Story Of Numbers, The

Asok Kumar Mallik

• Publisher: World Scientific Publishing Company
• Year: 2017
• ISBN:
  9789813222946
• Topic: Mathematics
• Subtopic: Games in Mathematics
• Index: Mathematics

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This book is more than a mathematics textbook. It discusses various kinds of numbers and curious interconnections between them. Without getting into hardcore and difficult mathematical technicalities, the book lucidly introduces all kinds of numbers that mathematicians have created. Interesting anecdotes involving great mathematicians and their marvelous creations are included. The reader will get a glimpse of the thought process behind the invention of new mathematics. Starting from natural numbers, the book discusses integers, real numbers, imaginary and complex numbers and some special numbers like quaternions, dual numbers and p -adic numbers.

Real numbers include rational, irrational and transcendental numbers. Iterations on real numbers are shown to throw up some unexpected behavior, which has given rise to the new science of "Chaos". Special numbers like e, pi, golden ratio, Euler's constant, Gauss's constant, amongst others, are discussed in great detail.

The origin of imaginary numbers and the use of complex numbers constitute the next topic. It is shown why modern mathematics cannot even be imagined without imaginary numbers. Iterations on complex numbers are shown to generate a new mathematical object called 'Fractal', which is ubiquitous in nature. Finally, some very special numbers, not mentioned in the usual textbooks, and their applications, are introduced at an elementary level.

The level of mathematics discussed in this book is easily accessible to young adults interested in mathematics, high school students, and adults having some interest in basic mathematics. The book concentrates more on the story than on rigorous mathematics.

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--> Contents:

• Introduction
• Integers
• Real Numbers
• Imaginary and Complex Numbers
• Special Numbers
• Appendix A:
  • Solution of Equation (1.2)
  • Brahmagupta's Equation and Its Solution
  • Solution to Equation (2.26)
• Appendix B:
  • Sum of Integral Powers of Natural Numbers
• Appendix C:
  • Origin of Curious Patterns

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--> Readership: Elementary and secondary education students as well as high school students and non-mathematicians. -->
Numbers;Integers;Real Numbers;Complex Numbers;Dual Numbers;Quaternions;p-adic Numbers Key Features:

• For the first time the readers are exposed to the joy of creating mathematics going beyond the problem-solving skills
• The story of inventing and using different numbers is told lucidly to generate interest in mathematics
• It combines serious and recreational mathematics; it also includes anecdotes for the students and general adult readers to enjoy mathematics as an entertaining human activity
• Lucid story form of presentation rather than a series of formula and techniques
• It avoids rigorous presentation and emphasizes an intuitive approach which is within the grasp of both elementary and secondary education students as well as high-school students and non-mathematician adults

Chapter 1

Introduction

Mathematics is a grand edifice of the human intellect constructed over a period of a few millennia, cutting across different civilisations. In natural and biological sciences, old theories make way for new ones, but in mathematics, new developments are added, sometimes connecting previous separate areas, without replacing them. Freeman Dyson in a lecture classified mathematicians into two groups, namely, birds and frogs. Birds take an overview of different areas and establish connections among them, whereas frogs continue to find new gems in one restricted area. The activities of both groups have enriched the subject. The ever increasing field of modern mathematics starts with counting. For counting we use positive integers 1, 2, 3, . . . , which are aptly called natural numbers. Everybody notices one Sun, two eyes, five fingers and so on in nature. Leopold Kronecker said, “God made the natural numbers, everything else is man’s handiwork” [9].

The sense of quantity of discrete objects (elements in a set) in terms of one, two, few and many is also possessed by some animals, birds and insects. Interesting experiments conducted on these species have confirmed that crows can keep a count of up to three [6]. In one such experiment, first, one person was sent up a tree where a crow couple was taking care of their offspring. The couple flew away to a nearby tree and kept watching their nest. They returned as soon as the person came down. Then two persons were sent up one after another. When the first person came down, the crows waited and returned only after the second person came down after sometime. Then three persons went up one after another and came down one after another with some time gap. The crows waited until all the persons came down. But with more than three persons sent up the tree one after another, the crows returned as soon as the third person came down implying that now they had lost count.

