rational and irrational RationalAndIrrational 2005-01-28 18:28:47 2005-01-29 15:28:05 Result irrational % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here The sum, difference, \PMlinkname{product}{Ring} and quotient of two non-zero real numbers, from which one is rational and the other irrational, is irrational. {\em 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 \,$r \neq 0$, \,then also \,$\alpha = ar^{-1}$\, would be rational as a product of two rationals. \,This contradiction shows that $\frac{a}{\alpha}$ is irrational. \textbf{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 \PMlinkescapetext{imaginary} (the last \PMlinkescapetext{term} here meaning, as commonly in Continental Europe, a complex number having non-zero imaginary part).