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Why is √2 irrational?

Published in Irrational Numbers 2 mins read

√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:

  1. 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).
  2. Square both sides: (√2)² = (a/b)² which simplifies to 2 = a²/b².
  3. Rearrange: 2 = . This tells us that is even (since it's equal to 2 times something).
  4. If is even, then a must also be even. We can therefore write a = 2k, where k is an integer.
  5. Substitute a = 2k back into the equation: 2 = (2k)² which simplifies to 2 = 4.
  6. Divide both sides by 2: = 2. This tells us that is also even.
  7. If 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.