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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.


Age 16 to 18 Challenge Level:

20 point star
To produce this star, twenty line segments of equal length are drawn in a continuous path, with equal angles between consecutive line segments.

Imagine instructing a small creature to walk along the path. You would give the instruction to walk forward a certain distance then to turn through a certain angle and to repeat the instruction over and over again.

To do this, you could use the Logo commands:
repeat 20 [forward 100 right $\theta$]

Experiment with the Logo program
repeat $q$ [forward 100 right $\theta$]

What shapes can you draw? Vary $q$ and $\theta$. For what values of $\theta$ can you find closed paths (returning to the starting point)?

Prove that the path is closed if and only if $\theta$ is a rational multiple of 360 degrees.

Compare this property to the results found in the problem Stars.