“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.   $x=\frac{a}{b}&space;\in&space;\mathbb{Q}&space;\iff&space;\mathrm{g.c.d.}&space;(a,b)&space;=1,&space;\&space;a&space;\in&space;\mathbb{Z},&space;\&space;b&space;\in&space;\mathbb{Z}&space;\setminus&space;\{0\}$.

Def. 2: Relatively Prime Numbers

Two integers $a$ and $b$ are said to be relatively prime to each other if the greatest common divisor of $a$ and $b$ is $1$ .
For example: The pairs (2, 9); (4, 7) etc. are such that each element is relatively prime to other.

Def. 3: Irrational Number

A real number, which does not fit well under the definition of rational numbers is termed as an irrational number.

A silly question: Let, in the definition of a rational numbers, $a=0$ and $b=8$ , then, as we know $\frac{0}{8}=0$ is a rational number, however $8$ can divide both integers $0$ and $8$ , i.e., $\mathrm{g.c.d.} (0,8) =8$ . (Why?) $\Box$

Primary ways to prove the irrationality of a real number

It is all clear that any real, if not rational, is irrational. So, in order to prove a (real) number irrational, we need to show that it is not a rational number (i.e., not satisfying definition 1). Most popular method to prove irrationality in numbers, is the Proof by Contradiction, in which we first assume the given (irrational) number to be ‘almost’ rational and later we show that our assumption was untrue. There are many more ways to prove the irrational behavior of numbers but all those are more or less derived from the proof by contradiction.
Some methods which I’ll discuss here briefly are:

1. Pythagorean Approach
2. Using Euclidean Algorithm
3. Power series expansion of special numbers
4. Continued Fraction representation of irrational numbers.

(1) Pythagorean Approach

This proof is due to Pythagoras and thus called Pythagorean Approach to irrationality. In this approach, we assume a number to be first. Later using the fundamental rules of arithmetic, we make sure whether or not our assumption was true. If our assumption was true, the number we took was rational, otherwise irrational.
For example:

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