Changes

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]].

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 involving 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 with a finite number of digits. Moreover the decimal expansion of an irrational number never repeats. For a rational number this is not true. For example <math>1/3=0.33333....</math>

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 (…):

It is common practice to write irrational numbers using only enough significant digits to solve the problem at hand, followed by an ellipsis (…):

:<math>\pi\ = 3.1415926...</math>

:<math>\pi\ = 3.1415926...</math>

==Formulations==

 

An irrational number is formally defined to be the limit of a [[Cauchy sequence]] of rational numbers.

==See also==

==See also==