is the hyperbolic sine, and hef
is the hyperbolic cosine. They are denoted in J by
5 with o. and 6 with o., respectively.
15A. Dot product and vector functions15A. Dot product and vector functions
S15A.
The expression +/w*x gives a sum over the
argument x weighted by the vector
w. For example:
x=: 2 3 5 7 11
w=: 0 1 2 3 4
w*x
0 3 10 21 44
+/w*x
78
The dot operator (.) applied to the
functions +/ and * produces an equivalent
function called the dot, inner, or matrix product.
Thus:
dp=: +/ . *
w dp x
78
Moreover, dp applies to a matrix (table)
left argument to produce a vector (list) of
results, using successive rows of the
matrix as weights. For example:
a=: 3 1 4
b=: 2 7 8
c=: 2 1 0
]m=: a,b,:c
3 1 4
2 7 8
2 1 0
m dp 2 3 5
29 65 7
b dp 2 3 5
65
g=: m with dp
g 2 3 5
29 65 7
If m is a square matrix (as in the present example),
then m with dp applies to a
vector to produce a vector of the same number of
elements. We will call such a
function a vector function.
15B. Dot product as a linear vector function
S15B.
The dot product with a vector or matrix left
argument distributes over addition,
and is therefore a linear function, as defined in Section
14F. For example:
x=: 2 3 5
y=: 0 1 2
f=: 2 7 8 with dp
((f x)+(f y)) , (f x+y)
88 88
g=: m with dp
((g x)+(g y)) ,: (g x+y)
38 88 8
38 88 8
The function I=: =/~ on i. yields a result
called an identity matrix, because
when used
with dp it produces an identity
function (that yields its argument
unchanged). For example:
I=: i. =/ i.
]i=: I 3
1 0 0
0 1 0
0 0 1
i with dp 2 3 5
2 3 5
A linear vector function need not be
expressed as a dot product, but if it is not, the
required matrix is easily determined.
For example, sop=: +/\ (sums over
prefixes) is a linear vector function,
and the matrix m for an equivalent function
m with dp may be determined as follows:
x=: 2 3 5 7 11
y=: 0 1 2 3 4
sop=: +/\
((sop x)+(sop y)) ,: (sop x+y)
2 6 13 23 38
2 6 13 23 38
]i=: I # x
1 0 0 0 0
0 1 0 0 0
0 0 1 0 0
0 0 0 1 0
0 0 0 0 1
i dp x
2 3 5 7 11
sop i dp x
2 5 10 17 28
(sop i) dp x
2 5 10 17 28
]m=: sop i
1 0 0 0 0
1 1 0 0 0
1 1 1 0 0
1 1 1 1 0
1 1 1 1 1
The resulting matrix provides an
interesting way of looking at the sums over
prefixes: the ones in the successive
rows indicate the elements included in the
successive sums.
15C. Matrix inverse
S15C.
The inverse of a linear vector
function m with dp is itself a linear vector
function, and can therefore be
expressed in the form mi with dp. The matrix
mi may be obtained from m by
applying the matrix inverse function %. . Thus:
]z=: m dp x=: 2 3 5 7 11
2 5 10 17 28
mi=: %.m
mi dp z
2 3 5 7 11
mi
1 0 0 0 0
_1 1 0 0 0
0 _1 1 0 0
0 0 _1 1 0
0 0 0 _1 1
The pairs _1 1 in the rows of the
inverse mi indicate that the inverse of the sum
over prefixes is a differencing operation,
and the matrix mi may be called a
difference matrix.
|.),
rotation (2 with |.) and 8 with A..
16A. Ascending and descending order of exponents16A. Ascending and descending order of exponents
S16A.
In conventional mathematics, polynomials
are sometimes expressed with the
exponents in ascending order (beginning with zero),
and sometimes in descending
order (beginning with the largest exponent). Thus:pa (for ascending
order), and pd (for descending). The
former is the function p. that we have used
in earlier chapters, and the latter
may be expressed in terms of it using the reversal
function (|.) to reverse the order of the
coefficients. Thus:
pa=: p.
pd=: |. on [ pa ]
x=: 0 1 2 3 4 5
1 2 3 pa x
1 6 17 34 57 86
1 2 3 pd x
3 6 11 18 27 38
1 2 3 pd 10
123
10
gives the value in decimal of the digits in
the list of coefficients. In fact, the decimal
number system (and number systems with bases
other than ten) are “polynomial
representations” of numbers, and we may
find that schemes for multiplying
polynomials lead to improved methods
of multiplying multi-digit decimal numbers.
16B. Products of polynomials
S16B.
The product of two polynomials c1 with p.
and c2 with p.
is itself a
polynomial, whose coefficients can be computed
from c1 and c2 by first forming
their multiplication table. For example:
c1=: 2 3 5
c2=: 4 1 0 3
c1 * table c2
+-+---------+
| | 4 1 0 3|
+-+---------+
|2| 8 2 0 6|
|3|12 3 0 9|
|5|20 5 0 15|
+-+---------+
The coefficient 8 in the upper left corner is
the coefficient of x^0 in the product
polynomial, the elements of the next
diagonal (2 12) are coefficients of x^1, the
next diagonal has coefficients of x^2,
and so on. The combined coefficients of
successive powers may therefore be
obtained by summing diagonals of the table to
obtain c3=: 8 14 23 11 9 15. Thus:
c3=: 8 14 23 11 9 15
x=: 0 1 2 3 4 5
(c1 p. x) * (c2 p. x)
8 80 840 4928 18800 54528
c3 p. x
8 80 840 4928 18800 54528
In the expression f/. the oblique
operator /. produces a function that applies f
to each of the diagonals of a table argument. Thus:
c1 */ c2
8 2 0 6
12 3 0 9
20 5 0 15
+//. c1 */ c2
8 14 23 11 9 15
c3 = +//. c1 */ c2
1 1 1 1 1 1
We may therefore define a function pc to
produce product coefficients as follows:
pc=: +//. on (*/)
c1 pc c2
8 14 23 11 9 15
1 1 pc 1 1
1 2 1
1 1 pc 1 1 pc 1 1
1 3 3 1
Similar reasoning will show that the
same function pc produces coefficients for
the product of descending polynomials.
This may be tested as follows:
c1 pd x
5 10 19 32 49 70
c2 pd x
3 8 39 120 275 528
(c1 pd x)*(c2 pd x)
15 80 741 3840 13475 36960
c3 pd x
15 80 741 3840 13475 36960
16C. Multiplication of decimal numbers
S16C.
The last result of Section 16A indicates
that the result of c pd 10 is the decimal
value of the number whose digits are the
elements of the list c. Thus:
c1=: 1 2 3
c2=: 1 2 0 3
(c1 pd 10),(c2 pd 10)
123 1203
c1 pc c2
represent the product of the numbers
123 and 1203. Thus:
]c3=: c1 pc c2
1 4 7 9 6 9
c3 pd 10
147969
123*1203
147969
Consequently, the product can be obtained
by making the table c1 * table c2
and summing its diagonals:
c1 * table c2
+-+-------+
| |1 2 0 3|
+-+-------+
|1|1 2 0 3|
|2|2 4 0 6|
|3|3 6 0 9|
+-+-------+
1,(+/2 2),(+/0 4 3),(+/3 0 6),(+/6 0),9
1 4 7 9 6 9
However, these diagonal sums for other
arguments may yield results greater than
9, and therefore themselves be multi-digit numbers.
