Proving √2 is Irrational

Intro to Proofs · Axiom Academy

A classic proof by contradiction that demonstrates fundamental proof techniques Prove that is irrational — that it cannot be written as a fraction of two integers. We work the proof one step at a time; at each step, decide what comes next. Excellent work! You've completed a classic proof by contradiction. Here's what makes this proof so important: Proof by Contradiction Strategy: Assume the opposite of what you want to prove, then show this assumption leads to a logical impossibility. The Power of "Lowest Terms": Assuming is what makes the contradiction bite — without it, "both even" wouldn't conflict with anything. Even/Odd Properties: If n^2 is even, then n must be even (an odd number squares to an odd number). This simple fact is the engine of the whole proof. The Contradiction: Both p and q being even means they share a factor of 2, directly contradicting the lowest-terms assumption. Historical Significance: Known since ancient Greece, this was one of the first proofs that not every number is a fraction — a revolutionary insight. The same technique extends to , , and many other roots. Proof by contradiction remains one of the most powerful tools in all of mathematics!

This is the written version of the interactive lesson above. See the full Intro to Proofs course.