Sounds hard, doesn’t it… After all it has the work ‘complex’ in the name. It’s true that you wouldn’t necessarily try to teach this to a six year old (well, not many, anyway), but give it a go. These things have been represented by a really bad PR department, so don’t let the name ‘complex number’ put you off.
The numbers that we use to quantify real world quantities are called real numbers. So when you say that there are 3 squirrels in a tree, or that the speed limits on French autoroutes is 130 km/h or that the temperature of your freezer/ice box is -18.3 °C, then those numbers (3, 130, -18.3) are all real numbers. So complex numbers must be something different, that don’t quantify ordinary quantities. On this page, we’ll introduce you to what complex numbers are, and then in Sections 12 – 14, we’ll see examples of how they are so useful in science.
Why are they useful?
Having said that, a very brief description of the usefulness of complex numbers might motivate you to make it to the bottom of the page. So here goes… Scientists love complex numbers because it turns out that they make life easier, via the following little ‘magic’ trick:
- Quantities can be described as complex numbers (we know we haven’t described what that means yet)
- Maths can be ‘done’ to the complex numbers – and the maths is often easier than if we tried to avoid the complex numbers and do it some other way
- The ‘complex number answer’ can be turned back into an ‘ordinary’ number
- And somehow, the ‘ordinary number answer’ describes what happens in reality
The universe seems to know that physical quantities that are quantified by real numbers can be more simply dealt with by using complex numbers. It’s sort of a miracle, or witchcraft, or magic, or something…
Quadratic equations in science
OK, so we are not quite telling you what complex numbers are yet. But we are leading up to it, we promise. And to do so, we’ll revisit a situation we have used for several purposes so far – projectile motion. In the video below, we work out how long it takes an object thrown off a cliff to reach two different heights. If you can’t be bothered with the video, then:
- Shame on you – sorry, just kidding!
- There’s a summary afterwards
To summarise the video:
- When asked the question “at what time was the vertical displacement -50 m, we got two solutions. Solutions with real numbers. One of those real number solutions was a negative number, which at first sight looks odd when dealing with time, but we could still attach a physical meaning to it
- When asked the question “at what time was the vertical displacement +50 m, we got NO solutions. There never was a time at which the object was 50 m above its starting point, because its initial velocity was too small to allow it to reach that height. We didn’t seek a solution in complex number form (which can be achieved – we’ll see how further down the page). We just said that the lack of a real solution means that a vertical displacement of 50 m is never achieved.
Quadratic equations in mathematics
Because we came at this from a ‘science angle’, when we set up the quadratic equations [latex]t^2-2t-10=0[/latex] and [latex]t^2-2t+10=0[/latex], we had in mind a real, physical meaning for the equations. However, these equations might well be of general interest to people learning mathematics rather than science – they certainly don’t have to refer to projectile motion. They might refer to something else, or nothing at all and just exist for their own sake. So let’s play mathematics for a while, and see where that takes us.
Just to persuade you that we are being more general than before (and also, spoiler alert, because we are going to create graphs on the [latex]xy[/latex]-coordinate plane), we will change the variable from [latex]t[/latex] to [latex]x[/latex], so that we are going to try to solve:
[latex]\displaystyle x^2-2x-10=0[/latex]
and
[latex]\displaystyle x^2-2x+10=0[/latex]
These are examples of the general quadratic equation, [latex]ax^2+bx+c=0[/latex], where [latex]a[/latex], [latex]b[/latex], and [latex]c[/latex] are the coefficients of the terms in [latex]x[/latex]:
- [latex]a[/latex] is the coefficient of the [latex]x[/latex]-squared term
- [latex]b[/latex] is the coefficient of the linear [latex]x[/latex] term
- [latex]c[/latex] is a constant term that does not depend on [latex]x[/latex]
The fact that there is an [latex]x[/latex]-squared term, but no higher terms (so, for example, no terms in [latex]x^3[/latex], [latex]x^4[/latex] etc), is what makes the equations ‘quadratic’. If there were an [latex]x^3[/latex] term, it would be a ‘cubic’ equation; if there were no [latex]x^2[/latex] term, the equation would be linear. As we saw in the video, the solutions to quadratics equations are given by
[latex]\displaystyle x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}[/latex]
Where does the quadratic formula come from? You can derive it yourself if you are familiar with the technique of ‘completing the square’. Otherwise, if you wish, you can let someone else do it for you – try this one at Maths Is Fun.
