Numbers and Numerals
Numbers and numerals are closely connected, but distinct, concepts. Numerals represent and symbolize numbers. Numbers themselves are abstract concepts that may or may not correlate to phenomena in the physical world.
Numerals are any symbols used to denote a number. Ancient societies began the practice of keeping written records thousands of years ago. Records indicating the quantities of things stored or sold would obviously have been of great use to the merchant class. A convenient and standardized system of symbols was, therefore, needed to meet such a requirement. Such a system is called a "numeral system". A numeral system is defined as a particular set of symbols and associated rules used to represent numbers. In past ages a variety of numeral systems have been employed, including "simple grouping systems" (e.g., Roman numerals), "multiplicative grouping systems" (e.g., Chinese-Japanese), and "ciphered systems" (e.g., Greek alphabetic). Modern numeral systems are place-value systems, meaning that the value of the symbol depends upon its position in the representation. For example, the "binary number system" is a positional system wherein any natural number is represented using only the symbols "0" and "1", and is used extensively in electronic computing. But by far the most familiar numeral system is the Hindu-Arabic numeral system, which is the decimal positional numeral system based on the symbols "0, 1, 2, 3, 4, 5, 6, 7, 8, 9" and the powers of ten. In this system any rational number is expressed using the base symbols given above and then as the sum of products involving powers of ten. For instance, "342.1" denotes the summation: (3 x 102) + (4 x 101) + (2 x 100) + (1 x 10-1).
In the broadest sense, a number is a unit of a mathematical system whose elements are subject to particular rules within the system. The bulk of modern mathematics can be made to rest upon the real number system, which in turn may be deduced from the whole or natural numbers. Through the use of set theory and logical methods, a rigorous mathematical construction of the infinite set of whole numbers can be attained, and through the whole numbers other number systems may be derived including the real and complex numbers.
The following list (and adjoining flow chart) describes various number systems:
- Complex numbers consist of the set {x | x is a real or imaginary number} in the form x = a + bi, where a and b are real numbers and i = (-1)1/2 is an imaginary number (e.g., x = 4 + 5i, x = 4, or x = 5i).
- Real numbers consist of the set {x | x is a rational or irrational number} in the form x = a + bi, where b = 0. (e.g., x = 4).
- Imaginary numbers consist of the set {x | x is a pure or nonpure number} in the form x = a + bi, where b 0. (e.g., x = 5i or x = 4 + 5i).
- Pure imaginary numbers consist of numbers in the form x = a + bi, where b 0, a = 0. (e.g., x = 5i).
- Nonpure imaginary numbers consist of numbers in the form x = a + bi, where b 0, a 0. (e.g., x = 4 + 5i).
- Irrational numbers consist of the set {x | x has a nonrepeating or nonterminating decimal expansion} in the form x = a + bi, where b = 0 (e.g., x = ).
- Rational numbers consist of the set {p / q | p and q are integers, a = p / q and q 0} or {x | x has a repeating or terminating decimal expansion} where x = a + bi, b = 0 (e.g., x = 4 / 1 or x = 4 / 3 = 1.333...).
- Integer numbers consist of the set {..., -3, -2, -1, 0, 1, 2, 3,...} where x = a + bi, b = 0.
- Natural numbers consist of the set {0, 1, 2, 3,...} where x = a + bi, b = 0.
- Whole (counting) numbers consist of the set {1, 2, 3,...} where x = a + bi, b = 0.
The simplest numbers are the whole numbers that consist of the set {1, 2, 3,...}. At the most fundamental level, a whole number is related to the quantity of elements composing a group or set. The sum and product of whole numbers is always a whole number. The difference and quotient may not result in a whole number. In order to make the subtraction of any whole numbers from itself a valid operation; it is necessary to introduce zero (0). This creates the natural numbers, consisting of the set {0, 1, 2, 3,...}. To make the subtraction of whole numbers always valid it is necessary to introduce negative integers. This creates the set of integer numbers denoted by {..., -3, -2, -1, 0, 1, 2, 3,...}. To make division of whole numbers always valid it is necessary to introduce the positive fractions. Because the difference of two positive fractions is not always a positive fraction, it is necessary to introduce the negative fractions. The positive and negative integers and fractions, and the number zero, comprise the rational numbers. The sum, difference, product, or quotient of two rational numbers is always a rational number (except for division by zero, which is undefined).
The ancient Greeks, through geometric methods, were able to construct polygons possessing incommensurable sides (i.e., line segments whose lengths cannot be expressed as a rational number). Hence, the existence of irrational numbers was demonstrated. When expressed in ordinary decimal format, irrational numbers, like the square root of 2 (21/2), have decimal expansions which are nonrepeating or nonterminating. The totality of the rational and irrational numbers makes up the real numbers.
Any real number multiplied by itself is 0 or positive. Therefore, "x2 = -1" has no real number solution. The creation of so-called "imaginary numbers" allows a solution to such an algebraic equation. All numbers of the form "x = a + bi", where a and b are real numbers and "i = (-1)1/2" is an imaginary number, belong to complex numbers.
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