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An irrational number is a real number which cannot be represented as a ratio of two integers. That is, if $x$ is irrational, then
$$ x \ne \frac{a}{b} $$
with $a,b \in \mathbb{Z}$ and $b \ne 0$
- $\sqrt[p]{2}$ is irrational for $p=2,3,\ldots$
- $\pi, e$ and $\sqrt[p]{2}$ for $p=2,3,\ldots$ are irrational,
- It is not known whether Euler's constant is rational or irrational.
- It $a$ is a real number and $a^n$ is irrational for some $n=2,3,\ldots$ then $a$ 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 (3) | owner history (1) ]
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Cross-references: numbers, quotient, product, difference, sum, rational, Euler's constant, integers, ratio, real number
There are 92 references to this entry.
This is version 7 of irrational, born on 2001-11-04, modified 2008-11-09.
Object id is 661, canonical name is Irrational.
Accessed 19647 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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