Proof by Contradiction Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Proof by Contradiction.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Proof by contradiction assumes the negation of the target claim and derives an impossibility.

Assume the opposite of what you want to prove, then follow the logic to a statement that is impossibly false — proving your assumption must have been wrong.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: A contradiction in assumptions validates the original statement.

Common stuck point: Students derive a surprising result, but not an actual contradiction.

Sense of Study hint: State the negated assumption clearly and identify what counts as impossible before starting.

Worked Examples

Example 1

medium
Prove by contradiction that \sqrt{2} is irrational.

Solution

  1. 1
    Assume for contradiction that \sqrt{2} is rational. Then \sqrt{2} = \frac{a}{b} where a, b are integers with \gcd(a,b)=1.
  2. 2
    Squaring: 2 = \frac{a^2}{b^2}, so a^2 = 2b^2. This means a^2 is even, so a is even. Write a = 2c.
  3. 3
    Then 4c^2 = 2b^2, so b^2 = 2c^2, meaning b is also even.
  4. 4
    But if both a and b are even, \gcd(a,b) \ge 2, contradicting \gcd(a,b)=1.

Answer

\sqrt{2} \text{ is irrational.}
Proof by contradiction assumes the negation of what we want to prove and derives a logical impossibility. The contradiction shows the original assumption must be false.

Example 2

medium
Prove by contradiction that there are infinitely many prime numbers.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

hard
Prove by contradiction: There is no largest integer.

Example 2

medium
Prove by contradiction that there is no smallest positive rational number.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

contradictionlogical statementdirect proof