You may also like
Tetra Inequalities
Can you prove that in every tetrahedron there is a vertex where the three edges meeting at that vertex have lengths which could be the sides of a triangle?
Staircase
Solving the equation x^3 = 3 is easy but what about solving equations with a 'staircase' of powers?
Proof Sorter - the Square Root of 2 Is Irrational
Try this interactivity to familiarise yourself with the proof that the square root of 2 is irrational. Sort the steps of the proof into the correct order.
Age 16 to 18
Challenge Level
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?
Copyright © 1997 - 2023. University of Cambridge.
All rights reserved.
NRICH is part of the family of activities in the Millennium Mathematics Project.
NRICH is part of the family of activities in the Millennium Mathematics Project.