Properties of Irrational and Rational Numbers

Question

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!

Answer 1

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).

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5. 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 :

$$P=X^4-10X^2+1$$

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