## Proof By ContradictionIt is sometimes difficult (or impossible) to prove that a conjecture
is true using direct methods. For example, to show that the square
root of two is irrational, we cannot directly test and reject the
infinite number of rational numbers whose square might be two. Instead,
we show that the assumption that root two is rational leads to a contradiction.
The steps taken for a proof by contradiction (also called - Assume the opposite of your conclusion.
- For “the primes are infinite in number,” assume that
the primes are a finite set of size
*n*. - To prove the statement “if a triangle is scalene, then no two of its angles are congruent,” assume that at least two angles are congruent.
- For “the primes are infinite in number,” assume that
the primes are a finite set of size
- Use the assumption to derive new consequences until one is the opposite
of your premise. For the two examples above, you would seek to establish:
- that there exists a prime not counted in the initial set of
*n*primes. - that the triangle cannot be scalene.
- that there exists a prime not counted in the initial set of
- Conclude that the assumption must be false and that its opposite (your original conclusion) must be true.
Why does this method make sense? One way to understand it is to note
that you are creating a direct proof of the contrapositive
of your original statement (you are proving Note that the contradiction forces us to reject our assumption because our other steps based on that assumption are logical and justified. The only “mistake”that we could have made was the assumption itself. An indirect proof establishes that the opposite conclusion is not consistent with the premise and that, therefore, the original conclusion must be true. ## Opposites
Sometimes, it can be a challenge determining what the opposite of a
conclusion is. The opposite of “all
## ResourcesSee Triangle with Restricted Angle Sum for a practice problem and Proof by Contradiction Class Activity for a lesson plan that introduces proof by contradiction. |

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