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The horizontal axis is the real numberline. The vertical axis is
the imaginary number line. Since complex numbers could be
treated as points on a graph it made them amenable for analysis by using geometry and trigonometry. It wasn’t long before
those branches of mathematics shed light on useful complex
numbers could be.
The most beautiful equation
Swiss mathematician Leonhard Euler (1707 - 1783) studied
complex numbers. Euler was aware that many functions could
be represented by infinitely long series of powers. For example
the exponential function e^x, which describes rapid (exponential) growth can be calculated by adding powers of x together.
Using the type of mathematical manipulation that is routine at
A-Level, he was able to show power series for sine and cosine
(from trigonometry) could combine with the imaginary unit to
give a power series for the exponential function. Euler uncovered the following relationship:
Here the symbol ? represents the angle that the line to the
complex number makes to the horizontal axis when it’s plotted
on the graph. Euler’s incredible equation links two previously
unconnected types of function: the exponential and trigonometric functions. The exponential function grows and grows.
Sine and cosine functions are oscillating waves. There was
no reason to think they should be related before complex
ICY SCIENCE | WINTER 2013- 2014