For example:
c4=: 1 2 3 4 5 6 7
c5=: 8 7 6 5 4
]c6=: c4 pc c5
8 23 44 70 100 130 160 126 92 59 28
c6 pd 10
108214735818
1234567*87654
108214735818
The result c6 does correctly represent
the product, but it is unsatisfactory in that
its multi-digit elements cannot be simply
squeezed together (as in 8234470100130160126925928)
to produce the correct product.
c6=: 8 23 44 70 100 130 160 126 92 59 28
c7=: 8 23 44 70 100 130 160 126 92 61 8
c8=: 8 23 44 70 100 130 160 126 98 1 8
c9=: 8 23 44 70 100 130 160 135 8 1 8
c10=: 8 23 44 70 100 130 173 5 8 1 8
c11=: 8 23 44 70 100 147 3 5 8 1 8
c12=: 8 23 44 70 114 7 3 5 8 1 8
c13=: 8 23 44 81 4 7 3 5 8 1 8
c14=: 8 23 52 1 4 7 3 5 8 1 8
c15=: 8 28 2 1 4 7 3 5 8 1 8
c16=: 10 8 2 1 4 7 3 5 8 1 8
c17=: 1 0 8 2 1 4 7 3 5 8 1 8
c6 pd 10
108214735818
c17 pd 10
108214735818
Moreover, the entire normalization
from c6 to c17 can be done easily by hand in a
single process. In summary, multiplication
of decimal numbers represented by the
lists of digits d1 and d2 can be performed
by hand by writing down the table
d1 * table d2, summing the diagonals,
and “normalizing” the results to single-digit
elements (written together without spaces).
This process may be compared with the
commonly-taught scheme shown below:
1234567
87654
------------
4938268
6172835
7407402
8641969
9876536
------------
108214735818
In contrast to the three distinct steps
of recording the multiplications in a table,
summing its diagonals, and a final
normalizion by carry, this conventional scheme
combines multiplication and carry
in each step of the production of each line, and
again uses carrying in the final
summation of the several lines.
16D. Other bases
S16D.
Just as c pd 10 gives the decimal (or base-10)
value of c as a weighted sum
with weights of decreasing powers of 10,
so c pd 8 gives the base-8 value,
using powers of 8 as weights. For example:
c1=: 1 2 3
c2=: 1 2 0 3
(c1 pd 8),(c2 pd 8)
83 643
]c3=: c1 pc c2
1 4 7 9 6 9
c3 pd 8
53369
(c1 pd 8)*(c2 pd 8)
53369
Again the result c3 can be normalized,
this time to the range 0 to 7. This requires
dividing by 8, “carrying” the integer
quotient, and leaving the remainder. Thus:
c3
1 4 7 9 6 9
1 4 7 9 7 1 pd 8
53369
1 4 8 1 7 1 pd 8
53369
1 5 0 1 7 1 pd 8
53369
16E. Remainder, Divisibility, and Integer part
S16E.
The carrying process used in preceding sections
made use of the remainder. For example, 8
divides into 24 exactly (that is, an
integer number of times), but it divides 27
leaving a remainder of 3.
| and is
illustrated by: | table i=: 1 2 3 4 5 6 7 8
+-+---------------+
| |1 2 3 4 5 6 7 8|
+-+---------------+
|1|0 0 0 0 0 0 0 0|
|2|1 0 1 0 1 0 1 0|
|3|1 2 0 1 2 0 1 2|
|4|1 2 3 0 1 2 3 0|
|5|1 2 3 4 0 1 2 3|
|6|1 2 3 4 5 0 1 2|
|7|1 2 3 4 5 6 0 1|
|8|1 2 3 4 5 6 7 0|
+-+---------------+
d|n is zero, we
say that n is divisible by d, as
illustrated by the following
divisibility table: =&0 on | table i
+-+---------------+
| |1 2 3 4 5 6 7 8|
+-+---------------+
|1|1 1 1 1 1 1 1 1|
|2|0 1 0 1 0 1 0 1|
|3|0 0 1 0 0 1 0 0|
|4|0 0 0 1 0 0 0 1|
|5|0 0 0 0 1 0 0 0|
|6|0 0 0 0 0 1 0 0|
|7|0 0 0 0 0 0 1 0|
|8|0 0 0 0 0 0 0 1|
+-+---------------+
n-d|n is divisible by d,
and the result of the division d%~n-d|n is
called the integer quotient. Thus: iq=: ([%~]-|)"0
8 iq 22 23 24 25 26 27
2 2 3 3 3 3
iq table i
+-+---------------+
| |1 2 3 4 5 6 7 8|
+-+---------------+
|1|1 2 3 4 5 6 7 8|
|2|0 1 1 2 2 3 3 4|
|3|0 0 1 1 1 2 2 2|
|4|0 0 0 1 1 1 1 2|
|5|0 0 0 0 1 1 1 1|
|6|0 0 0 0 0 1 1 1|
|7|0 0 0 0 0 0 1 1|
|8|0 0 0 0 0 0 0 1|
+-+---------------+
<. to
the result of division. For example: i%5
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
<.i%5
0 0 0 0 1 1 1 1
<. on (%~) table i
+-+---------------+
| |1 2 3 4 5 6 7 8|
+-+---------------+
|1|1 2 3 4 5 6 7 8|
|2|0 1 1 2 2 3 3 4|
|3|0 0 1 1 1 2 2 2|
|4|0 0 0 1 1 1 1 2|
|5|0 0 0 0 1 1 1 1|
|6|0 0 0 0 0 1 1 1|
|7|0 0 0 0 0 0 1 1|
|8|0 0 0 0 0 0 0 1|
+-+---------------+
16F. Notes
S16F.
In his Chapter 6: The Dawn of nothing, Hogben provides
an interesting discussion of number systems and
the role of zero in their development. For example:
Why this is necessarily true, is easy to see if we recall
the Roman representation of 32, i.e. XXXII. If the
Romans had written it in the form III II,
there would have been no way of distinguishing it from 302,
320, 3,020, 3,200, etc. The one simple way of escape
from this dilemma is to introduce a sign such as
the Maya lozenge ..., a dot or a circle for the
empty column of the abacus. We can then write
32, 302, 320, 3,020 as below:
17A. Introduction17A. Introduction
S17A.
In cooking, a list of one or more instructions is called
a recipe; in mathematics or computer science it is
called a program or algorithm.
odd=: 1:+even
even=: 2:*i.
odd makes
use of the function even, and each function may
be used independently: odd 5
1 3 5 7 9
even 5
0 2 4 6 8
(odd*even) 5
0 6 20 42 72
17B. Designing an algorithm
S17B.
Except for the very simplest cases, it is best to begin
by computing an example and assessing its suitability, and
only then composing the required definition or definitions.
We will illustrate two approaches to this final step:f
and g), and use them in a single definition
(such as h=: f+f on g).x. and y. .
x,
and applying sine and decay functions to it:
x=: 1r2*i.11
s=: 1&o. x
]n=: _1r5 * x
0 _1r10 _1r5 _3r10 _2r5 _1r2 _3r5 _7r10 _4r5 _9r10 _1
d=: ^ n
set 2
d*s
0 0.43 0.69 0.74 0.61 0.36 0.077 _0.17 _0.34 _0.4 _0.35
PLOT x;d,s,:d*s
17C. Explicit definition
S17C.
We will now illustrate the use of a form of definition in
which the arguments occur explicitly (using the names x.
and y.), and allows multiple lines in which names
may be assigned to intermediate results. Thus: b=: 4
b
4
f=: 4 : 0
a=: i. x.
b=: a ^ y.
c=: +/b
b;c
)
5 f 2
+----------+--+
|0 1 4 9 16|30|
+----------+--+
b
0 1 4 9 16
It is clear that the value originally assigned to b
(in order to illustrate this matter) has been changed by the execution of f, behaviour
that might cause the loss of valuable data. This unsuitable
behaviour can be avoided by using a different copula (=.)
to localize the names assigned within the function. Thus: f=: 4 : 0
a=.i. x.
b=.a ^ y.
c=.+/b
b;c
)
7 f 3
+-------------------+---+
|0 1 8 27 64 125 216|441|
+-------------------+---+
b
0 1 4 9 16
See Section S16C for a further example of explicit definition.
@. was
used to provide a conditional execution of functions
in its argument. In many programming languages, conditional
execution is provided by constructs such as if,
then, else, and while do. The form used in
explicit definition is illustrated
by: SUM=: 3 : 0
i=.0
a=.0
while. i<y.
do. a=.next a
i=.next i
end.
a
)
SUM 4
6
0+1+2+3
6
sum=: +/ on i.
sum 4
6
18A. Introduction18A. Introduction
S18A.
