Is √2 irrational?
umasundresh
Yes it is.
Hippasus, a Greek philosopher who was an early disciple of Pythagoras, is credited with being the first person to discover the existence of irrational numbers.
However, many of Pythagoras’s disciples couldn’t wrap their heads around the concept of irrational numbers and also couldn’t contradict Hippasus with reasoning.
He was drowned to death upon Pythagoras’ command.
The proof is by contradiction.
First let us assume that √2 is rational.
This means that, x/y = √2, where x and y have no common factors.
Squaring on both sides, we get:
x2 /y2 = 2
x2 = 2y2 (The number is even on the right-hand side. Because any number multiplied by 2 is always even.)
The only way the equation to be true is that x itself should be even.
Thus x2 is even and it is divisible by 4.
Therefore, x and y are even numbers with common factors. This contradicts our assumption that x and y have no common factors.
Therefore, √2 cannot be rational.
