An introduction to proof by contradiction, a powerful method of mathematical proof.

Show that there are infinitely many rational points on the unit circle and no rational points on the circle x^2+y^2=3.

The symbol [ ] means 'the integer part of'. Can the numbers [2x]; 2[x]; [x + 1/2] + [x - 1/2] ever be equal? Can they ever take three different values?

Solving the equation x^3 = 3 is easy but what about solving equations with a 'staircase' of powers?

Prove that sqrt2, sqrt3 and sqrt5 cannot be terms of ANY arithmetic progression.

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.

We are given a regular icosahedron having three red vertices. Show that it has a vertex that has at least two red neighbours.

Prove that in every tetrahedron there is a vertex such that the three edges meeting there have lengths which could be the sides of a triangle.