Just as the operator d.1 can be applied to
a function f to obtain its derivative,
so the operator d._1 can be applied to obtain
its anti-derivative, that is, a function whose
derivative is f. For example: derv=: d.1
anti=: d._1
c=: 1 2 3 4 5 6
c with p.
1 2 3 4 5 6&p.
c with p. derv
2 6 12 20 30&p.
c with p. anti
0 1 1 1 1 1 1&p.
c with p. anti derv
1 2 3 4 5 6&p.
derv and
anti on functions such as ^ and
1&o. and 2&o. and 6&o.1 2 3 4 5&p.
and 7 2 3 4 5&p. , and explain their agreement.18B. Area under a graph as a function
S18B.
A study of the graph of Section 3B (an approximation to
a circle) suggests that the area under the
graph of a function f from the
argument _1 (or any fixed point) to
an argument x is itself a function of x (that
depends upon the function f). a=: i:1j40
PLOT a;cir a
What is the rate of change of the area function? x=: 0.5, the spacing s=: 0.05, and
the next plotted point x+s. The change in area is due to
the quadrilateral with base s and heights cir x
and cir x+s, an area equal to s times
the average of the heights cir x and cir x+s.((cir x+s)-(cir x))%s is
therefore the average of cir x and cir x+s.
For small values of s, this average approaches cir x;
the derivative of the area under the graph of cir
is therefore the function cir itself.18C. Polynomial approximations
S18C.
As illustrated in Section 3B, the area under a curve can be
computed to give the value of the anti-derivative of a function
at a chosen point. But this does not yield a function for
the anti-derivative in the sense that the operator d._1
does for the functions to which it applies.
f
is provided by finding the coefficients of a polynomial that
approximates it, and then
using the fact that the anti-derivative (as well as the derivative) of
a polynomial is easily obtained.
(f x) %. x^/i.n yields the
coefficients of an n-term polynomial that best
fits the function f at the points x.
For example: ]x=: i:1j10
_1 _0.8 _0.6 _0.4 _0.2 0 0.2 0.4 0.6 0.8 1
]c=: (1&o. x) %. x ^/i.5
_3.46945e_17 0.997542 9.99201e_16 _0.156378 _7.77156e_16
c&p. _1 0 1
_0.841164 _3.46945e_17 0.841164
1&o. _1 0 1
_0.841471 0 0.841471
(c&p. x) ,. (sin x)
_0.841164 _0.841471
_0.717968 _0.717356
_0.564748 _0.564642
_0.389009 _0.389418
_0.198257 _0.198669
_3.46945e_17 0
0.198257 0.198669
0.389009 0.389418
0.564748 0.564642
0.717968 0.717356
0.841164 0.841471
The polynomial approximations may, however, be wildly wrong
for arguments outside of the list x to which it
was fitted.
der of Section 6B applied to a
list of polynomial coefficients yields the coefficients of
the derivative polynomial. A coresponding function for the
anti-derivative may be defined as follows: der=: 1: |. ] * i. on #
ant=: 0:,] % next i. on #
ant c
0 1 1 1 1 1 1
der ant c
1 2 3 4 5 6 0
19A. Imaginary numbers19A. Imaginary numbers
S19A.
In Chapter 1 we began with the counting numbers (1 2 3 4 etc.)
and found that the introduction of subtraction as the inverse
of addition led to a new class of negative numbers (when
we attempted to subtract a larger number from a smaller).
set=: 9!:11
set 4
sqr=: *:
sqrt=: sqr^:_1
a=: i.10
a
0 1 2 3 4 5 6 7 8 9
sqr a
0 1 4 9 16 25 36 49 64 81
sqrt a
0 1 1.414 1.732 2 2.236 2.449 2.646 2.828 3
b=: sqrt -a
b
0 0j1 0j1.414 0j1.732 0j2 0j2.236 0j2.449 0j2.646 0j2.828 0j3
sqr b
0 _1 _2 _3 _4 _5 _6 _7 _8 _9
19B. Complex numbers
S19B.
The sum of a real and an imaginary number is called a complex
number. For example:
a+b
0 1j1 2j1.414 3j1.732 4j2 5j2.236 6j2.449 7j2.646 8j2.828 9j3
sqr (a+b)
0 0j2 2j5.657 6j10.39 12j16 20j22.36 30j29.39 42j37.04 56j45.25 72j54
a + table b
+-+-------------------------------------------------------------+
| |0 0j1 0j1.414 0j1.732 0j2 0j2.236 0j2.449 0j2.646 0j2.828 0j3|
+-+-------------------------------------------------------------+
|0|0 0j1 0j1.414 0j1.732 0j2 0j2.236 0j2.449 0j2.646 0j2.828 0j3|
|1|1 1j1 1j1.414 1j1.732 1j2 1j2.236 1j2.449 1j2.646 1j2.828 1j3|
|2|2 2j1 2j1.414 2j1.732 2j2 2j2.236 2j2.449 2j2.646 2j2.828 2j3|
|3|3 3j1 3j1.414 3j1.732 3j2 3j2.236 3j2.449 3j2.646 3j2.828 3j3|
|4|4 4j1 4j1.414 4j1.732 4j2 4j2.236 4j2.449 4j2.646 4j2.828 4j3|
|5|5 5j1 5j1.414 5j1.732 5j2 5j2.236 5j2.449 5j2.646 5j2.828 5j3|
|6|6 6j1 6j1.414 6j1.732 6j2 6j2.236 6j2.449 6j2.646 6j2.828 6j3|
|7|7 7j1 7j1.414 7j1.732 7j2 7j2.236 7j2.449 7j2.646 7j2.828 7j3|
|8|8 8j1 8j1.414 8j1.732 8j2 8j2.236 8j2.449 8j2.646 8j2.828 8j3|
|9|9 9j1 9j1.414 9j1.732 9j2 9j2.236 9j2.449 9j2.646 9j2.828 9j3|
+-+-------------------------------------------------------------+
Since a complex number is a sum of a real and an imaginary
number, the product of complex numbers is treated consistently
as a product of sums. The steps in the multiplication may
be illustrated as follows:
c=: 2j3
d=: 4j5
c*d
_7j22
(2+0j3)*(4+0j5)
_7j22
(2*(4+0j5))+(0j3*(4+0j5))
_7j22
(2*4)+(2*0j5)+(0j3*4)+(0j3*0j5)
_7j22
8+0j10+0j12+_15
_7j22
_7+0j22
_7j22
Note that the product of two imaginaries (0j3
and 0j5) yields a negative real number.
19C. Division
S19C.
We begin by dividing the product e=: c*d by
c and then by d to get the
expected results d and c:
]e=: c*d
_7j22
e%c
4j5
e%d
2j3
To get a general procedure for division, we first note that
division by a real number is straightforward. For example:
12j22 % 2
6j11
Similarly, we note that multiplication of both numerator
and dednominator by the same number leaves the quotient
unchanged. For example:
e%c
4j5
g=: 2j_3
(e*g)%(c*g)
4j5
Finally, we note that the denominator c*g is
a real number, a result of choosing g as the
conjugate of c, that is, obtained
from c by reversing the sign of its
imaginary part: c*g
13
The function conj=: + yields the conjugate
of its argument, and it can therefore be used to provide
division by first producing an equivalent quotient with a
real denominator. For example:
conj=: +
p=: 2j5
q=: 3j_4
p%q
_0.56j0.92
conj q
3j4
p*conj q
_14j23
q*conj q
25
(p*conj q)%(q*conj q)
_0.56j0.92
NB. Compare with p%q
19D. The Exponential Family
S19D.
In Chapter 14 we showed that the hyperbolic sine and the
hyperbolic cosine belonged to the exponential family in the
sense that they could be obtained from the exponential
as its odd and even parts. Using imaginary numbers, we
can treat the sine and cosine similarly.
j. multiplies its argument
by 0j1 (the square root of negative one). For
example: j.i.10
0 0j1 0j2 0j3 0j4 0j5 0j6 0j7 0j8 0j9
Consequently, each term in the polynomial c with p. on j.
differs from that in the polynomial c with p.
by a power of 0j1 determined by its exponent.
