√2 is irrational because its decimal representation is infinite, non-terminating, and non-repeating. This characteristic definitively classifies it as an irrational number.
Understanding Irrational Numbers
Irrational numbers cannot be expressed as a simple fraction p/q, where p and q are integers and q is not zero. Their decimal expansions go on forever without repeating.
√2's Decimal Expansion
- Non-terminating: The decimal expansion of √2 continues infinitely.
- Non-repeating: There is no repeating pattern in the decimal expansion.
Proof by Contradiction (Simplified)
While the above explains why √2 is irrational based on its decimal expansion, a classic proof involves contradiction:
- Assume √2 is rational. This means we can write it as √2 = a/b, where a and b are integers with no common factors (simplified to lowest terms).
- Square both sides: (√2)² = (a/b)² which simplifies to 2 = a²/b².
- Rearrange: 2b² = a². This tells us that a² is even (since it's equal to 2 times something).
- If a² is even, then a must also be even. We can therefore write a = 2k, where k is an integer.
- Substitute a = 2k back into the equation: 2b² = (2k)² which simplifies to 2b² = 4k².
- Divide both sides by 2: b² = 2k². This tells us that b² is also even.
- If b² is even, then b must also be even.
Contradiction: We've shown that both a and b are even. This contradicts our initial assumption that a/b was simplified to its lowest terms (meaning a and b have no common factors). Therefore, our initial assumption that √2 is rational must be false.
Conclusion
Since our assumption that √2 is rational leads to a contradiction, √2 must be irrational. The fact that its decimal expansion is non-terminating and non-repeating further supports this conclusion.