A deductive proof using contradiction or elimination to rule out all cases except the desired conclusion.
An indirect proof is a way of proving something is true by showing that the opposite cannot be true.
Instead of building a direct path to the conclusion, we assume the conclusion is false and follow the logic until we reach a contradiction, something that breaks a known rule or fact. Since the assumption led to an impossibility, it must be wrong, which means the original conclusion must be right.
Here is a simple example: Suppose we want to prove that there is no largest even number. We assume the opposite: that there is a largest even number, call it n. But if n is even, then n + 2 is also even and larger than n, which contradicts our assumption. So no largest even number can exist.
Indirect proof comes in two common forms:
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Proof by contradiction: assume the opposite of what we want to prove, then show that assumption leads to a contradiction.
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Proof by elimination: consider all possible cases and rule out every one except the desired conclusion.
Both approaches reach the same destination; they just take different routes to get there.
When Do Students Learn About Indirect Proof?
Students build toward indirect proof through logical reasoning and an understanding of mathematical rules and properties.
Grades 6–8 – Logical Reasoning and Counterexamples
Students begin reasoning about what must be true or false in math, using counterexamples and elimination — early steps toward indirect proof thinking.
Grades 9+ – Formal Indirect Proof in Geometry and Beyond
Students encounter indirect proof formally in geometry and advanced mathematics, writing structured proofs by contradiction and elimination.