For example: 0j1^0 1 2 3 4 5 6 7 8
1 0j1 _1 0j_1 1 0j1 _1 0j_1 1
c=: 1 2 3 4 5 6 7
f=: c with p.
f t. i.10 NB. Taylor coefficients
1 2 3 4 5 6 7 0 0 0
f on j. t. i.10
1 0j2 _3 0j_4 5 0j6 _7 0 0 0
Consequently, the Taylor coefficients of the function ^ with j.
are the same as those of the sum cos+j. sin, and
the cosine and sine can be obtained from the even and odd parts
of ^ with j.
O=: .: -
E=: .. -
and experiment with their use on the function
f=: 0 1 2 3 4 5 with p. using expressions
such as f E and f E i:4
and (f O+f E) i:4^ and sin=: 1 with o.
and cos=: 2 with o. and cosh=: 6 with o. .
20A. Introduction20A. Introduction
S20A.
In stating the purpose of this book in the preface we said:
In the seventy-odd years since his publication, the
development of computer programming
has provided languages with grammars that are simpler and
more tractable than that of
conventional mathematical notation. Moreover, the
general availability of the computer
makes possible convenient and accurate experimentation
with mathematical ideas....
The unstated premise of this book - a premise that
virtually all mathematicians would agree to - is that
mathematics, like music, is worth doing for its
own sake. The author is, by profession, a medical man, but
he has a love for mathemtics and wants others to share
his enthusiasm.
The course title "Concrete Mathematics"
was originally intended as an antidote to "Abstract
Mathematics", since concrete classical results were
rapidly being swept out of the modern mathematical
curriculum by a new wave of abstract ideas popularly
called "The New Math." Abstract mathematics is a
beautiful subject ... But its adherents had become
deluded that the rest of mathematics was inferior
and no longer worthy of attention.
20B. Mathematics: from the Birth of Numbers
S20B.
We will comment on four topics covered by Gullberg,
primarily to show the form of their expression in J, and
the possibility for computer experimentation:
All real numbers can be written as periodic
or nonperiodic, terminating or nonterminating, decimal
fractions. Another way of writing a rational number
is in the form of a continued fraction,
This is equivalent to the function pr=: [+1:%]
applied over the list a0,a1,a2,a3, etc.,
and Gullberg's example of 221/41 may be treated
as follows: pr=: [+1:%]
2 pr 5
2.2
pr/5 2 1 1 3 2r1
221r41
pr/5 2 1 1 3 2
5.39024
pr/\5 2 1 1 3 2r1
5 11r2 16r3 27r5 97r18 221r41
The final expression illustrates how the operator \
may be used to obtain the partial sums or "convergents" to
the complete continued fraction.
pr/\1 1 1 1 1 1 1 1 1 1r1
1 2 3r2 5r3 8r5 13r8 21r13 34r21 55r34 89r55
pr/\1 1 1 1 1 1 1 1 1 1
1 2 1.5 1.66667 1.6 1.625 1.61538 1.61905 1.61765 1.61818
These decimal results may be recognized as increasingly
good approximations to the Golden Mean (or golden number),
also treated by Gullberg.
a=: 2 2 4 2 4 2 4 2 4
b=: pr/a
b
2.44949
b*b
6
DETERMINANTS
On pages 646,7, Gullberg states:A determinant is a value
representing sums and products of a square matrix.
The determinant of a matrix A, det A, is denoted as
arrays of numbers or algebraic quantities, called
entries or elements, disposed in horizontal rows and
vertical columns enclosed between single vertical bars:
| a11 a12 ... a1n |
det A = | ... ... ... ... |
| an1 an2 ... ann |
...
has 3! = 6 triple products of entries from every row
and column, of which no two triplets can be the same,
a11 a22 a33 a11 a23 a32 a12 a21 a33 a12 a23 a31 a13 a21 a32 a13 a22 a31
1-2-3 1-3-2 2-1-3 2-3-1 3-1-2 3-2-1
# of inversions:
none one one two two three
Correction factor:
+1 -1 -1 +1 +1 -1
In J the determinant function is defined as det=: -/ . *
which, applied to the three-by-three matrix (that is, table) of
Gullberg's example yields the following
results: det=: -/ . *
A=: 2 3 4,.0 1 2,.4 0 2
A
2 0 4
3 1 0
4 2 2
det A
12
Gullberg notes the following properties:
The value of a determinant is reversed in sign if any two
rows are interchanged.
For example: (1 A. A);(det 1 A. A);(det A)
+-----+---+--+
|2 0 4| | |
|4 2 2|_12|12|
|3 1 0| | |
+-----+---+--+
(1 3 1 * A);(det 1 3 1 * A);(3 * det A)
+-----+--+--+
|2 0 4| | |
|9 3 0|36|36|
|4 2 2| | |
+-----+--+--+
(2 3 4 * A);(det 2 3 4 * A);((*/2 3 4)*det A)
+------+---+---+
| 4 0 8| | |
| 9 3 0|288|288|
|16 8 8| | |
+------+---+---+
triangle=: 3 2$3 _2 , _4 1 , 8 3
bt=: triangle,.1
triangle;bt;(det bt);(1 A. bt);(det 1 A. bt)
+-----+-------+---+-------+--+
| 3 _2| 3 _2 1| | 3 _2 1| |
|_4 1|_4 1 1|_50| 8 3 1|50|
| 8 3| 8 3 1| |_4 1 1| |
+-----+-------+---+-------+--+
The last result agrees with Gullberg's statement that
the absolute value of the determinant yields twice
the area of the triangle. But, more generally, the determinant
yields a signed area that indicates whether the
vertices of the triangle (when plotted) occur in
clockwise or counter-clockwise order.
!3)
its signed volume. For example: ]tet=: 4 3$0 0 0 , 1 0 0 , 0 1 0 , 0 0 1
0 0 0
1 0 0
0 1 0
0 0 1
det tet,.1
_1
]tet2=: ?. 4 3$10
1 7 4
5 2 0
6 6 9
3 5 8
det tet2,.1
_106
LOGARITHMS
On page 154 Gullberg says:
that is, a logarithm is an exponent
which defines to what power the base a
must be raised in order to give x, called the
anti-logarithm.
In J the function a with ^. gives
the base a logarithm, and the function ^.
(or 1x1 with ^.) gives the natural log.
Hence: ln=: ^.
log=: 10 with ^.
lb=: 2 with ^.
x=: 1 2 4 10 100
ln x
0 0.693147 1.38629 2.30259 4.60517
log x
0 0.30103 0.60206 1 2
lb x
0 1 2 3.32193 6.64386
ln -1 2 4 10
0j3.14159 0.693147j3.14159 1.38629j3.14159 2.30259j3.14159
^ ln -x
_1 _2 _4 _10 _100
log -x
0j1.36438 0.30103j1.36438 0.60206j1.36438 1j1.36438 2j1.36438
10 ^ log -x
_1 _2 _4 _10 _100
THE PEANO AXIOMS
On page 157 Gullberg says:The natural numbers were
axiomatically defined in 1899 by Giuseppe Peano in
his ... Later, Peano modified his axioms to include
also zero:
The treatment in our Chapter 1 is based on Peano's original
formulation (which did not include zero as a natural number), and
denotes Peano's successor function by >: (and its
inverse, the predecessor, by <:).
20C. Concrete Mathematics
S20C.
The text Concrete Mathematics [2] will be referred to as GKP, and
will be discussed together with commentary on it (expressed in J)
from Concrete Math Companion [21], to be referred to as CMC.
Our generic generating function
has the form
In Section 7B, CMC discusses this as follows:
In GKP7.12 the limit (as
SUBSCRIPTS VERSUS FUNCTIONS In Section
2.1 GKP states that:n approaches infinity) of the
polynomial G=. (g i.n) p. ] is defined to be the
generating function for the function g.
