The logarithmic spiral has some very interesting properties and Bernoulli was especially fascinated by it.I’ll prove it’s most important property(the angle between the curve and the radius at every angle is constant) and proceed with an example.
In polar co-ordinates,the equation of the spiral is given by:
where are constants and
Now,to prove that any line from the origin which intersects the curve does so by making a constant angle(say ) with the curve(direction of tangent line),we consider the derivatives of the parameter equations which correspond to
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The irrationality of e can be proved using the infinite series expansion.
Assume that e is rational.i.e where and .
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