This is a staircase. Each stair is twice as big as the preceding one. The curve is the derivative and the sum of the boxes is called the Riemann integral. The same reasoning holds for rational numbers and algebraic numbers. If you are keen on mumbo-jumbo, you can say:
The derivative equals the function f=2*x (1)
If you make a similar staircase with rational numbers, and you want (1) to be sound, you have to replace "2" by "e". e is not a rational number, but a "transcendent" one.
Extending the staircase to algebraic or real numbers, won't give you very much new information. That is what I call 'Ockhams razor'. Stop when have got enough information! But I am just a metrologist, and not a "pure" mathematician.
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example:
1+1+½+1/2*3+1/2*3*4+.... will quickly converge to 2,7182818..., which is the logarithmic base e. As said afore, (e^x)'=e^x, that is quite handy. (e^îx)'=i*e^ix and e^ix=cos x+i*sin x. Thus we get a simple relation between e and pi. We can easily conclude why it is possible to construct a regular 17-gon, but not a 19-gon. It is so very easy that a child can do it.
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the sign ' means the derivative, that is the slope of the curve, but I am sure you knew that already
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