In other words, g k is the kth element
of the Taylor series for g. gex=. ^ t.
gsin=. 1&o. t.
gsinex=. (1&o. * ^) t.
]ce=. gex i. 8
1 1 0.5 0.166667 0.0416667 0.00833333 0.00138889 0.000198413
]cs=. gsin i.8
0 1 0 _0.166667 0 0.00833333 0 _0.000198413
]cse=. gsinex i. 8
0 1 1 0.333333 0 _0.0333333 _0.0111111 _0.0015873
We'll be working with sums of the general form
In Section 2A, CMC discusses this as follows:
We will therefore treat
STIRLING NUMBERS In Section 6.1, GKP defines Stirling
Numbers of the first and second kind
by:a
as a function, and the list of indices as a second
function, typically i. or the
function Ei=. i.@>: . For example Ei=. i.@>:
Ei 5
0 1 2 3 4 5
a=. *:
a Ei 5
0 1 4 9 16 25
+/ a Ei 5
55
...the number of ways to arrange n objects into k
cycles
and provides tables of them in Table 245 and Table 244.
^!._1 and uses it to define the
Stirling numbers as matrices that provide
the coefficients for linear transformations between the
coefficients of certain important polynomials.
s1=: |: on ((i. ^!._1/ i.) %. (i. ^/ i.))
s2=: %. on s1
(s1;s2) 8r1
+--------------------------------+-----------------------+
|1 0 0 0 0 0 0 0|1 0 0 0 0 0 0 0|
|0 1 0 0 0 0 0 0|0 1 0 0 0 0 0 0|
|0 _1 1 0 0 0 0 0|0 1 1 0 0 0 0 0|
|0 2 _3 1 0 0 0 0|0 1 3 1 0 0 0 0|
|0 _6 11 _6 1 0 0 0|0 1 7 6 1 0 0 0|
|0 24 _50 35 _10 1 0 0|0 1 15 25 10 1 0 0|
|0 _120 274 _225 85 _15 1 0|0 1 31 90 65 15 1 0|
|0 720 _1764 1624 _735 175 _21 1|0 1 63 301 350 140 21 1|
+--------------------------------+-----------------------+
Functions to provide Tables 245 and 244 of GkP are defined
in terms of these as follows S1=: | on s1
S2=: s2
(S1;S2) 8r1
+----------------------------+-----------------------+
|1 0 0 0 0 0 0 0|1 0 0 0 0 0 0 0|
|0 1 0 0 0 0 0 0|0 1 0 0 0 0 0 0|
|0 1 1 0 0 0 0 0|0 1 1 0 0 0 0 0|
|0 2 3 1 0 0 0 0|0 1 3 1 0 0 0 0|
|0 6 11 6 1 0 0 0|0 1 7 6 1 0 0 0|
|0 24 50 35 10 1 0 0|0 1 15 25 10 1 0 0|
|0 120 274 225 85 15 1 0|0 1 31 90 65 15 1 0|
|0 720 1764 1624 735 175 21 1|0 1 63 301 350 140 21 1|
+----------------------------+-----------------------+
20D. Computer Resources
S20D.
LABORATORIES Clicking the mouse on the menu item marked "Studio"
shows a menu that allows the selection of either
"Labs" or "Demos". Clicking on "Labs" provides an
opportunity to choose a variety of topics.,1, produces a matrix with an interesting
pattern. Moreover, it provides a function viewmat
that plots a view of the matrix as a grid of black and white
elements. Thus: f=: ,~ ,.~
f ,1
1 1
1 0
f f ,1
1 1 1 1
1 0 1 0
1 1 0 0
1 0 0 0
f f f ,1
1 1 1 1 1 1 1 1
1 0 1 0 1 0 1 0
1 1 0 0 1 1 0 0
1 0 0 0 1 0 0 0
1 1 1 1 0 0 0 0
1 0 1 0 0 0 0 0
1 1 0 0 0 0 0 0
1 0 0 0 0 0 0 0
load 'graph numeric trig'
viewmat f f f f f f f f ,1
o. will
show that the definition of the function 0 with o. (used
to graph circles in Sections 3B and 18B) is the square root of
one minus the square.20E. Conclusion
S20E.
Study of this book will not make a mathematician, but we hope
that the use of precise executable language possessing a
simple strict grammar will further the cause stated by Hogben in our preface
(and repeated here):
Perhaps the most important lesson for the layman is that
there is no need to be intimidated by mathematics or
mathematicians. On this matter, we give the final
word to Hogben:
There is a story about Diderot, the Encyclopaedist
and materialist, a foremost figure in the intellectual awakening
which immediately preceded the French Revolution. Diderot
was staying at the Russian court, where his elegant flippancy
was entertaining the nobility. Diderot was informed that a
mathematician had established a proof of the existence of God.
a+bn
------- = x, donc Dieu existe, repondez!
n
Algebra was Arabic to Diderot. Unfortunately, he did not
realize that was the trouble. Had he realized that algebra
was just a language in which we describe the sizes of
things, ... he would have asked Euler to
translate ... into French.S1A. The Counting (Natural) Numbers
This definition of the counting numbers in terms of a successor
function was made in 1899 by Giuseppe Peano. For further details,
see the discussion in Section 20B, and the Gullberg text to which
it refers.
S1B. Lists and Names
The function (or verb) denoted by ,. can be used to stitch
lists together to form tables, and lists and tables together to form
further tables, as illustrated by the following:
a=:1 4 7
b=:2 5 8
c=:3 6 9
a,.b
1 2
4 5
7 8
a,.b,.c
1 2 3
4 5 6
7 8 9
table=:a,.b,.c
table,.>:c
1 2 3 4
4 5 6 7
7 8 9 10
>:table
2 3 4
5 6 7
8 9 10
table,.>:>:>:table
1 2 3 4 5 6
4 5 6 7 8 9
7 8 9 10 11 12
The function denoted by , can be used to catenate lists
and tables as illustrated by the following:
a,b
1 4 7 2 5 8
a,b,c
1 4 7 2 5 8 3 6 9
table,a
1 2 3
4 5 6
7 8 9
1 4 7
table,>:>:>:>:>:>:>:>:>:table
1 2 3
4 5 6
7 8 9
10 11 12
13 14 15
16 17 18
The function denoted by |: can be used to transpose a
table as follows:
|:table
1 4 7
2 5 8
3 6 9
|:table,>:c
1 4 7 4
2 5 8 7
3 6 9 10
Write various expressions employing the functions used in the preceding
examples, and show the results produced.
S1C. Iteration (Repetition)
Write the result of the
expression next for list list , and compare your
result with that produced by entering it on the computer.
S1D. Inverse
While we had only the counting numbers (which begin at 1), our
lists could contain neither negative numbers nor zero; but now that we
have extended our domain to the integers this restriction is removed. We
may therefore introduce the functions i. and i: that produce
lists of integers as illustrated below:
i.5 NB. The first five non-negative integers
0 1 2 3 4
i:5 NB. The integers from _5 to 5
_5 _4 _3 _2 _1 0 1 2 3 4 5
In the foregoing we used NB. which indicates that what follows
it is a note or comment that has no effect on the meaning of the preceding sentence.
+: doubles its argument.
For example:
x=:i:5
x
_5 _4 _3 _2 _1 0 1 2 3 4 5
+:x
_10 _8 _6 _4 _2 0 2 4 6 8 10
double=:+:
double x
_10 _8 _6 _4 _2 0 2 4 6 8 10
y=:double for 2 x
y
_20 _16 _12 _8 _4 0 4 8 12 16 20
double for _1 y NB. Inverse of double
_10 _8 _6 _4 _2 0 2 4 6 8 10
double for _1 x
_2.5 _2 _1.5 _1 _0.5 0 0.5 1 1.5 2 2.5
Some of the results of the last expression are not integers. Just
as the inverse of the successor function led us out of the domain of
the counting numbers, so the inverse of double has led out
of the domain of integers -- this time into fractions or
rationals. These matters are treated further in Section 1J.
