Irrationality of the Square Root of Two

Answer

The proof that $\sqrt{2}$ is irrational is a classic example of a proof by contradiction. The process involves assuming that $\sqrt{2}$ is rational and then demonstrating that this assumption leads to a contradiction, thereby proving that $\sqrt{2}$ cannot be rational.

Step-by-Step Proof

1. Assumption of Rationality:

   • Assume $\sqrt{2}$ is a rational number. By definition, a rational number can be expressed as a fraction $\frac{a}{b}$, where $a$ and $b$ are integers with no common factors other than 1 (i.e., the fraction is in its simplest form), and $b \neq 0$.
   • Thus, we have: $\sqrt{2} = \frac{a}{b}$

2. Squaring Both Sides:

   • Square both sides of the equation to eliminate the square root: $2 = \left(\frac{a}{b}\right)^2$ $2 = \frac{a^2}{b^2}$

3. Clear the Denominator:

   • Multiply both sides by $b^2$ to get rid of the denominator: $2b^2 = a^2$
   • This equation implies that $a^2$ is even because it is equal to $2b^2$, which is clearly even.

4. Conclude $a$ is Even:

   • Since $a^2$ is even, $a$ must also be even (because the square of an odd number is odd).
   • If $a$ is even, then $a$ can be expressed as $2k$ for some integer $k$.

5. Substitute and Simplify:

   • Substitute $a = 2k$ into the equation $2b^2 = a^2$: $2b^2 = (2k)^2$ $2b^2 = 4k^2$ $b^2 = 2k^2$
   • This equation implies that $b^2$ is even, so $b$ must also be even.

6. Contradiction:

   • Since both $a$ and $b$ are even, they have a common factor of 2, which contradicts our initial assumption that $\frac{a}{b}$ is in its simplest form (i.e., $a$ and $b$ have no common factors other than 1).

7. Conclusion:

   • The assumption that $\sqrt{2}$ is rational leads to a contradiction. Therefore, $\sqrt{2}$ cannot be rational and must be irrational.

This proof elegantly demonstrates the power of proof by contradiction, showing that the assumption of rationality for $\sqrt{2}$ is inherently flawed, thereby confirming its irrational nature.