Books/MathForTheLayman/Further Topics
20: Further Topics
20A. Introduction
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....
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):
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.
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:
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.
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. 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:
CONTINUED FRACTIONS On page 127, Gullberg says:
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,
N = a0 + 1
-----------------------
a1 + 1
------------------
a2 + 1
-------------
a3 + 1
--------
a4 + ...
or, in a kind of accepted mathematical "shorthand",
N = [a0, a1, a2, a3 ...]
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.
The fibonacci numbers (treated by Gullberg on page 287) may also be obtained as continued fractions. For example:
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.
On page 143, Gullberg shows examples of continued fraction approximations to irrational numbers such as the square root of six:
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.
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.
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.
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 (!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.
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)
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:
1. Zero is a number.
2. Every natural number or zero, a, has an immediate successor a + 1.
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.
GENERATING FUNCTIONS and TAYLOR SERIES In Section 7.2 GKP states:
Our generic generating function has the form
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).
In Section 7B, CMC discusses this as follows:
In GKP7.12 the limit (as 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.
For example, the functions for the exponential, sine, and the product of the sine and the exponential may be generated as follows:
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
SUBSCRIPTS VERSUS FUNCTIONS In Section 2.1 GKP states that:
We'll be working with sums of the general form
a1 + a2 +...+ an
where each ak is a number that has been defined somehow.
In Section 2A, CMC discusses this as follows:
We will therefore treat 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
STIRLING NUMBERS In Section 6.1, GKP defines Stirling Numbers of the first and second kind by:
...the number of ways to arrange n objects into k cycles
...the number of ways to partition a set of n things into k nonempty subsets
and provides tables of them in Table 245 and Table 244.
In Section 5F, CMC introduces the falling factorial function ^!._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.
The required transformation matrices are given by:
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
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.
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 ,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
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 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.
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. 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):
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.
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.
.......
Before the assembled court, Euler accosted him with the following pronouncement, which was uttered with due gravity:
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.
.......
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.