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 fro3A. 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|)|
+---+-+-+-+-+-+-+-+-+-+-+-+