It has been confirmed that parrots, squirrels and chimps can be trained to keep a count of up to six. Pickover [27] mentions desert ants in Sahara that appear to have a built-in “computer” that counts their steps as they go out to a fairly large distance for collecting food. They bring back the collected food to their nests. Experiments were conducted by shortening and lengthening their strides after the food was collected. Towards this end, the leg lengths of the ants were manipulated, shortened by amputating and lengthened by adding stilts. With shortened strides it was noticed that the ants started searching for their nests way before reaching the destination. On the other hand, with longer strides they continued to move, going past their nests. Thus it appears that the process of counting in some form in terms of natural numbers exists in nature without human intervention.

One of the greatest advances in mathematics took place when the domain of positive integers, or natural numbers, was extended by introducing another integer as zero (with the current symbol 0). A vast literature exists on the history and significance of zero [17, 33]. Without getting into the intricacies of this great idea, we just notice that 0 is used both as an even number and a digit. As implied in the title of reference [17], zero as a number means “nothing”. Counting starts from 1 and 0 signifies absence. In a lighter vein, we refer to the following story of a kindergarten kid. When asked to name five animals of Africa, the kid answered three lions and two elephants. The teacher was not satisfied and said, “You have not named five animals.” The stubborn kid continued: “OK — three lions, two elephants, zero zebras, zero giraffe and zero deer and now I have named five animals”! Mathematically, the new answer just suggests that zebra, giraffe and deer are not to be found in Africa.

Ancient Indian mathematicians are credited for the invention of zero and the introduction of the now universally accepted Hindu–Arabic system of writing numbers which is so much superior to other systems that were used in different civilisations. The Indian mathematician Sridhara laid down the rules for arithmetical operations involving the number zero.

Natural numbers are unending or limitless. In mathematics, we use a symbol ∞ to express the unlimited infinity. This symbol was first used by Wallis in 1655 and mathematicians have used it ever since. Just as on zero, a vast literature also exists related to this abstract mathematical idea [8]. Even religious mysticism is connected to this mathematical concept of infinity [12]. Infinities appear in different contexts in different areas of mathematics. It has also been established that not all infinities are of the same order and there exist methods to compare different orders of infinities. Again without going into any further details, we now describe how zero and infinity are literally opposite poles in the theory of stereographic projections developed by Riemann [33]. Imagine a translucent sphere placed on an infinite horizontal plane. A point light source is placed at the top most point (the North Pole) of the sphere. Every point on the surface of the sphere casts its shadow on the horizontal plane. There exists a one-to-one correspondence between a point on the sphere and its shadow. The shadow of the bottom most point (the South Pole) is at the coincident point with the plane, i.e., the point of contact between the sphere and the plane. We consider this point as the origin or zero. The shadows of the points which are near the light source are very far off from this origin. The point corresponding to all the points at infinity (in every direction) is the North Pole. Thus the zero and the infinity are literally the opposite poles, zero at the South Pole and infinity at the North Pole.

It is mentioned in “The Book of Numbers”^a that many mathematicians, especially those working with infinities, prefer to start counting from zero rather than one. For example, Cantor did it while devising methods of counting different infinities. Automatic counting machines, unless otherwise taught, also start counting from zero. A three-digit ticket dispenser shows first 1000 tickets numbered as 000 to 999. The digital display of an odometer (used in a car) shows 00000 to 99999 to indicate the beginnings of the first to hundred thousandth kilometre.

