For those who don’t know, and correct me if I’m wrong, Euler’s Formula is e^(xi) = cos(x) + isin(x) and Euler’s Identity is where “x” is equal to pi, therefore e^(πi) = -1. It’s fascinating to me how something so complex can be equal to something so simple (talking about Euler’s Identity). But what I don’t get is how they’re practical. What are they used for, and why are they important?
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e-to-the-pi-i-equals-minus-one is just a pretty little consequence of the real “power tool”.
The real power is in exp(xi) = cos(x) + i*sin(x). This let’s you replace anything to do with rotation or oscillation with a simple algebraic formula.
Everything from the injector timings of an Internal Combustion Engine controlled dynamically by computer to alternating electrical current modelling is way way simpler this way.
Here’s a simple test: imagine you threw a hammer (eg. Thor’s Mijonir) upwards at 45-degress at 10 m/s and want to know if the head of the hammer or the handle will hit the ground? There’s an angular rate of rotation to consider! Try doing that with just cos() and sin()!!!
With exp(rateOfRotation * i * time) + ThorHeight you can solve it very very quickly and easily.
The identity is just kind of a beautiful fact. The formula can be used to work with complex numbers. It is why radians are so useful for describing angles.
Lots of things involve the unit circle which is what it describes.
You have several things in that formula:
* cos and sin are trigonometric functions, related to the circle, related to angles, very useful for geometry and also for “repeating patterns” a.k.a waves, frequencies, etc.
* i indicates complex numbers, and complex numbers can be used to represent “things” that may have two dimensions, such as vectors or, again, waves or frequencies. Physics and engineering use vectors a lot.
* e and pi are “special” numbers. Pi obviously relates to the circle, but also (again) to frequencies and waves, and e is used a lot in probability and statistics, as well as in “things that repeat” a certain way like arithmetic and geometric series.
So the applications can be broad. For example, the formula is used quite a bit for AC / power electricity calculations.
Pretty sure its used a lot in signal processing too.
Basically there’s a law where any signal, no matter how distorted or weird, it can be summed up as a series of added cosine and sinus waves with different periods and amplitudes.
So if you are looking for some specific signal you can essentially isolate it using math, which is super fucking useful for any kind of communication. Google fourier transform for more.
exp(i*x) is used in nearly every aspect of science and engineering. The general form is a solution to many differential equations, and it represents waveforms and signals. It also makes complex numbers neater to work with. Whenever there is modelling of real physical systems, it will almost always pop up. Physics (electromagnetics, quantum mechanics, etc.) engineering (power systems, structural design, etc.), it appears in almost every field that is math heavy.
The reason Euler’s formula is so important is because of its use in decomposing waveforms–i.e. fourier transforms.
Fourier transforms allow you form almost any realizable waveform into an infinite series of sines and cosines. Using Euler’s formula you compact those two functions into a single complex exponential function. Integrations and derivations of a simple exponential is MUCH easier than that of trigonometric functions.
e^πi = cos(x) + isin(x) can be used to present trigonometric functions as complex exponential functions.
sin(x) = ( e^(ix) – e^(-ix) ) / 2i
cos(x) = ( e^(ix) + e^(-ix) ) / 2
At first these may seem overly complex (heh). But when you have an equation that contains a combination of sin and cos functions being added and multiplied together and you want to simplify the mess they become very useful.
How do you even simplify trigonometric functions? They are so often treated as mysterious black boxes where you put numbers in and get other numbers out.
Sometimes you may be able to come up with some geometric reasoning for the simplification. But often the only practical way to do it is to use the complex exponents. (the other option is usually to use infinite series and hope that the sum/multiplication of those infinite series converges nicely)
Lets for example say that you have sin(x)*cos(x) in some formula that you want to simplify.
sin(x)*cos(x)
Substitute with complex exponentials.
= ( e^(ix) – e^(-ix) ) / 2i * ( e^(ix) + e^(-ix) ) / 2
Do the multiplication
= ( e^2ix + e^0 -e^0 -e^-2ix ) / 4i
= (1/2) * ( e^i(2x) – e^-i(2x) ) / 2i
Notice that this here is the complex exponent form of cosine
= (1/2) * cos( 2x )
Now we know that sin(x)*cos(x) = 0.5*cos(2x). Isn’t that nice.
This way you can simplify all sort of nasty trigonometric function containing statements. For example:
sqrt( 1-sin^(2)(x) ) = cos(x)
( 1 – 4 * sin^(2)(x) ) * cos(x) = cos(3x)
2 * cos( x/2 + y/2 ) * sin( x/2 -y/2 ) = sin(x) – sin(y)
The short answer is that it acts as a bridge between differential equation and trigonometry. It lets you use the tools of the one on the other and vice versa. It’s similar to the way DeCartes’ graphs were a bridge between algebra and geometry.
Let me copy a comment I mad recently in another thread here:
Already the formula
e^(ix) = cos(x) + i·sin(x)
can make your life much easier. For example,
e^(ix) · e^(iy) = e^(i·(x+y))
by a simple power law. But expand that using the formula you get
(cos(x) + i·sin(x)) · (cos(y) + i·sin(y)) = cos(x+y) + i·sin(x+y).
Now expand the left hand side and compare real/imaginary parts and you get
cos(x+y) = cos(x)·cos(y) – sin(x)·sin(y),
sin (x+y) = cos(x)·sin(y) + sin(x)·cos(y).
Hence the rather complicated two(!) laws of addition for (co)sine are the same as a simple power law when described via complex numbers.
Another consequence of the formula is
cos(x) = (e^(ix) + e^(-ix)) / 2,
sin(x) = (e^(ix) – e^(-ix)) / 2i.
This not only tells you how to define (co)sines of absolutely any complex number x, but also give you a way to calculate arcsin and arccos! For example, setting y = e^(ix) and thus x = log(y)/i, lets calculate an x such that cos(x)=a:
By the formula, we want 2a = e^(ix) + e^(-ix) = y + 1/y, i.e. after multiplying by y and getting everything to one side y² – 2a·y +1 = 0. The formula for quadratic equations now tells you
y = a ± sqrt(a²-1)
and hence by x = log(y)/i we arrive at
arccos(a) = x = log(a ± sqrt(a²-1)) / i.
Lastly, you can combine the formula(s) with the one for geometric series to quickly calculate sums such as
sin(1°) + sin(2°) + … + sin(179°). Try it!
To give a deep mathematical application as well:
A number a is called _algebraic_ if there are integers (or rationals, by clearing denominators) c0, …, cn such that
cn·α^n + c(n-1)·α^(n-1) + … + c1·α + c0 = 0.
For example α=sqrt(2) is algebraic because α^2 – 2 = 0. As are the golden ratio, the silver ratio, 0, 1, 65537/257 or any other rational number. A number that is not algebraic is called _transcendental_. Very sloppily, those are the numbers that are the most complictaed from the point of view of Algebra.
**Lindemann’s Theorem** now states that **if α is algebraic and not 0, then e^α is transcendental!**
Now, we know that e^(i·π) = -1 is as algebraic as it gets, thus there is no way i·π is algebraic. But a short argument then shows that this means that π cannot be algebraic either.
tl;dr: Euler’s identity is the reason why π is not the root of any rational polynomial, i.e. **π is transcendental**.
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