The first of our equations is readily solved by this method, by substituting in the values:
- [latex]a=1[/latex]
- [latex]b=-2[/latex]
- [latex]c=-10[/latex]
and we saw this in the video. Just to recap, and taking the algebraic steps very slowly, we get:
[latex] \displaystyle x=\frac{2\pm \sqrt{(-2)^2-4\times 1\times (-10))}}{2\times1}[/latex]
[latex] \displaystyle =\frac{2\pm \sqrt{(4-(-40))}}{2}[/latex]
[latex] \displaystyle =\frac{2+ \sqrt{44}}{2} \;\;\;\;or\;\;\;\; \frac{2-\sqrt{44}}{2}[/latex]
[latex] \displaystyle =4.43 \;\;\;\;or\;\;\;\; -2.32 (3.\mathrm{s.f.})[/latex]
Trying the same approach for the second equation, [latex]x^2-2x+10=0[/latex], leads us to change the value of the coefficient [latex]c[/latex] from -10 to +10, so that now:
- [latex]a=1[/latex]
- [latex]b=-2[/latex]
- [latex]c=10[/latex]
And this gives:
[latex] \displaystyle x=\frac{2\pm \sqrt{(-2)^2-4\times 1\times 10)}}{2\times1}[/latex]
This looks superficially similar to our previous working, but the change from -10 to 10 in the square root makes an enormous difference, because we end up requiring the square root of a negative number.
[latex] \displaystyle =\frac{2\pm \sqrt{(4-40)}}{2}[/latex]
[latex] \displaystyle =\frac{2 \pm \sqrt{-36}}{2}[/latex]
There is no real number that, when squared, gives -36. The reason for this is that any real number when multiplied by itself gives a positive result, whether it is itself positive or negative. So, for example [latex]4^2=4\times4=16[/latex], and also [latex](-4)^2=(-4)\times(-4)=16[/latex]. In turn, this means that the equation [latex]x^2-2x+10=0[/latex] has no real solutions, because there is no real number that ‘makes it work’.
The square root term in the quadratic formula then dictates whether a quadratic has real solutions; the mathematical difference between the two equations above is that:
- In the first, [latex]b^2-4ac>0[/latex] and
- In the second, [latex]b^2-4ac<0[/latex].
The quantity [latex]b^2-4ac[/latex] is sometimes called the ‘discriminant of a quadratic equation’, because it discriminates between equations having real solutions, and those not having real solutions.
So far, we have visualised the lack of a real solution to certain quadratic equations algebraically; we can also imagine the same point graphically, as shown in the next video.
Introducing [latex]i[/latex]
We have already seen that our problem in solving [latex]x^2-2x+10=0[/latex] is that the use of the quadratic formula requires us to evaluate the square root of a negative number:
[latex]\displaystyle x=\frac{-2\pm\sqrt{-36}}{2}[/latex]
and that an alternate viewpoint of the problem is that a graph of the function [latex]f(x)=x^2-2x+10[/latex] ‘has no roots’, that is, it does not cross the [latex]x[/latex]-axis. This led us to say that the equation has no real solutions. But this isn’t a page on quadratic equations – it’s a page on complex numbers, and all of the above was to lead you to the point of wanting to be able to find the square root of a negative number, so that we can show you what a complex number is! So, at last, it is time to introduce ‘[latex]i[/latex]’.
