rational and irrational

The sum, differencePlanetmathPlanetmath, productPlanetmathPlanetmath (http://planetmath.org/Ring) and quotient of two non-zero real numbers, from which one is rational and the other irrational, is irrational.

Proof.  Let a be a rational and α irrational number.  Here we prove only that aα is irrational — the other cases are similar.  If  aα were a rational numberPlanetmathPlanetmathr0,  then also  α=ar-1  would be rational as a product of two rationals.  This contradictionMathworldPlanetmathPlanetmath shows that aα is irrational.

Note.  In the result, the words real, rational and irrational may be replaced resp. by the words complex, algebraic and transcendental or resp. by the words complex, real and (the last here meaning, as commonly in Continental Europe, a complex numberMathworldPlanetmathPlanetmath having non-zero imaginary part).

Title rational and irrational
Canonical name RationalAndIrrational
Date of creation 2013-03-22 14:58:33
Last modified on 2013-03-22 14:58:33
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 7
Author pahio (2872)
Entry type Result
Classification msc 11J72
Classification msc 11J82
Related topic ExamplesOfPeriodicFunctions
Related topic CommensurableNumbers
Related topic SolutionsOfXyYx