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
The expanded form of Euler’s Identity gives a hint to what is going on:
e^ix = cos x + i sin x
If you plug pi for x, you get the form you know. Cosine(pi) is -1 and Sine(pi) is 0.
Now think about a 2-dimensional plane. Normally, we’d label the axes as x and y. But we could just as easily label them ‘real’ and ‘imaginary’. So the point you think of as (2,3) could also be considered 2 + 3i.
The advantage of doing this is that multiplying those coordinates represented by complex numbers performs a rotation (as well as a linear scaling – we’ll be using the unit circle to make the scaling remain 1).
For example, if you have an angle of 30 degrees with a magnitude of 1 that’s the same as coordinates (sqrt(3)/2, 0.5) – or sqrt(3)/2 + 0.5i. We know this because that (cos(30),sin(30)) or cos(30) + i sin(30).
If we want to rotate that point another 30 degrees, we can simply add the angles. Our magnitude will remain 1, but now we have 30 + 30 = 60 degrees. Using the complex plane, we can also multiply the complex representation of our coordinate by itself to get 0.5 + sqrt(3)/2i – which happens to be cos(60) + i sin(60).
Now, exponentiation is just repeated multiplication. Moreover, we can convert the base of any exponent to the base of any other exponent by the use of a constant factor.
So if we start with the form b^ix, what we’re really doing is expressing a set of coordinates in 2-dimensional space where x represents some amount of ‘rotation’ of the basic vector b^i. Depending on what we choose for b, it’s simply a constant factor of a logarithm off of being converted to any other base. That magnitude will be a scaling factor (just like multiplying rectangular coordinates by a constant will push those coordinates away from the origin by the constant factor).
If you’re willing to take a bit of a leap of faith, it turns out that e is the base we need to use to avoid scaling. So we can represent any angle by e^ix, where x is the angle in radians.
Euler’s Identity just expresses that these two forms – the complex rectangular coordinates and the ‘phasor’ form – are the same point on the plane.
The way it’s customarily presented at the secondary school level just involves plugging pi in for x. Which is the same as saying that the point on a unit circle 180 degrees counter-clockwise is (-1,0).
Edit: Changed Cosine/Sine values for pi to reflect correct values.
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