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`radius vector' from 0 to z(t) and has length 1. Therefore the solution z(t) always lies in the unit circle, and the point z(t) moves round this circle with angular velocity 1 measured in radians per second. (The radian measure of an angle is the length of the arc of the unit circle corresponding to that angle.) The circumference of the unit circle is 2p, so t ¼ p is halfway round the circle. But halfway round is visibly the point z ¼ À1. Therefore eip ¼ À1, which is Euler's formula.

All the ingredients of this proof are well known, but the overall package seems not to get much prominence. Its big advantage is to explain why circles (leading to p) have anything to do with exponentials (defined using e). So given the right

........................................... background, Euler's formula ceases to be mysterious.

Your Call May be Monitored for Training Purposes `The number you have dialled is imaginary. Please rotate your

........................................... phone 90 degrees and try again.'

Archimedes, You Old Fraud! `Give me a place to stand, and I will move the Earth.' So, famously, said Archimedes, dramatising his newly discovered law of the lever. Which in this case takes the form

Force exerted by Archimedes

6distance from Archimedes to fulcrum

equals

Mass of Earth6distance from Earth to fulcrum

The fulcrum is the pivot ­ the black triangle in the picture:

two page view?

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