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numbers were discovered. It’s like finding out that two of your friends, who didn’t previously know each
other are actually related to each other. It’s difficult to convey how shocking that result must have seemed
to mathematicians at the time.
What follows from Euler’s equation is both trivial and profound. Trivial to demonstrate: when the angle ? is
180° (or ? radians in mathematical currency) the formula becomes
But the sine part disappears at this angle, and the equation simplifies to e^i?=-1. Rearranging this so that
all the terms are on the left side of the equation gives us one the most profound and beautiful mathemati-
cal results of all time
This is a single equation that captures the five most important numbers in mathematics. The Nobel prize-winning
physicist Richard Feynman (1918 - 1988) described it as “one of the most remarkable, almost astounding, formulas
in all of mathematics.”
Real applications for imaginary numbers
We’re almost at the end of this real and imaginary journey. Despite their name, imaginary (and complex) numbers
have found very real applications in science and engineering. For electrical engineers complex numbers are a useful
computational tool for dealing with frequencies and time varying voltages and resistances. You can find the imaginary unit at the heart of quantum mechanics in the Schrodinger equation. The most iconic image of 20th century
mathematics, the Mandelbrot set, is constructed from simple rules applied to complex numbers. My own research
background is image processing – particularly improving noisy radiological images. The techniques used in that field
(Fourier transforms) have imaginary numbers embedded within them.
We might not be able to imagine what the square-root of minus one looks like but we need it to fully capture of the
essence of reality
Words: Adrian Jannetta
ICY SCIENCE | WINTER 2013- 2014