In Section 06 (Complex Numbers), we saw that a complex number on the Argand diagram can be expressed in Cartesian coordinates as [latex]x+iy[/latex], or alternatively in polar coordinates [latex](r,\theta)[/latex] such that [latex]x=r\cos{\theta}, \;y=r\sin{\theta}[/latex]. (If that sentence makes no sense to you, you should really go and have a look at Section 06…).
For example, in the picture below, the complex number represented by the red dot can be expressed as [latex]3+4i[/latex] or alternatively as [latex]5(\cos{53.1^{\circ}}+i\sin{53.1^{\circ}})[/latex].
On this page we will extend that understanding, and arrive at Euler’s formula, named after Leonhard Euler, an 18th century Swiss mathematician. Then we will explain just why scientists love it so much (there are at least two reasons, but they are related, so maybe it’s one big reason…).
Arriving at Euler’s formula
To do this, we are going to need some knowledge of the series expansions of the sine, cosine and exponential functions. But that’s fine, because you have obviously memorised Section 07 – Power series!
Well, just in case you haven’t, here they are again:
[latex]\sin{x}=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\cdots=\sum_{k=0}^{\infty}\frac{(-1)^kx^{2k+1}}{(2k+1)!}[/latex]
And
[latex]\cos{x}=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\cdots=\sum_{k=0}^{\infty}\frac{(-1)^kx^{2k}}{(2k)!}[/latex]
And
[latex]e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots+\frac{x^n}{n!}+\cdots=\sum_{k=0}^{\infty}\frac{x^n}{n!}[/latex]
Again, if your reaction is “I’ve never seen that before!”, then you should go and have a look at Section 07 before you carry on…
You might notice that the three expansions look quite similar. In particular the sine and cosine expansions appear to have alternate terms from the [latex]e^x[/latex] expansion. It’s almost as though we could add the sine and cosine expansions to produce the exponential function. It’s not quite as straightforward as that though, because in the trigonometric expansions, alternate terms are negative, whereas in the exponential series, all the terms are positive.
You’d be right, though, to think that the three functions must be intimately connected. In the video below, we’ll work out the relationship between them.
So, to summarise the video,
[latex]\displaystyle{e^{i\theta}=\cos{\theta}+i\:\sin{\theta}}[/latex]
[latex]e^{i \theta}[/latex] is called a complex exponential. (It’s an exponential function with a complex term in the power). And this is Euler’s formula.
So, big deal? Well, yes, actually…
Why is Euler’s formula so important? Part 1
We hinted at the answer to this question in the section on complex numbers. We said there:
“Scientists love complex numbers because:
- Quantities can be described as complex numbers
- Maths can be ‘done to them’ – and the maths is often easier than if we tried to avoid the complex numbers and do it some other way.
- And then the ‘real’ part of the answer can be retained and the imaginary part thrown away.
- And somehow, the answer describes what happens.
- The universe seems to know that physical quantities that are quantified by real numbers can be more simply dealt with by using complex numbers.”
Now you can see what we meant there.
- Number 1 means that we can take a sine or (cosine) function, and rewrite it as a complex exponential. A sine becomes the imaginary part of the complex exponential, and a cosine becomes the real part.
- Number 2 means that it is often easier to ‘do stuff’ to exponentials than to trigonometric functions. For example, it’s really easy to differentiate exponentials… (see section 05 if you can’t remember why)
- Number 3 means that if you get an answer in the form of a complex exponential, then you can write it in the form [latex]\cos{\theta}+i\:\sin{\theta}[/latex]. And then you can take the real or imaginary part, depending on which one you need (which might be linked to step 1).
- And number 4 and 5 are just magic…!
In fact, in some situations there are two related real quantities, both of which vary sinusoidally but quarter of a revolution out of phase (like sine and cosine). We can then encode information about both quantities in a single complex quantity. By taking the real or imaginary part of the complex quantity, we regain the two separate quantities. AC magnetic susceptibility is an example of this. Don’t worry – we aren’t expecting you know what that is. But if you meet it, maybe you’ll be reminded of this page and this message…
Why is it so important? Part 2
We are used to the Cartesian and polar representations of complex numbers.
But Euler’s equation gives us a third… Well maybe it’s not really a third – maybe it’s just an extension of the polar representation, but it’s still pretty useful – so here it is.
We have already said that [latex]3+4i[/latex] can be written as [latex]5(\cos{53.1^{\circ}}+i\:\sin{53.1^{\circ}})[/latex]. And now we know from Euler’s formula that we can write:
[latex]\cos{53.1^{\circ}}+i\:\sin{53.1^{\circ}}=e^{(i\times53.1^{\circ})}[/latex]
And that means that [latex]3+4i[/latex] can also be written as [latex]5e^{i\times53.1^{\circ}}[/latex].
Here are three more diagrams.
The diagram on the left shows the complex number [latex]e^{i\theta}[/latex]. It has a modulus (magnitude) of 1. That is, the vector representing it on the complex plane has a radius of 1.
The middle diagram shows the complex number [latex]e^{i(\theta+\alpha)}[/latex], because the angle of the purple arrow from the real axis is [latex]\theta+\alpha[/latex]. The purple arrow looks like the red arrow has been rotated by an angle [latex]\alpha[/latex]. On a seemingly different note, the way that powers multiply is that you add the powers, which means that [latex]e^{i(\theta+\alpha)}[/latex] is the same thing as [latex]e^{i\theta}\times e^{i\alpha}[/latex].
Now read the previous paragraph again – it contains two ideas. And the connection between the ideas means that multiplying by [latex]e^{i\alpha}[/latex] is equivalent to rotating an arrow on the complex plane by an angle [latex]\alpha[/latex]. This might not seem that useful at the moment, but we will use this result in Section 13 when we look at circular motion and oscillations.
And finally the third diagram shows a blue dot rotating round the orange circle at a constant rate. If the rate of rotation is [latex]\omega[/latex] amount of angle per unit time, then the angle [latex]\theta[/latex] is given by [latex]\theta=\omega t[/latex]. And the position of the blue dot on the complex plane at time [latex]t[/latex] is given by [latex]e^{i \omega t}[/latex]. This kind of representation of a rotating object occurs all the time in science. Some people call it a phasor. It’s a rotating vector in the complex plane, but it can be used to describe real rotating/oscillating phenomena, such as vibrating objects, circular motion, and waves. We’ll see how phasors can be used in these contexts in Sections 13 and 14.
Possibly more famous than Euler’s formula (in ‘popular maths’ books anyway) is Euler’s identity: [latex]e^{i\pi}+1=0[/latex]. It’s famous because it links arguably the five most important numbers in maths in one identity! And it is pretty amazing that they are linked in one identity!
However, Euler’s identity is just one special case of Euler’s formula. And the formula is much more useful…
Matt Parker at standupmaths does a great job of explaining this in his YouTube video “What does i^i = ?“
Link to Section 05: What ‘exponential’ means
If you need a reminder of the basics of complex numbers, this is the place to go.
Link to Section 06: Complex numbers
If you need a reminder of the basics of complex numbers, this is the place to go.
Link to Section 07: Power series
This page has assumed a knowledge of power series. If you need a reminder, try this page.
Link to Section 13: Circular motion and phasors
Where we will show how complex exponentials can be a very useful way to represent rotating and oscillating phenomena.
Link to Home
And you can always go here…