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Be Reasonable

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

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The Root Cause

Prove that if a is a natural number and the square root of a is rational, then it is a square number (an integer n^2 for some integer n.)

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Good Approximations

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


Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

First prove that the line segments are chords of a circle. Then prove that the angle of turn between one chord and the next is equal to the angle subtended by the chord at the centre of the circle.

You do not need to have met the Logo programming language previously in order to do this question. You can download the free MSWlogo program from

If you have not used Logo before then, from the first part of this simple introduction to Logo for beginners , you can learn enough in five minutes to use Logo to experiment in drawing the spirostars.

Try repeat 1000 [fd 200 rt 360*sqrt 2]

What happens?