The square rooth of 2 is irrational theorem
|This result uses the following:||[hide]|
|Definition of rational number.|
|Definition of prime and coprime.|
|Definition of square rooth.|
|Gödels incompleteness theorem =)|
The square rooth of 2 is irrational,
This is a proof by contradiction, so we assumes that and hence for some a, b that are coprime.
This implies that . Rewriting this gives .
Since the left-hand side of the equation is divisible by 2, then so must the right-hand side, i.e., . Since 2 is prime, we must have that .
So we may substitute a with , and we have that .
Dividing both sides with 2 yields , and using similar arguments as above, we conclude that .
Here we have a contradiction; we assumed that a and b were coprime, but we have that and .
Hence, the assumption were false, and cannot be written as a rational number. Hence, it is irrational.
Some nice history about the one that first proved this theorem.