The Euler (pron. oiler) Equation must surely rate as one of the most elegant, beautiful and awe-inspiring formula in maths. Its consequences are diverse and shocking when you consider its simplicity.
The Euler Equation is often given by the following: etheta i = cos theta + i sin theta
Proof of the Euler Equation
Let us consider the function y = cos x + i sin x. Continuing to treat i like any other number, we have, by differentiation:
dy/dx = -sin x + i cos x = i(cos x + i sin x)
Hence dy/dx = iy
=> i dx/dy = 1/y
=> ix = ln y + c
But when x = 0, y = 1. So c = 0.
=> ix = ln y
=> y = eix
So cos x + i sin x = eix
It may be objected that we have nowhere defined the meaning of a number such as ez when z is complex; but the reader should not be deterred by such inhibitions! Indeed, the above paragraph may be regarded as providing, if not a definition, at least a reasonable exposition of the meaning of eix, making it consistent with the familiar processes of mathematics.
Consequences of the Euler Equation
One of the most fundemental consequences of the Euler Equation is shown by taking theta to equal pi radians. When this is done then the equation (once rearranged slightly) gives the following:
epi i + 1 = 0
This unites the five most important numbers in maths; pi, i, e, 1 and 0 into one relation and so the Euler Equation is taken as one of the most important points of unification. From this equation one gains a glimmer of how the whole of maths fits together.
This glimmer is reinforced if one takes the natural log of both sides of the equation (after subtracting one from both sides). This allows us to define the natural log of the negative numbers in the complex plane as follows:
ln -1 = i pi
A far reaching result which defines a whole family of hyperbolic (or modular) forms (strongly related to hyperbolic functions) based around two mutually perpendicular complex planes (represented commonly by Argand Diagrams) sharing no axii. These were linked to another, seemingly unrelated, part of maths known as eliptic curves by the Taniyama-Shimura theorum which (as the first part of the Langland's Programme) became part of the quest for a Grand Unified Mathematics and forms the basis for some of the most important maths today. Indeed, the famed proof of Fermat's Last Theorum by Andrew Wiles was in fact also the proof of Taniyama-Shimura and so became dually celebrated as a triumph over an amateur tease-artist and as the stabilisation of an increasingly shaky foundation of an entire branch of maths.
Not bad for such a simple equation.