Number Sets

Numbers are at the heart of mathematics and have been a source of fascination and curiosity for mathematicians, astronomers, scientists, and even theologians, magicians, and astrologers. Some mathematicians pursue this fascination in a field of mathematics called number theory, in which number sets are used.

The emergence of the whole numbers 0, 1, 2, 3, 4, … naturally stems from the fundamental idea of counting. If a group of bananas and pineapples can be paired off, one banana for every pineapple, then the two groups, or sets, have the same "number" of objects. The members of the two sets do not resemble each other in appearance—one is the set of bananas, whereas the other is a set of pineapples—but both sets share a common property.

The common property that results from comparing the "size" of sets leads to the counting numbers. Any two sets whose members can be paired off has the same "number" of elements and this equality in "size" of the two sets is denoted by an appropriate numeral: 1, 2, 3, ….

Whole numbers are adequate for counting, but they are not adequate for many other purposes. For instance, dividing an odd number of objects, like 3, into an even number of parts is not possible with just whole numbers. The need to evenly divide odd number of objects leads to fractions like , and . The fraction means dividing 7 pieces in 5 equal parts. But what does ¾ indicate? Does this mean dividing 3 pieces in 4 equal parts? Another interpretation is to think of ¾ as taking 3 equal pieces of something that has already been divided into 4 equal parts.

In mathematics, numbers are removed from particular descriptions of objects or sets. For instance, 7, ¾, and are numbers that can stand alone without describing objects or sets quantitatively. Some primitive languages have different names for a number of particular objects. For example, in a language spoken by Fiji Islanders, ten boats is "bolo," but ten coconuts is "koro." The idea of removing number from particular objects is called "abstraction"—the essential idea of number is removed, or abstracted, from counting a group of particular objects.

The Greeks made another important discovery that forced mathematicians to extend the number system beyond fractions. The length of the diagonal line of a square with sides of 1 unit cannot be expressed as a whole number or fraction. They called the length of the diagonal an "incommensurable quantity."

Using the Pythagorean Theorem , the length of the diagonal line equals . Mathematicians further extended the number set to include numbers of the form , and so on. In decimal notation, these numbers have a nonterminating and nonrepeating form; is 1.41421… and the numbers after the decimal point do not end or contain a repeating pattern. These numbers are called irrational because they cannot be written as the ratio of two integers. A simple nonterminating but repeating decimal is 0.333…. In fractions, 0.333… is expressed as ⅓, and therefore it is a rational number.

The set of all rational and irrational numbers is called the real number set, R. Are real numbers adequate for all mathematical needs? Consider the equation: x ² = −1. The equation requires a number x whose square is negative one. But the square of all real numbers is positive. Therefore, the equation, x ² = −1 has no solution in the real number set. But that did not stop mathematicians from further extending the number set.

Suppose i is a number whose square is negative 1. The number i, with the defined property of i ² = −1, is called an imaginary number. This name is unfortunately misleading and an historic accident. The number i is no less a number than −3, , . In fact, the name "real" numbers is also misleading because numbers that are not in the set of real numbers are just as "real" as any other number. The following is a summary of the sets of numbers.

Whole Numbers

The set of whole numbers, W, consists of the counting numbers and zero: {0, 1, 2, 3, …}. Every whole number can be generated from 1 and the operation of addition: 2 = 1 + 1, 3 = 2 + 1, and so on. The sum of two whole numbers is also a whole number. Mathematicians describe this property by saying that the whole number set is closed under addition. In symbols, the closure property is described as the following: If a, b ∈ W, then a + b ∈ W. The whole number set is also closed under multiplication, but it is not closed under subtraction because the result of 3 − 7 is not a whole number.

Integers

The integer number set contains the positive counting numbers, their corresponding negative numbers, and 0. The integer number set I is written as {…, − 2, − 1, 0, 1, 2,…} or {0, ± 1, ± 2,…}. In other words, the set of integers consists of the set of whole numbers and their negatives. Like the whole number set W, I is closed under addition and multiplication. But I is also closed under subtraction. For instance, 3 − 9 is − 6, which is an integer. However, I is not closed under division. Dividing 8 by 4 equals 2, which is an integer, but dividing 7 by 2 does not equal an integer.

Rational Numbers

All numbers of the form a /b where a and b are integers and b is not equal to 0 are called rational numbers. Rational numbers can be written as the ratio of integers. The rational number set, Q, is closed under all the four operations: addition, subtraction, multiplication, and division (provided division by 0 is excluded). All rational numbers have a terminating or nonterminating-but-repeating decimal form.

Irrational Numbers

Numbers like and that cannot be expressed as a ratio of integers are called irrational numbers. In decimal form, irrational numbers have a nonterminating and nonrepeating form. The sum of two irrational numbers is also an irrational number, so the irrational set is closed under addition. But the irrational number set is not closed under multiplication; for example, consider the product of two irrational numbers, and . Their product is or 6, which is not an irrational number.

Real Numbers

The real number set, R, is constructed by combining the sets of rational and irrational numbers. In the language of sets, R is the union of the rational number set and irrational number set.

Complex Numbers

A number of the form a + bi, where a and b are real numbers and i ² = −1, is called a complex number. For a = 0 and b = 1, the result of a + bi is the imaginary number, i. All real numbers are also complex numbers. For instance, can be expressed as + 0i. Therefore, by taking b = 0, all real numbers can be expressed as complex numbers. A nontrivial example of a complex number is 5 + 6i.

Imaginary Numbers

An imaginary number is that part of a complex number which is an even root of a negative number. Examples are or . Imaginary numbers typically are represented by the constant, i. Like real numbers, imaginary numbers have relative magnitude and can be plotted along a number line. Imaginary numbers may also be negative, with 3i > i > − i > - 3i.

see also Integers; Numbers, Complex; Numbers, Irrational; Numbers, Rational; Numbers, Real; Numbers, Whole.

Rafiq Ladhani

Bibliography

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Conway, John H., and Richard K. Guy. The Book of Numbers. New York: Copernicus, 1996.

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Miller, Charles D., Vern E. Heeren, and E. John Hornsby, Jr. Mathematical Ideas, 9th ed. Boston: Addison-Wesley, 2001.