Difference between revisions of "Irrational number"
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| − | An '''irrational number''' is a number that cannot be expressed as the ratio of two [[integers]]. Irrational numbers together with [[rational | + | An '''irrational number''' is a number that cannot be expressed as the ratio of two [[integers]]. Irrational numbers together with [[rational number]]s make up the set of [[real numbers]]. |
Irrational numbers often arise as solutions to problems involoving rational numbers. For example, the square root of 2 is irrational. Other irrationals, such as [[pi]], serve as fundamental constants in many mathematical problems. | Irrational numbers often arise as solutions to problems involoving rational numbers. For example, the square root of 2 is irrational. Other irrationals, such as [[pi]], serve as fundamental constants in many mathematical problems. | ||
| − | Irrational numbers can never be expressed exactly using decimal notation. Instead it is common to write them using only enough significant digits to solve the problem at hand, followed by an | + | Irrational numbers can never be expressed exactly using decimal notation with a finite number of digits. Instead it is common to write them using only enough significant digits to solve the problem at hand, followed by an ellipsis (…): |
| − | :<math>\pi\ = 3. | + | :<math>\pi\ = 3.1415926...</math> |
Simply, an irrational number can be defined as a non-repeating and non-terminating number, usually a decimal. | Simply, an irrational number can be defined as a non-repeating and non-terminating number, usually a decimal. | ||
| + | |||
==Formulations== | ==Formulations== | ||
| − | An irrational number is formally defined to be the limit of a [[ | + | |
| + | An irrational number is formally defined to be the limit of a [[Cauchy sequence]] of rational numbers. | ||
==See also== | ==See also== | ||
| + | |||
[[Transcendental numbers]] | [[Transcendental numbers]] | ||
| − | [[ | + | |
| + | [[Category:mathematics]] | ||
Revision as of 17:45, July 14, 2007
An irrational number is a number that cannot be expressed as the ratio of two integers. Irrational numbers together with rational numbers make up the set of real numbers.
Irrational numbers often arise as solutions to problems involoving rational numbers. For example, the square root of 2 is irrational. Other irrationals, such as pi, serve as fundamental constants in many mathematical problems.
Irrational numbers can never be expressed exactly using decimal notation with a finite number of digits. Instead it is common to write them using only enough significant digits to solve the problem at hand, followed by an ellipsis (…):
- <math>\pi\ = 3.1415926...</math>
Simply, an irrational number can be defined as a non-repeating and non-terminating number, usually a decimal.
Formulations
An irrational number is formally defined to be the limit of a Cauchy sequence of rational numbers.