Prove that √2 is an irrational number.

The √2 is an irrational number because it cannot be expressed as a fraction of two integers and does not terminate or repeat in its decimal representation.

The proof that √2 is an irrational number typically follows a proof by contradiction.

Assume, for the sake of contradiction, that √2 is rational.

So, we have ​. Squaring both sides, we get ​. Rearranging, we find a²=2b².

This implies that a² is even, and consequently, a must also be even (since the square of an odd number is odd). Let a = 2k, where k is another integer.

Substituting this back into a² = 2b², we get 4k² = 2b², and simplifying gives b² = 2k².

Now, this implies that b² is also even, and therefore b must also be even.

However, if both a and b are even, they have a common factor of 2, which contradicts our initial assumption that a and b have no common factors (other than 1).

This contradiction arises from assuming that √2 is rational, leading to the conclusion that √2 must be irrational. Therefore, √2 cannot be expressed as a fraction of two integers, confirming its irrationality.