gen-briefly-explain-proof-by-contradiction-and-one-common-miscon-2a94

Briefly explain proof by contradiction and one common misconception about it.

# Proof by Contradiction **The method:** To prove a statement P is true, you assume P is false (assume "not P"), then show this assumption leads to a logical contradiction. Since the assumption produced an impossible result, P must be true. **Example:** Proving √2 is irrational - Assume √2 IS rational (can be written as a/b in lowest terms) - Then 2 = a²/b², so a² = 2b² - This means a² is even, so a is even - If a is even, then a² is divisible by 4, making b² even, so b is even too - But if both a and b are even, they weren't in lowest terms—contradiction! - Therefore √2 must be irrational ## Common Misconception **The fallacy of "proving" by showing what you want is true:** Some people think proof by contradiction means "assume the opposite, then show the original statement is true." This is circular reasoning! Instead, you must derive a *contradiction* (like "2 = 3" or "a number is both even and odd")—something impossible within the logical system itself. You don't just re-demonstrate what you wanted to prove; you show the opposite assumption breaks mathematics.