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An irrational number is a real number which cannot be represented as a ratio of two integers. That is, if  is irrational, then
with
 and  .
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![$ \sqrt[p]{2}$](http://images.planetmath.org:8080/cache/objects/661/l2h/img5.png "$ \sqrt[p]{2}$") is irrational for
 ,
 , and
![$ \sqrt[p]{2}$](http://images.planetmath.org:8080/cache/objects/661/l2h/img8.png "$ \sqrt[p]{2}$") for
 , are irrational,- It is not known whether Euler's constant is rational or irrational.
- It
 is a real number and  is irrational for some
 , then  is irrational (proof).
- The sum, difference, product, and quotient (when defined) of two numbers, one rational and another irrational, is irrational. (proof).
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"irrational" is owned by yark. [ full author list (2) | owner history (1) ]
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(view preamble)
Cross-references: quotient, product, difference, sum, rational, Euler's constant, integers, ratio, real number
There are 78 references to this entry.
This is version 6 of irrational, born on 2001-11-04, modified 2005-04-10.
Object id is 661, canonical name is Irrational.
Accessed 15454 times total.
Classification:
| AMS MSC: | 11J72 (Number theory :: Diophantine approximation, transcendental number theory :: Irrationality; linear independence over a field) | | | 11J82 (Number theory :: Diophantine approximation, transcendental number theory :: Measures of irrationality and of transcendence) |
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Pending Errata and Addenda
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