This is true for a frictionless spring. The force applied by the spring is F = -kx where k is a constant and x is the displacement. By writing Newton''s law of dynamics on the x axis, m a = F which means mx'''' = -kx, ie

x'''' + (k/m) x = 0

This differential equation is solved like this: first, analyze the associated equation : r² + (k/m) = 0 whose solutions are

r1 = i * sqrt(k/m)
r2 = - i * sqrt(k/m)

Maths tell us that solutions are of the form x = A * exp(r1 * t) + B * exp(r2 * t), where A and B are constants. Which means that if A = 1 and B = 0, then x = exp(r1 * t) will be a solution. Which is why it works.

And this is why each time we see x'''' + w²x = 0 we write x = X cos(wt + phi)...



There are even more magical things in maths... For instance, consider a sequence Un = card { (a,b) in Z² | a² + b² <= n }
What does Un / n tend towards when n teds towards infinity?
Try it out, you''ll be surprised :-)

And if you takle 2 random positive integers, the chance of their gcd being 1 is equal to pi² / 6 (don''t remember how to prove that, though).





ToohrVyk
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2 things (I''ve mostly missed the boat though ):

1)
i*i = j*j = k*k = ijk = -1

This equation has no solutions if you restrict yourself to using complex numbers. (However it is part of the definition for quaternions)

2)
There was a time when x = 4 - 5 had no solution. Then someone decided that negative numbers would fix it. Then there was a time when 4 * x = 5 had no solution, so someone decided that fractional numbers would fix it. Then there was a time when x * x = 2 had no solution, so someone decided that irrational numbers would fix it. Then there was a time when x * x = -1 had no solution, so someone decided that complex numbers would fix it. Complex numbers are just an extension of the numbering system, but we generally find it a bit hard to start with because you can''t imagine what an imaginary number looks like - all the other numbers have been 1 dimensional, and now this is 2 dimensional.

I''m not sure how useful this would be, but the idea that an irrational number could be thought of as 2 dimensional is possible:

a + b * n where n = Sqr(3)

It wouldn''t have all the nice properties of having n*n = -1, but it is still a nice analogue.

Trying is the first step towards failure.