Why Square Root 2 is Irrational

A solution to a 2300 year old proven problem with new perspectives

But before proving, let’s take a look at what the rational and irrational numbers mean.

Rational Numbers

A Rational Number can be written as a Ratio of two integers (is a simple fraction).

Example: 3,5 is rational, because it can be written as the ratio 7/2

Example: 10 is rational, because it can be written as the ratio 10/1

Example: 0,999… (9 repeating) is also rational, because it can be written as the ratio 1/9

Irrational Numbers

But some numbers cannot be written as a ratio of two integers …

…they are called Irrational Numbers.

Example: π (PI) is the most famous irrational number in the world.

π = 3,14159265358979323846264…

Example: e(Euler Number)

e = 2.7182818284590452353602874…

Or famous Golden Ratio :

φ = 1,61803398874989484820…

So now we can show that the number √2 is not rational. In other words, there are no 2 integers that we can write the √2 with a / b

Proof 1

Lets try to write √2 as a / b then cross-multiply and square both sides.

Since the left side is even the right side must also be even too.

If we substitute C then simplify the produces

So as we can see that A and B is even, contraducting our first assumption that they have no common factors. This means that we cannot write √2 as a / b

Veysel Guzelsoz