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so the dervative of a function is a just a plug and chug. its how newtion and leibnitz found the instantanious velocity of a falling object and do it again then you have where it is at a certain time then nothing.

but before all of this in the 1400's people were solving things at the same time and not knowing it because of the communication gap.

so the euclidians , bruts that they were and secretive, came across imaginary numbers i.e. irrational, 1/3 = .333333333333, pi etc. namely a right triangle with two equal sides it follows that 1^2 + 1^2 = (the hypotenuse)^2 yielding sqrt of 2 a never ending number.

well then later some people all over the world started working on this so

the theory of algebra or a line y=a +mb so lets try a+ib i= any imaginary number or formof imaginary irattional function)

a + ib (a +ib) + (id)= i(b+d) which is the same thing as a real function or real number

so they made a real number out of imaginary ones in a rational way

ttthhheennnn

they mutiplied them

(a + ib)(c + id)= ac + ibc +aid+idib = ac +i(bc +ad)+i^2(bd) which is a simple basic foundation of algebra turning an imaginary number (over 350 years of proving) into a real quadratic equation where you can find the slope of an imaginary function and make it real.

this gave rise to those "power series" written at first and convergencce and divergence of these numbers and then mandelbrot with his fractals. pretty long time to come down to just that.

forgot..they took the inverse power series too as greeks found number first only through comparison x/b was the first types of real numbers. the whole evolution of it is amazing especially how far we have come in the past 100 years (actually 106). BUT newton and leibnitz, those guys knew how to party man. in their wont of a way to decribe how things worked they had to make up some pretty crazy stuff to explain everything "on the shoulders giants". the physics needed math to explain it. so the characters were less important then the part they played. yet how can a person be less important than the work they provided for us? yet...this is true. i would say THAT was the first thought of quantum theory...super position. they can both be true at the same time. d(newtons work) < (the nature of things mathematically) and d(newtons work) also> (the narture of things mathematically). Shroedingers newton.