irrational number

Part of the Mathematics glossary:

An irrational number is a real number that cannot be reduced to any ratio between an integer p and a natural number q . The union of the set of irrational numbers and the set of rational number s forms the set of real numbers. In mathematical expressions, unknown or unspecified irrationals are usually represented by u through z . Irrational numbers are primarily of interest to theoreticians. Abstract mathematics has potentially far-reaching applications in communications and computer science, especially in data encryption and security.

Examples of irrational numbers are 2 1/2 (the square root of 2), 3 1/3 (the cube root of 3), the circular ratio pi , and the natural logarithm base e . The quantities 2 1/2 and 3 1/3 are examples of algebraic number s. Pi and e are examples of special irrationals known as a transcendental number s. The decimal expansion of an irrational number is always nonterminating (it never ends) and nonrepeating (the digits display no repetitive pattern).

If x and z are irrationals such that x < z , then there always exists an irrational y such that x < y < z . The set of irrationals is "dense" like the set Q of rationals. But theoretically, the set of irrationals is "more dense." Unlike Q , the set of irrationals is nondenumerable . There are more nonterminating, nonrepeating decimals than is possible to list, even by implication. To prove this, suppose there is an implied list of all the nonterminating, nonrepeating decimal numbers between 0 and 1. Every such number consists of a zero followed by a decimal point, followed by an infinite sequence of digits from the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. Suppose the elements of the list are denoted x 1 , x 2 , x 3 , ... and the digits in the numbers are denoted a ii . The list can be written like this:

x 1 = 0. a 11 a 12 a 13 a 14 a 15 a 16 ...
x 2 = 0. a 21 a 22 a 23 a 24 a 25 a 26 ...
x 3 = 0. a 31 a 32 a 33 a 34 a 35 a 36 ...
x 4 = 0. a 41 a 42 a 43 a 44 a 45 a 46 ...
x 5 = 0. a 51 a 52 a 53 a 54 a 55 a 56 ...
x 6 = 0. a 61 a 62 a 63 a 64 a 65 a 66 ...

Even though we don't know the actual values of any of the digits, it is easy to imagine a number between 0 and 1 that can't be in this list. Think of a number y of the following form:

y = 0. b 11 b 22 b 33 b 44 b 55 b 66 ...

such that no b ii in y is equal to the corresponding a ii in the list. The resulting number y is nonterminating and nonrepeating, is between 0 and 1, but is not equal to any x i in the list, because there is always at least one digit that does not match.

The non-denumerability of the set of irrational numbers has far-reaching implications. Perhaps most bizarre is the notion that "not all infinities are created equal." Although the set of rationals and the set of irrationals are both infinite, the set of irrationals is larger in a demonstrable way.

This was last updated in September 2005
Posted by: Margaret Rouse

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