April 14, 2025: The irrationality of √2
Theorem: √2 is an irrational number meaning it can’t be written as a/b.
Proof:
Assume √2 is a rational number. Then √2 can be written as a/b where a and b are relatively prime. When we square both sides we get:
2 = a²/b²
This implies that 2 * b² = a².
This means a² is even, therefore a has to be an even number. We can represent a as 2m.
Substitute 2m for a:
2 * b² = 4m² b² = 2m²
Similarly, this implies that b² is even which can only happen if b is an even number. So, a has to be even and b has to be even but their gcd(a,b) = 1, which is impossible.
Therefore, by using proof by contradiction, we have proved that √2 is not a rational number; instead, it is an irrational number.