Rational and irrational numbers

Find the sum of this series of surds.

Can you express every recurring decimal as a fraction?

Can you work out the irrational numbers that belong in the circles to make the multiplication arithmagon correct?

Prove that sqrt2, sqrt3 and sqrt5 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.

What fractions can you find between the square roots of 65 and 67?

How many differently shaped rectangles can you build using these equilateral and isosceles triangles? Can you make a square?

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

A spiropath is a sequence of connected line segments end to end taking different directions. The same spiropath is iterated. When does it cycle and when does it go on indefinitely?