mathbase
Learn/indirect proofs/Classic Examples (Irrationality of √2)

Lesson subsection

Classic Examples (Irrationality of √2)

Read the explanation, try the on-paper prompts, then explain the idea in your own words. Use AI feedback as a mentor, not a shortcut.

10–20 min focusProof-first mindset

Best flow: read → think on paper → write a short explanation → refine with feedback.

Reading

Core explanation

One of the most famous contradiction proofs shows that √2 is irrational.

Goal: Prove √2 is not a rational number.

Assume the opposite: √2 = a/b in lowest terms (a and b have no common factors).

Then:

  1. Square both sides: 2 = a² / b² → a² = 2b².
  2. This implies a² is even → a is even → let a = 2k.
  3. Substitute: (2k)² = 2b² → 4k² = 2b² → b² = 2k² → b² is even → b is even.
  4. But if a and b are both even, the fraction was not in lowest terms.

This contradiction proves √2 is irrational.

Other famous contradiction proofs include:

  • Infinitely many primes.
  • √3, √5, √6 are irrational.
  • No rational solution to x² = 3.

These proofs show contradiction is essential for major results.

TL;DR — key idea

Classic contradiction proofs often assume a minimal or reduced form and then show that assumption violates its own conditions.

Try these in your notebook

Don’t skip this – writing proofs or explanations on paper is where most of the learning actually happens.

  • 1

    Explain in your own words where the contradiction arises in the proof that √2 is irrational. Then outline how the argument would look for √3.

Once you’ve sketched some ideas, summarize the main insight in the reflection box on the right.

Check your understanding

In 3–6 sentences, explain the core idea of this subsection as if you were teaching a friend who hasn’t seen it. Focus on the logic, not just the final statements.

AI is optional. Use it to spot gaps and sharpen your wording, not to replace your own thinking.