Numbers

Type of physical science: Mathematical methods

Field of study: General topics in mathematics

Numbers and language are the two most basic educational needs of society. Numbers are universal in all of their characteristics and manifestations. The scientist is filled with intellectual insight by their relationship with magnitude, and the mathematician is fascinated by the level of abstraction achieved in them and by the enormous wealth of their sovereign structures.

Overview

The most commonly used symbols for representing numbers are the Hindu numerals 1, 2, 3, 4, 5, 6, 7, 8, 9, 0. The concept of zero, which is strangely present to show that nothing is there, is very significant. The system without it is imperfect and grossly inadequate for scientific purposes. An important concept in common-day usage of numbers is their place-value representation (like 365 for the number of days in a year). In this, the place and the value of a digit both have significance. By changing places or by changing the values, the numbers are changed. This representation, developed in India, is considered to be the most perfected system ever invented by human minds. Mathematics and the physical sciences could not have advanced with other representations, such as roman numbers, which are totally unable to take care of any advancement beyond simple counting. The simplest of the rules for addition and multiplication cannot be applied to roman numbers.

The simple-looking numbers have been and continue to be the subject matter of great mathematical studies and scientific applications. In fact, starting from basic numerals/numbers, the domain of numbers has grown tremendously and is still growing. In mathematics, there are several ways to classify the numbers. Practically all "abstract" algebraic structures are rooted in the different classes of the number system.

The counting and the natural numbers are represented by the symbols {1, 2, 3, 4,. . .}.

Conceptually speaking, the counting numbers arise from man's quantitative experience, while the natural numbers are obtained from a set of abstract postulates such as "Peano's axioms" of Giuseppe Peano. The positive integers have the same notations and properties as the counting and natural numbers. The operations of addition and multiplication on positive integers result in a positive integer. In this sense, mathematicians say that the set of positive integers is "closed" under the operations of sum and product. Nevertheless, the same statement cannot always be made with respect to subtraction and division, because there are numbers (-3, 2/3, and so many others) that are not positive integers.

Of special importance among the class of natural numbers are the prime numbers. Each natural number is divisible by the number 1 and itself and therefore has at least two divisors. Yet, some numbers such as 6 (whose divisors are 1, 2, 3, and 6) have more than two divisors. Those natural numbers that have only two divisors are called prime and the others are called composite.

Some of the prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, and so forth. Aside from 2, they are all odd.

Prime numbers play a key role in number theory and are the building blocks of natural numbers. The fundamental theorem of arithmetic, or the unique factorization theorem, states that every natural number can be uniquely written as a product of the powers of the primes, for example, 6 = 2 x 3, 12 = 22 x 3, 100 = 22 x 52.

Mathematicians have been particularly fascinated by the prime numbers and their properties; their properties are interesting to others also. For example:

sum of the first two = 2 + 3 = 2² + 1

sum of the first three = 2 + 3 + 5 = 3² + 1

sum of the first four = 2 + 3 + 5 + 7 = 4² + 1

sum of the first twenty = 20² + 1

The set of integers, namely, -3, -2, -1, 0, 1, 2, 3, and so on, contains negative integers, zero, and positive integers. The idea of representing negative numbers is traced back to China and is commonplace for purposes of representing temperature below zero or loss in business. In the domain of sets of integers, one can freely perform addition, subtraction, and multiplication, where the product of two negative integers, (-2) x (-3) = 6, is a positive integer. To understand such rules, one must understand negative numbers beyond mere intuition.

Rational numbers or fractions are represented as ratios of two integers, such as 3/2 or 355/113, or 22/7. Rational numbers help one to measure things in terms of units or fractional parts of the units. The collection of all numbers, p/q, where p and q are integers, is called the set of rational numbers. In this set, one can successfully perform all the four operations, namely addition, subtraction, multiplication, and division (except division by zero).

