irrational
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≠ab |
with a,b∈ℤ and b≠0.
Examples
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1.
p√2 is irrational for p=2,3,…,
-
2.
π,e, and p√2 for p=2,3,…, are irrational,
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3.
It is not known whether Euler’s constant is rational or irrational.
Properties
-
1.
It a is a real number and an is irrational for some n=2,3,…, then a is irrational (proof (http://planetmath.org/IfAnIsIrrationalThenAIsIrrational)).
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2.
The sum, difference, product, and quotient (when defined) of two numbers, one rational and another irrational, is irrational. (proof (http://planetmath.org/RationalAndIrrational)).
Title | irrational |
Canonical name | Irrational |
Date of creation | 2013-03-22 11:55:59 |
Last modified on | 2013-03-22 11:55:59 |
Owner | yark (2760) |
Last modified by | yark (2760) |
Numerical id | 12 |
Author | yark (2760) |
Entry type | Definition |
Classification | msc 11J82 |
Classification | msc 11J72 |
Synonym | irrational number |
Related topic | TranscedentalNumber |
Related topic | AlgebraicNumber |
Related topic | Integer |
Related topic | LindemannWeierstrassTheorem |
Related topic | GelfondsTheorem |
Related topic | ProofThatTheRationalsAreCountable |