What is the properties that helps us when we are proving a number whether rational or irrational? It is better if you can give several properties.

If you can give links, that will helps too.

Thanks in advance!


closed as too broad by Hagen von Eitzen, Juniven, Rohan, Leucippus, user91500 Feb 13 '17 at 4:47

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Partial (and elementary) answer

  1. If $x,y\in\mathbb{Q}$ then $x+y\in\mathbb{Q}$ and $xy\in\mathbb{Q}$

  2. If $x\in\mathbb{Q}^\times$ then $x^{-1}\in\mathbb{Q}$

  3. If $x\in\mathbb{Q}$ and $y\in\mathbb{R}-\mathbb{Q}$ then $x+y\in\mathbb{R}-\mathbb{Q}$

  4. If $x\in\mathbb{Q}^\times$ and $y\in\mathbb{R}-\mathbb{Q}$ then $xy\in\mathbb{R}-\mathbb{Q}$

1) and 2) are straightforward. 3) and 4) are consequences of 1) and 2).

  1. If $n\in\mathbb{N}^\times$, if $a_0,\cdots a_{n-1}$ are integers and if $\alpha$ is a real root of $P(X)=X^n+a_{n-1}X^{n-1}+\cdots+a_1X+a_0$ which is not an integer, then $\alpha\in\mathbb{R}-\mathbb{Q}$.

For example, let $\alpha=\sqrt2+\sqrt3$. We see that $\alpha^2=5+2\sqrt6$, thus $(\alpha^2-5)^2=24$, hence $\alpha$ is a root of :


Since $\alpha\not\in\mathbb{Z}$ (which follows from trivial upper and lower bounds), we conclude that $\alpha$ is irrational.

  • $\begingroup$ I like the fifth property, thanks for your answer. $\endgroup$ – Mathelogician Feb 12 '17 at 14:06

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