Irrational numbers were first understood in Greece; they arose in response to the inadequacy of rational numbers in making all the measurements. For example, the diagonal of a unit square cannot be represented by a rational number of the form p/q. Similarly, the length of the circumference of a circle is not a rational number. Thus, in nature, one finds numbers that are not rational. These are called irrational numbers. There are an infinite number of rational numbers and an infinite number of irrational numbers. Interestingly, the set of irrational numbers, in a way, is more numerous than the set of all the rational numbers.

With rational and irrational numbers, different measurements can be represented by numbers. The class of numbers--rational and irrational together--form the set of real numbers.

The set of real numbers is also closed under the four basic operations. Additionally, in this class, a square root of any positive real number can be found. An important achievement in the class of real numbers is that they are identical (have one-to-one correspondence) with all the points on a line, and this provides a sound basis for interaction between geometry and arithmetic/algebra.

This observation is basic for many abstract ideas in functional analysis and topology, two important branches of modern mathematics.

Since the square of a real number, positive or negative, is a nonnegative number, the square root of a negative number, such as -1, cannot be a real number. By including a number i = √-1, the class of numbers is extended to what are called "complex numbers."

Another way to classify real numbers is to consider equations of classical algebra.

Classical algebra is the workshop of mathematics. The simplest equations are the algebraic equations in integral powers of a variable with coefficients as integers. Such equations are sometimes called polynomial equations. The numbers that satisfy (or can be obtained as roots of) a polynomial equation are called algebraic numbers. Yet, not all real numbers satisfy an algebraic equation. Therefore, the numbers that are not algebraic are called transcendental numbers. Some of the very important numbers such as pi, the ratio of the circumference of a circle to its diameter, and the number e, which is the base of natural logarithms, are examples of transcendental numbers. The proof for their transcendental nature is complex. It can be seen that all rational numbers are algebraic, but the converse is not true. For example, √2, an irrational number, is algebraic, being a root of the algebraic equation x² - 2 = 0. What is true, on the other hand, is that all transcendental numbers are irrational. Thus, π and e are both transcendental and irrational numbers.

Decimal representation is in common use. An amount of 2 dollars and 37 cents is written in decimal form as $2.37. This representation is very revealing in the study of numbers.

Some fractions can be expressed as terminating decimals, some cannot. For example, 5/4 = 1.25 has a terminating decimal representation, there being no decimal digit after 5. Yet 4/3 = 1.333. . ., with the digit 3 repeating indefinitely, has a nonterminating decimal representation.

Among the nonterminating representations, there can be two types: those in which a digit or a finite number of digits recur or repeat over and over, and others in which there are no repeating patterns of digits. A significant observation is that all rational numbers or fractions can be represented not as terminating or nonterminating but as recurring decimal forms. The nonterminating, nonrepeating decimals, such as 1.4142. . . for and 3.14159265358. . . for pi, represent irrational numbers. In number theory, the study of continued fractions also reveals some interesting characteristics of numbers.

More generally, decimal representation can be seen as a series in the integral powers of the base number 10. Thus, 127.82 = 1 x 10² + 2 x + 7 x ⁰ + 8 x ^-1 + 2 x ^-2. This is a way of representing numbers in a base other than 10. The most important among other bases is 2, because computer languages use only two symbols: 0 and 1. The base 2 representation, called binary representation, can be achieved by expressing any number as a series in powers of 2 with coefficients of either 0 or 1, as shown in the following example: 7 = 1 × 2² + 1 × 2¹ + 1 × 2⁰ = 111₂ 25 = 1 × 2⁴ + 1 × 2³ + 0 × 2² + 0 × 2¹ + 1 × 2⁰= 11001₂

Another base that has been used, but is not of much interest, is 12. The numbers are then called duodecimal. From classical algebra, one can go to abstract algebra. Abstract algebra deals with various types of structures of sets of objects under mathematical operations. Number sets are rich for their structural properties. Structural properties of number systems have actually led to practically all the ideas in abstract algebra. These structures are given special names such as group, ring, field, integral domain, vector space, to name a few. The operations of addition, subtraction, multiplication, and division are, in general, well known in numbers, but they may mean different things for other types of objects.

