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Proofs of Irrationality

“Irrational numbers are those real numbers which are not rational numbers!”

Def.1: Rational Number

A rational number is a real number which can be expressed in the form of \frac{a}{b} where a and b are both integers relatively prime to each other and b being non-zero.
Following two statements are equivalent to the definition 1.
1. x=\frac{a}{b} is rational if and only if a and b are integers relatively prime to each other and b does not equal to zero.
2. x=\frac{a}{b} \in \mathbb{Q} \iff \mathrm{g.c.d.} (a,b) =1, \ a \in \mathbb{Z}, \ b \in \mathbb{Z} \setminus \{0\}.

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