## Proof That the Square Root of 2 is Irrational

Rrational numbers are fractions where both numerator and denominator are integers. Irrational numbers are conversely, any number that cannot be written as a fraction with one integer divided by another.

Suppose then we want to prove that is irrational. We can prove this by contradiction.

Suppose that where both and are numbers and suppose that and have no factors in common, so cannot be simplified. If could be simplified then could written where and and relabelled Square both sides of to give and multiply both sides by to obtain (1)

This means that must be even, since is a whole number so is even.

This means we can write where is an integer, so that (1) becomes Cancelling 2 from both sides gives (2)

This means that must be even, since is an integer so is even.

Write and subsitute into (2) to give Cancelling 2 from both sides gives hence and this is a contradiction since it is an assumption that cannot be simplified, so that is irrational. 