In studying structures, an attempt is made to find what kind of properties a certain type of set of elements always satisfies under these operations. For example, if addition of two objects of a set always produces a third element of the set, one can say the set is closed under addition.

To present the ideas precisely in a general setting, let a, b; and c be three numbers of a set. The four conditions that mathematicians check with regard to addition are closure property--a + b is a member of the set; associativity--(a + b) + c = a + (b + c); existence of the identity element--there is a member 0 in the set such that a + 0 = a; and existence of additive inverses--there is a member -a in the set for each a, such that a + (-a) = 0. When these four properties are satisfied by the set, it is said to have the algebraic structure called "group" under the operation of addition. If it additionally satisfies the condition of commutativity--a + b = b + a--it is called an "abelian" or a "commutative group."

The set of numbers {1, 2, 3, . . . ,} or {0, 1, 2, 3, . . . ,} does not satisfy all the above conditions and therefore does not form a group. The set {. . . ,-2, -1, 0, 1, 2, 3, . . .} satisfies all the conditions and is therefore an abelian group under addition.

The corresponding five conditions under product (x) are as follows: a x b is a member of the set; (a x b) x c = a x (b x c); there is a number 1 in the set such that a x 1 = a; for every a in the set there is a number a to the power of -1 in the set such that a x a^-1 = 1; and a x b = b x a.

There are also properties combining + and x, as follows: left distributivity--a x (b + c) = a x b + a x c; and right distributivity--(b + c) x a = b x a + c x a.

The sets that satisfy these conditions are said to form a "ring" under + and x. One finds that {. . . ,-2, -1, 0, 1, 2, 3, . . .} is a ring; the sets of rational numbers and real numbers are also rings. Further, if a set of objects satisfies these properties, it is called a "field," and if additionally it satisfies commutativity properties, an abelian or commutative field. It can be easily checked that the set of rational numbers and the set of real numbers each form a commutative field.

The numbers used for order (such as first, second, third, twentieth, and hundredth) are referred to as ordinal numbers. Numbers used for signifying the size of something, such as the population of New York, are called cardinal numbers. The idea of cardinal numbers is very important in set theory, a foundation of modern mathematics. It brings one to the study of infinite numbers. When mathematicians think of an infinite number, the sense is entirely different from what is meant in the physical sense. For example, the total number of atoms in the universe is infinite in the physical sense but quite finite, less than 3⁷⁴, mathematically.

In general, cardinal numbers that are not finite are called "transfinite numbers." The smallest infinite number in mathematics is the cardinal number of the set of all natural numbers. It is denoted by the letter α. In fact, any set having one-to-one correspondence with this set of natural numbers shares the cardinal number α. A simple but significantly distinguishing property of an infinite set is that it possesses infinite subsets bearing one-to-one correspondence with the set itself. This property is not shared by finite sets. Further, infinite does not mean incomprehensibly large and useless in the physical sense. The set of all the points on a line segment of unit length is infinite--in fact, bigger than the infinity that represents the cardinal number α of the set of natural numbers. If the cardinal number of the line segment of unit length or any length is denoted by the letter c, then c = 2 α. One of the basic open problems in mathematics--called continuum hypothesis--is related to the ordinal number c of the real line or real numbers. Georg Cantor and John von Neumann are credited with developing the infinite cardinal numbers.

Mathematicians have looked at rigorous ways of defining the numbers. Apart from Peano's axioms, there are two other elegant methods, one by Richard Dedekind through cuts of the set of rational numbers, and the other through convergent sequences of rational numbers.

The introduction of infinite numbers has made mathematics more sound, unified, and applicable. For example, it is in terms of infinite series that all special functions of physics and engineering are best studied.

