{"id":63694,"date":"2026-06-16T18:17:18","date_gmt":"2026-06-16T10:17:18","guid":{"rendered":"https:\/\/blog-admin.thethinkacademy.com\/?p=63694"},"modified":"2026-06-16T18:17:21","modified_gmt":"2026-06-16T10:17:21","slug":"math-proof-by-contradiction","status":"publish","type":"post","link":"https:\/\/blog-admin.thethinkacademy.com\/blog\/2026\/06\/16\/math-proof-by-contradiction\/","title":{"rendered":"Math Proof by Contradiction: Classic Examples and How It Works (Euclid Contest Guide)"},"content":{"rendered":"\n

Proof by contradiction is one of the most powerful tools in a mathematician’s toolkit, and one of the least intuitive when students first encounter it. The idea \u2014 assume the opposite of what you want to prove, then show that assumption leads to something impossible \u2014 feels backwards at first. But it is the standard technique behind some of the most famous results in mathematics, and it appears regularly in Euclid Contest<\/a> problems that ask students to prove something cannot be true. This guide explains the method clearly, works through the classic examples, and shows how the technique applies to Ontario Grade 11\u201312 math and Euclid Contest preparation.<\/p>\n\n\n\n

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What is proof by contradiction?<\/h2>\n\n\n\n

Proof by contradiction (also called indirect proof, or reductio ad absurdum) is a method of proving a statement by assuming it is false and showing that this assumption leads to a logical impossibility.<\/p>\n\n\n\n

The structure is always the same:<\/p>\n\n\n\n

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  1. Assume the opposite<\/strong> of what you want to prove<\/li>\n\n\n\n
  2. Reason logically<\/strong> from that assumption<\/li>\n\n\n\n
  3. Reach a contradiction<\/strong> \u2014 something that is impossible, or that contradicts the original assumption itself<\/li>\n\n\n\n
  4. Conclude<\/strong> that the assumption must be false, which means the original statement must be true<\/li>\n<\/ol>\n\n\n\n

    If a statement takes the form “If P, then Q,” proof by contradiction works by assuming P is true and Q is false, then deriving a contradiction. If a statement is simply “X is true,” the method assumes “X is false” and works from there.<\/p>\n\n\n\n

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    Why proof by contradiction works<\/h2>\n\n\n\n

    This method relies on a basic principle of logic: a statement and its negation cannot both be true. If assuming a statement is false leads to something impossible, the only way to resolve the contradiction is to conclude the original statement must be true after all.<\/p>\n\n\n\n

    Proof by contradiction is particularly useful for proving:<\/p>\n\n\n\n