updated: 2022-01-23_12:32:30-05:00

sqrt(2) is irrational

A number is rational if it is a fraction of form $m/n$, where m and n are integers

Proof by Contradiction:

Assume $\sqrt(2)$ is rational

sqrt(2) = m/n where m and n are integers

  • m and n are divisible by some integer > 1

  • reduce fraction:

  • if reduced as much as possible, at least one is odd

  • sqrt(2)=m/n

  • square both sides

  • 2 = m^2/n^2

  • multiply n^2 on both sides

  • 2n^2 = m^2

    • implies m^2 is even
    • if m^2 is even, m is also even
      • (square of an odd number is always odd)

  • Since m is even we can write it as m = 2k

  • substitute m into 2n^2 = m^2

  • 2n^2 = (2k)^2

  • n^2 = 2k^2

  • therefore n is even

both m and n are even

which contradicts one having to be odd