S1E. Addition and Subtraction (+ and -)
Study the following function tables, and try to state the definitions of
each of the functions denoted by = > <. <: | and !
a=:i.6
a
0 1 2 3 4 5
=table a
+-+-----------+
| |0 1 2 3 4 5|
+-+-----------+
|0|1 0 0 0 0 0|
|1|0 1 0 0 0 0|
|2|0 0 1 0 0 0|
|3|0 0 0 1 0 0|
|4|0 0 0 0 1 0|
|5|0 0 0 0 0 1|
+-+-----------+
(=table a),.(>table a),.(<.table a)
+-+-----------+-+-----------+-+-----------+
| |0 1 2 3 4 5| |0 1 2 3 4 5| |0 1 2 3 4 5|
+-+-----------+-+-----------+-+-----------+
|0|1 0 0 0 0 0|0|0 0 0 0 0 0|0|0 0 0 0 0 0|
|1|0 1 0 0 0 0|1|1 0 0 0 0 0|1|0 1 1 1 1 1|
|2|0 0 1 0 0 0|2|1 1 0 0 0 0|2|0 1 2 2 2 2|
|3|0 0 0 1 0 0|3|1 1 1 0 0 0|3|0 1 2 3 3 3|
|4|0 0 0 0 1 0|4|1 1 1 1 0 0|4|0 1 2 3 4 4|
|5|0 0 0 0 0 1|5|1 1 1 1 1 0|5|0 1 2 3 4 5|
+-+-----------+-+-----------+-+-----------+
(<:table a),.(|table a),.(!table a)
+-+-----------+-+-----------+-+------------+
| |0 1 2 3 4 5| |0 1 2 3 4 5| |0 1 2 3 4 5|
+-+-----------+-+-----------+-+------------+
|0|1 1 1 1 1 1|0|0 1 2 3 4 5|0|1 1 1 1 1 1|
|1|0 1 1 1 1 1|1|0 0 0 0 0 0|1|0 1 2 3 4 5|
|2|0 0 1 1 1 1|2|0 1 0 1 0 1|2|0 0 1 3 6 10|
|3|0 0 0 1 1 1|3|0 1 2 0 1 2|3|0 0 0 1 4 10|
|4|0 0 0 0 1 1|4|0 1 2 3 0 1|4|0 0 0 0 1 5|
|5|0 0 0 0 0 1|5|0 1 2 3 4 0|5|0 0 0 0 0 1|
+-+-----------+-+-----------+-+------------+
= is a truth function that compares its arguments for
equality, giving 1 if the relation is true, and 0
if it is false.> is a truth function called greater than.<. is called lesser-of or minimum.
<: is a truth function called less than or equal.
| is remainder on dividing by the left argument.
! is the binomial coefficient (m!n
is the number of different ways of choosing m things from n things).
This table is commonly shown without the zeros, and called Pascal's Triangle.
S1F. Bonding
Apply the bond (with=:&) to some of the functions
= < >. <: | introduced in Section S1E, and show
their results. Compare your results with the following:
with=: &
a=:i:3
a
_3 _2 _1 0 1 2 3
= with 2 a
0 0 0 0 0 1 0
< with 2 a
1 1 1 1 1 0 0
<: with 2 a
1 1 1 1 1 1 0
<. with 2 a
_3 _2 _1 0 1 2 2
2 with | a
1 0 1 0 1 0 1
3 with | a
0 1 2 0 1 2 0
S1G. Multiplication or Times (*)
Comment on the pattern of positive and negative numbers in the four
quadrants defined by the row and column of zeros in the
following multiplication table:
* table i:4
+--+-----------------------------+
| | _4 _3 _2 _1 0 1 2 3 4|
+--+-----------------------------+
|_4| 16 12 8 4 0 _4 _8 _12 _16|
|_3| 12 9 6 3 0 _3 _6 _9 _12|
|_2| 8 6 4 2 0 _2 _4 _6 _8|
|_1| 4 3 2 1 0 _1 _2 _3 _4|
| 0| 0 0 0 0 0 0 0 0 0|
| 1| _4 _3 _2 _1 0 1 2 3 4|
| 2| _8 _6 _4 _2 0 2 4 6 8|
| 3|_12 _9 _6 _3 0 3 6 9 12|
| 4|_16 _12 _8 _4 0 4 8 12 16|
+--+-----------------------------+
The idea of multiplication can be introduced at an elementary level
by pictures of rectangles. For example:
a=:3 4 $ 1
a
1 1 1 1
1 1 1 1
1 1 1 1
,a
1 1 1 1 1 1 1 1 1 1 1 1
+/,a
12
+/a
3 3 3 3
+/+/a
12
b=:|:a
b
1 1 1
1 1 1
1 1 1
1 1 1
,b
1 1 1 1 1 1 1 1 1 1 1 1
+/,b
12
+/b
4 4 4
+/+/b
12
Discussion of the notation used here (such as $)
may be found in Sections 2E and 2F.
1 used above, but uses dots:
c=: a { ' .'
c
....
....
....
|:c
...
...
...
...
c='.'
1 1 1 1
1 1 1 1
1 1 1 1
+/+/c='.'
12
Moreover, he draws a tied bag (rather similar to &) to denote
an unknown quantity, and translates to algebraic expressions
that denote unknowns by letters such as x and y.
For example: g=:'&&...'
g
&&...
translates to 2x+3.
g,g
&&...&&...
which is denoted by (2x+3)+(2x+3) and
is (pictorially) equivalent to that shown above, or to
&&&&...... which, in turn, translates
to 4x+6.
S1H. Power and Exponent
See Section S1J for an extension of the power table to negative arguments.
S1I. Monomials and Polynomials (p.)
Write various expressions using the polynomial function p. and
show the results they produce. Include cases with negative coefficients and
negative arguments, and compare your results with the following:
b=:i:4
b
_4 _3 _2 _1 0 1 2 3 4
1 3 3 1 p. b
_27 _8 _1 0 1 8 27 64 125
_1 3 _3 1 p. b
_125 _64 _27 _8 _1 0 1 8 27
(b-1)^3
_125 _64 _27 _8 _1 0 1 8 27
1 _4 6 _4 1 p. b
625 256 81 16 1 0 1 16 81
(b-1)^4
625 256 81 16 1 0 1 16 81
S1J. Division (%)
Comment on the pattern in the rows and columns of the division table. Then
make a division table for both positive and negative arguments, and compare
your results with the following:
c=:i:5r1
c
_5 _4 _3 _2 _1 0 1 2 3 4 5
%table c
+--+------------------------------------------------+
| | _5 _4 _3 _2 _1 0 1 2 3 4 5|
+--+------------------------------------------------+
|_5| 1 5r4 5r3 5r2 5 __ _5 _5r2 _5r3 _5r4 _1|
|_4| 4r5 1 4r3 2 4 __ _4 _2 _4r3 _1 _4r5|
|_3| 3r5 3r4 1 3r2 3 __ _3 _3r2 _1 _3r4 _3r5|
|_2| 2r5 1r2 2r3 1 2 __ _2 _1 _2r3 _1r2 _2r5|
|_1| 1r5 1r4 1r3 1r2 1 __ _1 _1r2 _1r3 _1r4 _1r5|
| 0| 0 0 0 0 0 0 0 0 0 0 0|
| 1|_1r5 _1r4 _1r3 _1r2 _1 _ 1 1r2 1r3 1r4 1r5|
| 2|_2r5 _1r2 _2r3 _1 _2 _ 2 1 2r3 1r2 2r5|
| 3|_3r5 _3r4 _1 _3r2 _3 _ 3 3r2 1 3r4 3r5|
| 4|_4r5 _1 _4r3 _2 _4 _ 4 2 4r3 1 4r5|
| 5| _1 _5r4 _5r3 _5r2 _5 _ 5 5r2 5r3 5r4 1|
+--+------------------------------------------------+
Comment on the patterns in the following extension of the power table
to negative arguments. In particular, note the value of 0^0 (which
may have surprised you in the power table in Section 1H).
