How does Euler’s identity work?

Question

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

Answer 1

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).