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# Be Reasonable

Why do this problem?

It is an exercise in proof by contradiction.

Possible approach

First discuss proof by contradiction so that students appreciate how the logic of arguments by contradiction work. You can draw on this article on Proof by Contradiction.

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Age 16 to 18

Challenge Level

- Problem
- Getting Started
- Student Solutions
- Teachers' Resources

Why do this problem?

It is an exercise in proof by contradiction.

Possible approach

First discuss proof by contradiction so that students appreciate how the logic of arguments by contradiction work. You can draw on this article on Proof by Contradiction.

Then discuss the proof that $\sqrt 2$ is irrational.

The students can work with the interactivity
Proof Sorter and perhaps some of them might read
this article which was written by two undergraduates.

Key Question

If the difference between$\sqrt 2$ and $\sqrt 3$ is an integer
multiple of the common difference in an arithmetic series, and the
difference between $\sqrt 3$ and $\sqrt 5$ is also an integer
multiple of that common difference, can you use these two facts to
write down two expressions, eliminate the unknown common difference
and then find an impossible relationship?

Possible support

Proof Sorter and
article

Possible
extension

Can you prove that $\sqrt{1}$, $\sqrt{2}$ and $\sqrt{3}$
cannot be terms of ANY arithmetic progression?

Solve quadratic equations and use continued fractions to find rational approximations to irrational numbers.

Given that a, b and c are natural numbers show that if sqrt a+sqrt b is rational then it is a natural number. Extend this to 3 variables.