Starting from non-negative integers, using one or more of the direct (like addition, multiplication, exponentiation) and/or inverse (like subtraction, division, root extraction) arithmetical operations on these, different types of other numbers have been created. These “unnatural” numbers, not to be found anywhere in nature, are abstract products of the human imagination. For example, negative integers −1, −2, −3, . . . have been created to make sense of subtracting a bigger positive integer from a smaller one. Arithmetic rules for handling positive and negative numbers were laid down by the Indian mathematician Brahmagupta. Of course, a negative number signifying the quantity of a natural object, like the number of bees, was regarded as meaningless. But in one instance, a negative distance was interpreted as a distance measured in the opposite direction.

Besides negative integers, these man-made numbers include rational, irrational, transcendental, imaginary, complex, dual, hyperreal, p-adic numbers and quaternions, and so on. Mathematicians unravel the mysteries of all such numbers. They also enjoy deciphering special characteristics of certain numbers in each category. The language of mathematics, created by using different types of numbers, has been most profitably used by natural scientists. They have used this language to express their ideas clearly, concisely and quantitatively. Famous physicist Wigner was surprised at this “unreasonable success of (man-made) mathematics towards describing and understanding the working of nature”.

In this short story of numbers, we discuss various types of numbers and mention special types and characteristics of some well-known and some not-so-well-known numbers. The objective is not to discuss serious mathematics, but point out curious things about numbers which are expected to be recreational for lay persons (amateurs) having an interest in something mathematical. High school students, who regularly use numbers, may find some new and interesting information which is not available in textbooks of mathematics.

It must be admitted that serious mathematics requires rigorous treatment of every concept. Even natural numbers have Dedekind’s set-theoretic and Peano’s axiomatic definitions and the naïve set theory may not suffice. The development of every concept follows a strict logical sequence. But sometimes this process is abandoned. In the introduction of a classical text in mathematics, entitled “Numbers”,^b it was conceded that logically the last chapter of the book should have been the first chapter. But the decision was to follow an age-old advice of not to begin at the beginning, as it is always the most difficult part. About rigour in mathematics, sometimes famous mathematicians also make jokes. It is said that too much rigour may cause rigor mortis to set in stopping all movements. To quote V. I. Arnold, “mathematicians do not understand a sentence like ‘Bob washed his hands’. They would rather simply say, ‘There exists a time interval (t₁, 0) in which the natural mapping t → Bob(t) represents a set of people with dirty hands and there is another interval [0, t₂) where the same mapping denotes a set complementary to the one considered earlier.’ ”

Jokes apart, it may be mentioned at this stage, that numbers were not always at the centre of all of mathematics. Greek geometers, like Pythagoreans, founded their mathematics on the abstract concepts of points and lines. They were interested in investigating the properties of shapes and sizes. Numbers were used to express various measures like the length of a line or the area enclosed by a geometrical figure or the volume of a solid.

Geometry was held in very high esteem by great minds. Galileo said that the secrets of nature are written in the language of geometry. At the entrance of Plato’s academy it was inscribed “Let no one ignorant of geometry enter here”. In a late nineteenth century senate meeting of Yale University, some members were criticising replacement of courses in classics and languages by those in mathematics and science. Gibbs, a man of few words, got up and said “Gentlemen what are we discussing? Mathematics is a language.” However, it may also be mentioned that not all great men were so kind to mathematics and mathematicians. Goethe said, “Mathematicians are like Frenchmen. Whatever you say to them, they translate into their own language, and forthwith it is something entirely different.” It is not clear whether he was more disgusted with Frenchmen or mathematicians.

Geometry and not arithmetic being the basis of Greek mathematics, geometrical construction with a compass and a straight edge was a necessary precondition for any arithmetic operation to be acceptable. Even with the idea of a point, the idea of zero was absent. Negative numbers were not i...

Table of contents

• Cover
• Halftitle
• Series Editors
• Title
• Copyright
• Dedication
• Preface
• About the Author
• Contents
• 1. Introduction
• 2. Integers
• 3. Real Numbers
• 4. Imaginary and Complex Numbers
• 5. Special Numbers
• Appendix A
• Appendix B
• Appendix C
• References
• Index