I have looked up countless explanations for this equation and none of them make sense to me. The equation just seems like putting random numbers that have no relation together and somehow producing a beautiful outcome. If I substituted pi for another real number, would the equation still produce 0? Just please explain, how does this all work?
In: Mathematics
if you substitute pi for any other number you will get a different result (unless that number is a multiple of pi)
first things first:
complex numbers are weird, but they can be viewed as a “2-dimensional plane” (think x-y-plane, with the real part being the x-part and the imaginary part being the y-part).
the key here is that exp (i * x) is related to good old sinus/cosinus on this x-y-plane (sin x = (exp(ix)-exp(-ix))/2i).
WHY this expression holds requires a bit more math and I think is not necessary here.
essentially exp (ix) is the function that tells you your position if you were “walking” on the unit circle (the circle with a radius 1 unit around the origin) for x units.
Now we know the circumference of this circle is 2 pi, so if you walk 2 pi units you’re back where you started.
after 1 pi you have walked “half the circle”. meaning if you start a the coordinate 1/0 (x-y-plane or also phrased “1 real, 0 imaginary”)
then you’ll be at -1/0 (drawing this circle helps visualizing it).
so exp( i 2 pi) = 1, exp (i 3 pi) -1, etc…
*With all this in mind you can formulate Eulers Identity the following way*
**”If you turn around, you look in the opposite direction”**
(because you turn 180° or 1 Pi and then your orientation goes from +1 to -1)
or
exp (i pi) = -1
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