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 thatis irrational. We can prove this by contradiction.

Suppose thatwhere bothandare numbers and suppose thatandhave no factors in common, so cannot be simplified. Ifcould be simplified thencould writtenwhereandand relabelled

Square both sides ofto giveand multiply both sides byto obtain(1)

This means thatmust be even, sinceis a whole number sois even.

This means we can writewhereis an integer, so that (1) becomes

Cancelling 2 from both sides gives

(2)

This means thatmust be even, sinceis an integer sois even.

Writeand subsitute into (2) to give

Cancelling 2 from both sides gives

henceand

this is a contradiction since it is an assumption thatcannot be simplified, so thatis irrational.