1 for
the value of the "indeterminate" expression 0^0, see
page 785 of Gullberg [20].
^table c
+--+-----------------------------------------------------+
| | _5 _4 _3 _2 _1 0 1 2 3 4 5|
+--+-----------------------------------------------------+
|_5|_1r3125 1r625 _1r125 1r25 _1r5 1 _5 25 _125 625 _3125|
|_4|_1r1024 1r256 _1r64 1r16 _1r4 1 _4 16 _64 256 _1024|
|_3| _1r243 1r81 _1r27 1r9 _1r3 1 _3 9 _27 81 _243|
|_2| _1r32 1r16 _1r8 1r4 _1r2 1 _2 4 _8 16 _32|
|_1| _1 1 _1 1 _1 1 _1 1 _1 1 _1|
| 0| _ _ _ _ _ 1 0 0 0 0 0|
| 1| 1 1 1 1 1 1 1 1 1 1 1|
| 2| 1r32 1r16 1r8 1r4 1r2 1 2 4 8 16 32|
| 3| 1r243 1r81 1r27 1r9 1r3 1 3 9 27 81 243|
| 4| 1r1024 1r256 1r64 1r16 1r4 1 4 16 64 256 1024|
| 5| 1r3125 1r625 1r125 1r25 1r5 1 5 25 125 625 3125|
+--+-----------------------------------------------------+
S1K. Review and Supplement
Experiment with moving around in the text as follows:
S1L. Notes
S2A. Introduction
To gain further familiarity with the computer, enter some of the expressions from the
supplemental sections for Chapter 1.
S2B. Number of places
The !: used in the expression set=:9!:11 to
define the function for setting the number of digits displayed
is called the foreign conjunction, because it concerns matters other than
the normal verbs such as + * and %.
set. Also click on the button marked >> to
see further pages.
S2C. Displays
Sawyer [22] emphasizes
the use of trees and tables for insight, and introduces an
interesting non-numeric table as follows:
In the Japanese Hiragana alphabet, each character
represents a sound such as ka, ru, si, or o. There are 10
possible beginnings - the consonants k, s, t, n, m, y, r, w, and
'blank'; ... The ending is always one one of the five sounds
a, e, i, o, u. How many characters are there in this alphabet?
We could show this as a tree with 10 branches and 5
twigs on each ... This problem could equally well be solved by
the rectangle ...
Beginnings=:' KSTNHMYRW'
Endings=:'aeiou'
{ Beginnings;Endings
+--+--+--+--+--+
| a| e| i| o| u|
+--+--+--+--+--+
|Ka|Ke|Ki|Ko|Ku|
+--+--+--+--+--+
|Sa|Se|Si|So|Su|
+--+--+--+--+--+
|Ta|Te|Ti|To|Tu|
+--+--+--+--+--+
|Na|Ne|Ni|No|Nu|
+--+--+--+--+--+
|Ha|He|Hi|Ho|Hu|
+--+--+--+--+--+
|Ma|Me|Mi|Mo|Mu|
+--+--+--+--+--+
|Ya|Ye|Yi|Yo|Yu|
+--+--+--+--+--+
|Ra|Re|Ri|Ro|Ru|
+--+--+--+--+--+
|Wa|We|Wi|Wo|Wu|
+--+--+--+--+--+
The catalogue function ({) used above may
be further illustrated as follows:
{ 0 1 2;0 1 2;0 1 2
+-----+-----+-----+
|0 0 0|0 0 1|0 0 2|
+-----+-----+-----+
|0 1 0|0 1 1|0 1 2|
+-----+-----+-----+
|0 2 0|0 2 1|0 2 2|
+-----+-----+-----+
+-----+-----+-----+
|1 0 0|1 0 1|1 0 2|
+-----+-----+-----+
|1 1 0|1 1 1|1 1 2|
+-----+-----+-----+
|1 2 0|1 2 1|1 2 2|
+-----+-----+-----+
+-----+-----+-----+
|2 0 0|2 0 1|2 0 2|
+-----+-----+-----+
|2 1 0|2 1 1|2 1 2|
+-----+-----+-----+
|2 2 0|2 2 1|2 2 2|
+-----+-----+-----+
{ 'cht';'oa';'gtw'
+---+---+---+
|cog|cot|cow|
+---+---+---+
|cag|cat|caw|
+---+---+---+
+---+---+---+
|hog|hot|how|
+---+---+---+
|hag|hat|haw|
+---+---+---+
+---+---+---+
|tog|tot|tow|
+---+---+---+
|tag|tat|taw|
+---+---+---+
S2D. Integer Lists
Explain the difference between the following tables:
%table i:4
+--+--------------------------------------------------+
|_4| 1 1.33333 2 4 __ _4 _2 _1.33333 _1|
|_3| 0.75 1 1.5 3 __ _3 _1.5 _1 _0.75|
|_2| 0.5 0.666667 1 2 __ _2 _1 _0.666667 _0.5|
|_1| 0.25 0.333333 0.5 1 __ _1 _0.5 _0.333333 _0.25|
| 0| 0 0 0 0 0 0 0 0 0|
| 1|_0.25 _0.333333 _0.5 _1 _ 1 0.5 0.333333 0.25|
| 2| _0.5 _0.666667 _1 _2 _ 2 1 0.666667 0.5|
| 3|_0.75 _1 _1.5 _3 _ 3 1.5 1 0.75|
| 4| _1 _1.33333 _2 _4 _ 4 2 1.33333 1|
+--+--------------------------------------------------+
%table i:4r1
+--+--------------------------------------+
| | _4 _3 _2 _1 0 1 2 3 4|
+--+--------------------------------------+
|_4| 1 4r3 2 4 __ _4 _2 _4r3 _1|
|_3| 3r4 1 3r2 3 __ _3 _3r2 _1 _3r4|
|_2| 1r2 2r3 1 2 __ _2 _1 _2r3 _1r2|
|_1| 1r4 1r3 1r2 1 __ _1 _1r2 _1r3 _1r4|
| 0| 0 0 0 0 0 0 0 0 0|
| 1|_1r4 _1r3 _1r2 _1 _ 1 1r2 1r3 1r4|
| 2|_1r2 _2r3 _1 _2 _ 2 1 2r3 1r2|
| 3|_3r4 _1 _3r2 _3 _ 3 3r2 1 3r4|
| 4| _1 _4r3 _2 _4 _ 4 2 4r3 1|
+--+--------------------------------------+
S2E. Vocabulary
The Vocabulary provides two definitions for the symbol -;
one (called Negate) for use with one argument,
and the other (called Minus) for use with two arguments. For example:
- i.7 NB. Monadic use
0 _1 _2 _3 _4 _5 _6
3 - i.7 NB. Dyadic use
3 2 1 0 _1 _2 _3
Other functions also have both monadic and dyadic cases. For example, a * b
denotes the product of a and b, while *b denotes
the sign of b.
S2F. Functions over lists
Just as the over adverb / modifies a verb to which it applies,
the prefix adverb \ modifies its argument verb, applying
it to each prefix of the eventual argument. For example:
a=:1 2 3 4 5
+/a
15
+/\a
1 3 6 10 15
sum=:+/
sum\a
1 3 6 10 15
(sum 1),(sum 1 2),(sum 1 2 3),(sum 1 2 3 4),(sum 1 2 3 4 5)
1 3 6 10 15
b=:1 2 3 4 5 6r1
*/\b
1 2 6 24 120 720
!b
1 2 6 24 120 720
%/\b
1 1r2 3r2 3r8 15r8 5r16
Consulting the Vocabulary if necessary, state the results of
the following expressions, and check your results by executing them
on the computer:
c=:18 3 96 17 24 12
<./c
<./\c
>./\c
+./c
+./\c
c % +./\c
*./\c
(*./\c) % c
S2G. Notes
S3A. Plotting
The link function (denoted by ;) used in the expression
PLOT x;f x in Section 3A boxes its arguments and
catenates them to form a two-element list. Together with the laminate
function also introduced in Section 3A and the , and
,. introduced in Section S1B, they provide four ways
of forming lists and tables.
