The integers are the positive and negative whole numbers... -4, -3, -2, -1, 0, 1, 2,.... The name "integer" comes directly from the Latin word for "whole." The set of integers can be generated from the set of natural numbers by adding zero and the negatives of the natural numbers. To do this, one defines zero to be a number which, added to any number, equals the same number. One defines a negative of a given number to be a number which, plus the given number, equals zero. Symbolically, for any number n: 0 + n = n (additive identity law) and -n + n = 0 (additive inverse law). Because arithmetic is done with natural numbers, one needs rules which will convert integer arithmetic into natural-number arithmetic. This is true even with a calculator. Most simple four-function calculators have no easy way of entering negative numbers, and the user has to apply the rules for
|ab is a unique integer.||a + b is a unique integer.||Closure law|
|ab = ba||a + b = b + a||Commutative law|
|a(bc) = (ab)c||a +(b + c)= (a + b) + c||Associative law|
|(1)(a) = a||0 + a = a||Identity law|
|-a + a = 0||Inverse law|
|If ac = bc (c = 0), then a = b.||If a + c = b + c, then a = b||Cancellation law|
|a(b + c) = ab + ac||Distributive law|
himself. Rules are often stated using the concept of absolute value. The absolute value of a number is the number itself if it is positive and its opposite if it is negative. For example, the absolute value of +5 is +5, or 5, while the absolute value of -3 is +3, or 3. Absolute values are always positive or zero.
There are two basic rules for addition: 1) To add two numbers with like signs, add their absolute values and give the answer the common sign. 2) To add two numbers with opposite signs, subtract the smaller absolute value from the larger and give the answer the sign of the larger.
For example: -4 + (-7) is -11, and -8 + 3 is -5.
There is a single rule for subtraction. It does not give the result directly but converts a difference into a sum: To subtract a number, add its opposite. For example, -8 - 9 becomes -8 + (-9), and 4 - (-2) becomes 4 + 2. This latter example uses the fact that the negative (or opposite) of a negative number is positive.
Division and multiplication have two simple rules: 1) The product or quotient of two numbers with like signs is positive. 2) The product or quotient of two numbers with unlike signs is negative. For example. (-30)(18) is -540; (-6)/(-3) is 2; and 20/(-4) is -5.
Because the integers include negative numbers, it is possible for every subtraction, as well as every addition and multiplication, to be completed using only integers. The set of integers is therefore "closed" with respect to subtraction, addition, and multiplication. It is not closed with respect to division, however. Three divided by seven is not an integer.
The set of integers form an "integral domain." This is a mathematical system governed by these laws for all integers a, b, and c. Notice that there is no inverse law for multiplication. Integral domains do not necessarily have multiplicative inverses, and, consequently, division is not always possible.
Integers are useful in business, where an amount of money can be a loss as well as a gain. They are useful in science when a quantity can be negative or positive, as in the charge borne by electrons, protons and other elementary particles, or in temperatures above and below zero. They show up in games, even, where one can be a number of points ahead or "in the hole." And they are absolutely necessary in mathematics, which would otherwise be incomplete and of little interest.
Gelfond, A.O. Transcendental and Algebraic Numbers. Dover Publications, 2003.
Klein, Felix. "Arithmetic." In Elementary Mathematics from an Advanced Standpoint. New York: Dover, 1948.
Rosen, Kenneth. Elementary Number Theory and Its Applications. 4th ed. Boston: Addison-Wesley, 2000.
Stopple, Jeffrey. A Primer of Analytic Number Theory: From Pythagoras to Riemann. Cambridge: Cambridge University Press, 2003.
Van Niven, I. Numbers: Rational and Irrational. New Mathematical Library, Washington, DC: The Mathematical Association of America, 1975.
Weisstein, Eric W. The CRC Concise Encyclopedia of Mathematics. New York: CRC Press, 1998.