# Famous Theorems of Mathematics/√2 is irrational

The square root of 2 is irrational,

## Proof[edit]

Assume for the sake of contradiction that . Hence holds for some *a* and *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 . However, we assumed that such that that *a* and *b* were coprime, and have now found that and ; a contradiction.

Therefore, the assumption was false, and cannot be written as a rational number. Hence, it is irrational.

## Another Proof[edit]

The following reductio ad absurdum argument is less well-known. It uses the additional information √2 > 1.

- Assume that √2 is a rational number. This would mean that there exist integers
*m*and*n*with*n*≠ 0 such that*m*/*n*= √2. - Then √2 can also be written as an irreducible fraction
*m*/*n*with*positive*integers, because √2 > 0. - Then , because .
- Since √2 > 1, it follows that
*m*>*n*, which in turn implies that*m*> 2*n*–*m*. - So the fraction
*m*/*n*for √2, which according to (2) is already in lowest terms, is represented by (3) in strictly lower terms. This is a contradiction, so the assumption that √2 is rational must be false.

Similarly, assume an isosceles right triangle whose leg and hypotenuse have respective integer lengths *n* and *m*. By the Pythagorean theorem, the ratio *m*/*n* equals √2. It is possible to construct by a classic compass and straightedge construction a smaller isosceles right triangle whose leg and hypotenuse have respective lengths *m* − *n* and 2*n* − *m*. That construction proves the irrationality of √2 by the kind of method that was employed by ancient Greek geometers.

## Notes[edit]

- As a generalization one can show that the square root of every prime number is irrational.
- Another way to prove the same result is to show that is an irreducible polynomial in the field of rationals using Eisenstein's criterion.