In mathematics, generalizations of numbers can also be considered. One such generalization, called q-numbers, arises in a totally different way. Consider that there is a base number q, the number n to the base q is defined as [n]_q = (qⁿ - 1)/(q - 1). Thus, []_q = (q² - 1)/(q - 1) = q + 1, []_q = (q³ - 1)/(q - 1) = q² + q + 1, and so forth.

What makes these numbers generalizations of known numbers is that Lim_q → 1 [n]_q = n, the usual number. Thus, by the limiting process, which is the basis of mathematical analysis and calculus, these q-numbers give the usual numbers. The study of special functions--when usual numbers are replaced by q-numbers--has been an interesting area of study.

In a physical sense, a number represents a quantity or a count. When more than one number is required for the purpose, such as height and weight, and the addition of different measurements does not make sense, a need arises for a representation that can have several numerical entries. This is achieved by a class of mathematical objects called matrices. Thus, matrices, with their entries as numbers, are generalizations of numbers, with laws such as addition and multiplication defined for them suitably. Quaternions, given by Carl Friedrich Gauss, are another interesting generalization of numbers.

Applications

Numbers play a significant role in all branches of science. In physics, the laws of conservation of mass, energy, and momentum need and employ mathematical analysis, which deals with phenomena of continuous type. On the other hand, industry, economics, planning, and war and defense efforts use mathematics of numbers called discrete mathematics.

The first significant application of numbers is in mathematics itself. The properties and structures of numbers have led to abstract algebraic concepts of group theory, ring theory, theory of finite fields, and linear spaces, which have found applications in physics and other areas.

Number theory, which was long considered to be the study of numbers for their own sake, is no longer so exclusive. The theory of finite fields provides a major base for practically all coding situations for reliable communication. Practical codes for communication use some of the most abstruse results of the theory. Quadratic residue codes play an important role in the study of error-correcting codes. Quadratic residue codes are preferred in communication because they can be developed quite systematically, tend to have high minimum distance, and possess easy decoding procedures.

Cryptography, the theory of secret codes, is largely based on the properties of numbers and many interesting results of number theory. An application to cryptography relies on the disparity in computer time required to find large primes and to factor large integers. A secret code maker may choose two primes p₁ and p₂, which require only a few minutes of computer time; their product may be made public, but without knowledge of the two primes used, they cannot be decoded.

There is a recreational use of numbers in mathematical puzzles and games. Magic squares have become amusing and useful mathematical designs. Mathematical puzzles and games have become very popular and productive with the advent of computer games.

The theory of combinatorial designs used for quality control in industry, and for statistically analyzing agricultural production problems, form an important study in discrete mathematics. A large class of problems of traffic control, city planning, and social interactions can be formulated in graph theory, another important branch of discrete mathematics.

Operations research was developed during World War II as a mathematical method for solving problems arising in human operations of a large order. Mathematical programming is a widely applicable area. Computer programs, based on these techniques, are developed for airline reservations and routing telephone calls. Integer programming techniques are used in production engineering to find integer optimal solutions to a programming problem. The use of computers in solving some practical problems of the physical and natural world has introduced numerical methods.

Context

Numbers are quite indispensable for society in every walk of life, but cultures and societies of the West lived practically without them up to the twelfth century. The roman numerals, which are still customarily used for some applications, were the best that developed in Greece and were passed on to other European countries. The present number system was developed in India. Scholars believe the system was developed by 200 B.C. without any external stimulus. The place value system made numbers workable. The concept of zero may be traced to Hindu philosophical traditions originating with the Vedas, and its representation by a symbol first liberated the digits from the counting board and enabled them to stand alone.

From India, numbers, algebra, and astronomy migrated to the neighboring Arabs. In 773, in the court of Caliph al-Mansur of Baghdad, a Brahman from India brought the Sanskrit treatise, the "Siddhanta" by Brahmagupta, on astronomy. It was translated into Arabic.

