Proof by Contradiction: Definition & Examples

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  • 0:01 Proof by Contradiction
  • 1:18 Steps
  • 1:42 Examples
  • 5:33 Lesson Summary
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Lesson Transcript
Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

One of several different ways to prove a statement in mathematics is proof by contradiction. Learn the definition of this method and observe how it is applied to proving a statement's truth value through examples and exploration.

Proof By Contradiction

In the book A Mathematician's Apology by G.H. Hardy (pictured below), he describes proof by contradiction as 'one of a mathematician's finest weapons.' He went on to say, 'It is a far finer gambit than any chess play: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game.'

Sometimes, when it is difficult or downright impossible to prove the truth of a statement directly, we may turn to proof by contradiction. We know that a statement cannot both be true and false - it has to be one or the other. Proof by contradiction uses this fact to prove something is true by showing that it cannot be false. When proving something is true using proof by contradiction, you assume the statement to be false, and as you proceed with the proof, you run into a contradiction, making the assumption of your original statement being false impossible, thus it must be true.

When deciding if proof by contradiction is the best way to prove a given statement, it is a good idea to ask yourself, what would happen if the statement weren't true? If the result of the statement not being true leads to something that makes no sense, such as 1 = 5 or p is even and p is odd, then proof by contradiction is a good way to proceed.

Steps

To clarify the process of proof by contradiction further, let's break it down into steps. When using proof by contradiction, we follow these steps.

  1. Assume your statement to be false.
  2. Proceed as you would with a direct proof.
  3. Come across a contradiction.
  4. State that because of the contradiction, it can't be the case that the statement is false, so it must be true.

Examples

A very common example of proof by contradiction is proving that the square root of 2 is irrational. Before looking at this proof, there are a few definitions we will need to know in order to understand the proof:

  • Even number: a number m that can be written as m = 2n where n is an integer
  • Odd number: a number r that can be written as r = 2s + 1, where s is an integer. For example, 14 is an even number because 14 = 2 * 7. Similarly, 101 is an odd number because 101 = 2 * 50 + 1. Notice that if an integer is not even, then it is odd. The reverse is also true. If an integer is not odd, then it is even.
  • Rational number: a number that can be written as p/q where p and q are integers. For example, 3 and 0.9 are rational numbers because we can write 3 as 3/1 and we can write 0.9 as 9/10.

Now, let's take a look at this proof that the square root of 2 is irrational.

Statement: The square root of 2 is irrational.

Proof by contradiction:

Assume not. That is, assume that the square root of 2 is rational. Then the square root of 2 can be written as p/q where p and q are integers, and p/q is reduced as much as possible (p and q don't share any common factors). Observe:

proof that square root of 2 is irrational - 1

Therefore, p^2 is even, so p is even. Since p is even, the following holds true:

proof that square root of 2 is irrational - 2

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