We have had good discussions on mathematical proofs, so I am planning to create a mathematical proof series that will discuss the basics such as direct proof, indirect proof, and proof by mathematical induction. But before I do that, let me continue with more examples of proof by contradiction.

Proof by contradiction, as we have discussed, is a proof strategy where you assume the opposite of a statement, and then find a contradiction somewhere in your proof. Finding a contradiction means that your assumption is false and therefore the statement is true. Below are several more examples of this proof strategy.

Example 1: irrational.

Example 2: is irrational. The proof of this is basically the same as example 1, so it is left as an exercise.

Example 3: Proof that there are infinitely many primes.

Example 4: Knights and Liars

Example 5: is irrational.

Proof*: *

* *Suppose the statement is false. Then there is a rational number such that . Now, squaring both sides we have . This means that

Now, from example 2, is irrational, and we know that is rational since is rational. This makes the left hand side of the equation irrational and the right hand side rational — a contradiction!

Therefore, our supposition is false, which implies that is irrational.

Example 6: If a line is tangent to a circle, then it is perpendicular to the radius at the point of tangency.

Proof:

Consider a circle with center , radius , and with line tangent to the circle at . We have to show that .

Suppose is not . Then there exists a segment which is . If so, it follows that (Why?). But lies outside the circle since is a tangent line. Therefore . But this contradicts the previous statement that . Hence, our supposition is false. Therefore, .

Mathematical proofs is one of the most difficult topics in mathematics, but you can only see a few examples in books. In the Introduction to Mathematical Proofs series that I am planning to write, I am going to give as many and as varied examples as possible. This series is intended to give an intuitive introduction to mathematical proofs for Grades 9-12 students.

*Credit: Example 5 was adapted from Numbers: Rational and Irrational by Ivan Niven*

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It’s refreshing to some some posts on proofs – I’m definately going to keep reading these posts. It’s quite a coincidence that only yesterday did I realise that the proof that a tangent to a circle is perpendicular to the radius would have to be by contradiction (it took me over a week to figure this out!).

I recently wrote two posts on proving a result which a friend of mine noticed about powers of 2 – the link to the first post is below. The secind post is less about proof and more about not noticing a major part in the first post.

http://danpearcymaths.wordpress.com/2012/03/25/exxploring-numbers-thinking-like-a-mathematician-and-proof/

Hi Dan. I read your post and I particularly like the proof by exhaustion. I can’t comment in your post though. It requires to have an account. Are you aware of it?