Al-Khwarizmi, around 820, wrote the famous book KITAB AL-JABR WA AL-MUQUABALAH, explaining the numbers and methods he learned from Indian writings.

This book was translated into Latin with the title ALGEBRA ET ALMUCABALA in the twelfth century, and the discipline came to be known as algebra. Magic squares originated in India and China. The Chinese used the philosophical concepts "yen-ying" to develop magic squares successfully.

Number theory, which is also called higher arithmetics, is the branch rich in mathematics of numbers. It has fascinated amateurs as well as professional mathematicians; the names of the greatest mathematicians such as Leonhard Euler, Karl Friedrich Gauss, and Srinivasa Aaiyangar Ramanujan are associated with this theory. The theory owes much to the seventeenth century French amateur mathematician Pierre de Fermat. His famous last theorem is: There are no positive integers x, y, z, and n greater than 2 such that xⁿ + yⁿ = zⁿ. This theorem still remains unproved, and in the process of discovering its proof, several systems of mathematics have been created.

The proof that π is a transcendental number, in 1882, by the German mathematician Ferdinand von Lindemann settled a major problem coming from the time of Euclid, the "impossibility of squaring the circle."

Infinity and infinite numbers have also fascinated mathematicians. A distinguishing property of infinite sets is that they are equivalent to their own subsets. Philosophically, this idea was known to Hindus from Vedic (prehistoric) times.

Until the mid-twentieth century, number theory was considered the purest of the pure mathematics, but with no applications. The advent of digital computers and digital communication theory has revealed that this branch has unexpected answers to real-world situations.

Principal terms

ALGEBRAIC EQUATION: an equation in positive integer powers of a variable x, with coefficients as integers

ALGEBRAIC NUMBERS: numbers that satisfy an algebraic equation

CARDINAL NUMBERS: the number of elements in a finite set represents its cardinal number; the cardinal number of the infinite set {1,2,3. . .}, and of all sets having a one-to-one correspondence with it is denoted by the symbol α

INTEGERS: numbers such as 0, 1, -1, 2, and -2

IRRATIONAL NUMBERS: the real numbers that are not rational

NATURAL NUMBERS: numbers 1, 2, 3,. . ., 9, 10,. . . are called natural numbers or counting numbers

PRIME NUMBERS: a positive integer that has exactly two divisors, namely, the number 1 and itself; examples are 2, 3, 5, 7, 11

RATIONAL NUMBERS: fractions such as 1/2 and -2/3 that can be expressed as ratios of two integers

REAL NUMBERS: numbers that are either rational or irrational

Bibliography

Baker, Alan. A CONCISE INTRODUCTION TO THE THEORY OF NUMBERS. Cambridge, England: Cambridge University Press, 1985. A nontechnical discussion of the theory of numbers.

Conway, J. H. ON NUMBERS AND GAMES. London: Academic Press, 1976. For the reader with an interest in the creative axiomatic aspects of the theory of transfinite numbers. Some background in mathematics is needed.

Menninger, Karl. NUMBER WORDS AND NUMBER SYMBOLS: A CULTURAL HISTORY OF NUMBERS. Translated by Paul Broneer. Cambridge, Mass.: MIT Press, 1969. An excellent book for one who appreciates intellectual and cultural history. Presents the history of numbers in the ancient cultures of Babylonians, Chinese, Egyptians, Indians, and Romans, and traces the journey of the number system from the subcontinent of India to the West.

Niven, Ivan, and Herbert Zuckermann. AN INTRODUCTION TO THE THEORY OF NUMBERS. 4th ed. New York: Wiley, 1980. A very thorough presentation of the topics of number theory for the mathematician.

Welsh, Dominic. CODES AND CRYPTOGRAPHY. Oxford: Oxford University Press, 1988. This text may be consulted for many applications of number theory in coding and cryptography.

Sets and Groups