$ to examine the shapes of results.
S3B. Local behaviour and area
Use the Vocabulary to examine the definition of the circle
function o. used in Section 3 B. In particular,
experiment with the monadic case in expressions such as
o.1 and o.2 and o.1r2 .
sin=:1 with o.)
to determine the approximate area under one of its arches. Then define
cos=:2 with o. and use plot
to plot each of sin and cos against the argument,
and against one another.
+: (the double function) and plot
sin on +: against sin .
5 with o. against 6 with o. on
the argument 1r12*i:10 to draw a hyperbola.
S3C. Graphs versus function tables
0 0 1 with p. and 2 0 1 with p. and 1 4 6 4 1 with p. .
S3D. Notes
S4A. Bonding
c and d are the same length
(that is, have the same number of elements), then the polynomial (c+d) with p.
is equivalent to the sum c with p. + d with p. . If one is shorter
than the other, it must be extended by trailing zeros before adding it.c and d. with=:&
x=:i:4
x
_4 _3 _2 _1 0 1 2 3 4
0 0 1 with p. x
16 9 4 1 0 1 4 9 16
x^2
16 9 4 1 0 1 4 9 16
0 0 3 with p. x
48 27 12 3 0 3 12 27 48
3*x^2
48 27 12 3 0 3 12 27 48
e is the diagonal sum(s) of the multiplication table
c */ d, then the function e with p. is
equivalent to the product g=: c with p. * d with p. . Test
this assertion for various values of the coefficients c and
d.
1 1 with p. x yields the result
1+x, and (1 1 with p. * 1 1 with p.) x
yields (1+x)^2. Determine (by hand and by computer) the
coefficients of polynomials equivalent to further powers of 1+x.
Compare your results with the table of binomial coefficients !table i.8.
d obtained by multiplying a list of
coefficients c by its indices i.#c and then deleting
the leading element gives the polynomial d with p. that is the
derivative or rate of change of the function c with p. . Thus:
c=:1 2 1
c
1 2 1
i.#c
0 1 2
c*i.#c
0 2 2
d=:2 2
x=:i:3
x
_3 _2 _1 0 1 2 3
(c with p. ,: d with p.) x
4 1 0 1 4 9 16
_4 _2 0 2 4 6 8
Use the expression PLOT x;(c with p. ,: d with p.) x to plot
the function together with its derivative, and observe that the derivative does
represent the rate of change. Repeat for other polynomials.
d.1 (discussed in Chapter 6) as
illustrated below:
c with p. d.1 x
_4 _2 0 2 4 6 8
c with p. d.1 t. i.5
2 2 0 0 0
d.2 gives the second derivative, the
derivative of the derivative. Experiment with its use.
t. to determine the coessicients of sums,
differences, products, and composition of various polynomials.
d._1 yields the anti-derivative or
integral discussed in Chapter 18. Experiment with its use on various polynomials.
S4B. Notes
S5A. Introduction
! | _1: 3: ] used in the definition of the
power series function ps . Then experiment with each of the
following "components" of ps, and explain their behaviour:
a=:4: | ]
b=:3: = a
c=:_1: ^ b
d=:2 with |
e=: d * c
f=:% on !
g=:f * e
S5B. Truncated power series
The function pg=:% on ! is the power series function for the
growth or exponential function discussed in Chapter 7. Evaluate,
plot, and otherwise experiment with, its use.
g=:(pg 0 1 2 3 4 5 6) with p.
S5C. Notes
S6A. Approximation to the derivative (rate of change)
S6B. Derivatives of polynomials
DER=: 1: }. ] * i. on #
der=: 1: |. ] * i. on #
Note that their results are equivalent as polynomial coefficients, but
that the latter has the (sometimes convenient) property of yielding a result
with the same number of elements as the argument.
f reaches its lowest
point (and therefore changes from decreasing to increasng) is called a
point of inflection of f. Enter the following expressions, and
comment on the resulting graphs:
der=: 1: |. ] * i. on #
c=: 4 _3 _2 1
PLOT x2;(c with p.,(der c)with p.,:(der der c)with p.)x2
S6C. Taylor coefficients
Experiment with the approximations to the exponential, sine, and cosine functions
and their derivatives given in Section 6C, by applying them to the following argument:
x=: i: 1j5
x
_1 _0.6 _0.2 0.2 0.6 1
S6D. Notes
S7A. Growth polynomials
Execute the following expressions, and comment on the results:
x=:i:2
x
^x
^ - x
%^x
(^x)*(^-x)
(^*^ on -) x
S7B. The name Exponential
S8A. Introduction
Plot the sine and cosine functions together, using the argument y=:i:3j100.
S8B. Harmonics
For a discussion of Harmonic Analysis see Chapter 31 of Gullberg [20].
S8C. Decay
S9A. Inverse
Experiment with the use of the inverse adverb on other functions
in expressions such as:
INV=:^:_1
a=:i.5
a
0 1 2 3 4
^&3 a
0 1 8 27 64
^&3 INV a
0 1 1.25992 1.44225 1.5874
^&3 ^&3 INV a
0 1 2 3 4
^&1r3 a
0 1 1.25992 1.44225 1.5874
^ a
1 2.71828 7.38906 20.0855 54.5982
^INV ^ a
0 1 2 3 4
a+2
2 3 4 5 6
!a+2
2 6 24 120 720
! INV ! a+2
2 3 4 5 6
S9B. Equations
Repeat the first few steps of Section 9B to find approximations to the root of
the function g defined therein.
S9C. The method of false position
Define the function g as follows:
c=: 4 6 _8 2
g=: c with p.
Use the function tighten of Section 9C with the initial bracket
ab=: _1 1 to get approximations to one of the roots of g.
g to the results to confirm that they are indeed (near) roots. Compare
your roots with the result of p. c .
p. c shows each complex root with a j
separating its real and imaginary parts.
S9D. Newton's method
Apply Newton's method to obtain the roots of various polynomials.
S9E. Roots of Polynomials
Pages 87-88 of Gullberg [20] give a brief history of complex numbers.
Gullberg uses the conventional notation i for
the imaginary square root of negative one, and a+bi
for the complex number obtained by adding the real number
a to the product of the real number b with the
imaginary i.
j. which multiplies
its single argument by the square root of negative one, and
which forms a complex number from two arguments. For example:
j.4
0j4
(j.4)*(j.4)
_16
3+j.4
3j4
3 j. 4
3j4
3j4 * 5j12
_33j56
(3*5)+(3*0j12)+(0j4*5)+(0j4*0j12)
_33j56
The last result illustrates the fact that the multiplication
of complex numbers follows the normal rules for the product
of the sums of their components (3 0j4 5
and 0j12).
3j4 with
its conjugate 3j_4 yields a real number.
+ yields the conjugate
of its argument. For example:
+5j12
5j_12
5j12 * +5j12
169
%: 5j12 * +5j12
13
| 5j12
13
The foregoing illustrates that the square root of the product of
a number with its conjugate is its magnitude (given
by the function |). Verify that the magnitude of a real
number is itself, as is its conjugate.
S9F. Logarithms
S10A. Introduction
S10B. Word formation
S10C. Parsing
S10D. Conventional Mathematical Notation
S11A. Introduction
S11B. Informal proofs
Write an informal proof that +/i.n (the sum
of the first n non-negative integers) is equivalent
to n*(n-1)%2.
+/*:i.n)
and try to find an equivalent expression that is easier to
calculate. Then write an informal proof of the discovered
relation.
S11C. Formal proofs
Make formal proofs of the informal proofs of Section S11B.
S11D. Inductive proofs
Make inductive proofs for the relations found in Section S11B.
S11E. Recursive definition
Enter the following recursive definitions and apply the
resulting functions to integer arguments to learn what
you can about them:
f=:1:`(+//.@(,:~)@($:@<:))@.*
g=:1:`((],+/@(_2&{.))@$:@<:)@.*
See page 286 of Gullberg for a discussion of Fibonacci numbers.