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.
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
It’s because the numbers aren’t random. They define each other. Like saying “a kidney bean is a bean shaped like a kidney”.
“E” is a number that’s often used when defining the result of counting things to infinity.
A circle is just a polygon with infinite sides. And “Pi” deals with circles. So “Pi” and “E” are related.
“I” is a number that’s often used when dealing with a formula that goes up and down over and over and over. Like a wave. And the sine and cosine functions are waves. And the sine and cosine functions are built using circles. So “I” and “Pi” are related, and by extension, “I” and “E” are related.
And the way that they’re related is that e^(i*pi) = -1
The best one I’ve seen is on YT by 3Blue1Brown ([https://www.youtube.com/watch?v=v0YEaeIClKY](https://www.youtube.com/watch?v=v0YEaeIClKY))
Complex numbers are very counterintuitive. In this case, e^i corresponds to a circle in the complex plane. Multiplying i will always give you a point, but multiplies of pi happen to be on the real axis so that they produce a real number for the Euler’s identity.
Okay real simple now.
Imaginary numbers are no more imaginary than real numbers. Real numbers represent quantities. Imaginary numbers represent rotation.
There are many ways we can use imaginary numbers to represent rotation. We can use addition (a + bi). We can use multiplication (3i * 4i). We can also use exponentials ( e^ix ).
Lets discuss the exponential case.
Exponential numbers grow proportional to their own value. “e” in particular grows at a rate that is equal to its value. It’s easiest to think about a doubling population. Every year, take the current population and add it to itself. Its growth is proportional to its population.
But when we take that exponent to an imaginary number, it has the effect of turning that growth 90° sideways. What this results in, is a circle!
At 0° your growth is facing straight up
At 90° your growth is facing straight left
At 180° your growth is facing straight down
At 270° your growth is facing straight right
You just trace out a circle as you continue to grow.
That is what e^ix represents.
But how quickly does that travel around a circle?
Well, if we replace “x” with “90°”, it will rotate our circle by 90°.
If we replace “x” with “180°”, it will rotate our circle by 180°.
And so on and so forth.
But degrees are not really a “real” unit. We made it up because it’s convenient to split a circle into 360 really small chunks. The more natural unit is the “radian”, which is the angle you get when you trace out one radius length along the arc of the circle. It’s about 57°.
When we convert between degrees and radians, we find that 180° is about 3.14 radians, or what we call “pi”.
So e^iPI is a statement that says “rotate around the circle 180°”.
If you consider that you start at “1”, and you rotate around the circle by 180° (a half rotation), then the number you arrive at is going to be -1.
Any other number will land you somewhere on the circle. Most numbers will land you somewhere “complex” (consisting of a real and an imaginary component). At 0 and 180° you will have purely real numbers (1 and -1). At 90° and 270° you will have purely imaginary numbers (i and -i).
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