rational and irrational

The sum, difference, product (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 $\alpha$ irrational number.  Here we prove only that $\frac{a}{\alpha }$ is irrational — the other cases are similar.  If  $\frac{a}{\alpha }$ were a rational number  $r\ne 0$,  then also  $\alpha =a{r}^{-1}$  would be rational as a product of two rationals.  This contradiction shows that $\frac{a}{\alpha }$ 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 number 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