We can solve the equation above if we ‘invent’ a new non-real number equal to the square root of 1. Meet [latex]i[/latex]. The number [latex]i[/latex] is defined as [latex]i=\sqrt{-1}[/latex], or, alternatively, [latex]i^2=-1[/latex]. The [latex]i[/latex] stands for ‘imaginary’, which is where the comment about complex numbers having a bad PR department comes in, because mathematicians might argue that [latex]i[/latex] is just as good a number as the ‘real numbers’ – we’ll leave that argument until the end of the page.
Whatever [latex]i[/latex] is, though, it is NOT the unit vector [latex]\mathbf{i}[/latex] in the [latex]x[/latex]-direction in Cartesian coordinates (see Section 11)!
The imaginary number [latex]i[/latex] is certainly different from real numbers, in that it does not fit on the number line we are taught at school (real numbers do). The most obvious numbers on the number line are the natural numbers – the numbers we count with. 1, 2, and 3 are examples of natural numbers, and natural numbers are a subset of real numbers.
3/2, or 1.5, isn’t a natural number, but most people don’t have trouble extending their mental picture of the number line to deal with such fractional numbers.
Likewise, at some point in human history, people realised that they needed numbers less than zero. Admittedly if you have three apples, it is quite difficult to take away five apples, but if it is 3 °C, it is perfectly possible for it to get 5 °C colder. It will then be -2 °C. And you could argue that -2 apples does make sense if you take it to mean “in debt by 2 apples”. After all, that’s how company balance sheets operate. So the number line was extended to the left past zero. When we are young, many of us do have a little trouble assimilating this idea into our worldview, but we get there while still children. Lots of school pupils happily operate with negative numbers on a day to day basis.
Other extensions of the number line became necessary. It might seem that, in creating fractions with any numerator and denominator you like (chosen from the infinity of natural numbers), we should be able to create every number on the number line between the natural numbers. Not so! [latex]\pi[/latex] and [latex]e[/latex] are two examples of real numbers that cannot be expressed as fractions. Such numbers are called irrational.
The point, though, is that natural, fractional, negative and irrational numbers can all be placed on the number line of real numbers, whereas [latex]i[/latex] cannot. There is no number on that number line which, when multiplied by itself, gives the result -1. Hence the use of the word ‘imaginary’ to describe [latex]i[/latex].
In the meantime, here’s a funnier number line…
What complex numbers are…
Complex numbers are numbers with a real part and an imaginary part. The complex number [latex]3+4i[/latex] has a real part ([latex]3[/latex]) and an imaginary part ([latex]4[/latex]). Both parts are described by real numbers (3 and 4 in this case). However, the ‘parts’ do not have to be natural numbers: they can be fractional, negative and even irrational. They can also be zero. So [latex]-45[/latex], [latex]0[/latex], [latex]3i[/latex], [latex]3+4i[/latex], [latex]1.726-16i[/latex] are all complex numbers. Seen in this way, real numbers are just complex numbers with a zero imaginary part.
Something instructive has happened here! We used some brainpower to generalise from our everyday experience (real numbers) to invent a more abstract quantity (complex numbers). And then we said that the concrete quantity (real numbers) is just a special case of the more general quantity. Did we ‘invent’ or ‘discover’ the more abstract quantity?
This happens all the time in science, and especially in maths. It is why some mathematicians say that complex numbers get an unfair press – they are just numbers, and real numbers are just some of them! However, whereas most of us have managed to extend our mental number line to include fractional and negative numbers, and some of us may be happy to accept irrational numbers, many people have much more trouble extending their worldview to include complex numbers, and there are good reasons for that. After all, temperatures of -2 °C and 3/2 °C can be visualised in a way that has physical meaning, whereas (3+4i) °C really can’t…
Armed with complex numbers, we can solve the equation [latex]x^2-2x+10=0[/latex]. We have already seen that the solution will be
[latex]\displaystyle x=\frac{-2 \pm \sqrt{(-36)}}{2}[/latex]
And we can now make further progress.
[latex]\displaystyle x=\frac{-2 \pm \sqrt{(-36)}}{2} = \frac{-2 \pm \sqrt{36 \times -1}}{2} = \frac{-2 \pm 6i}{2} = -1\pm3i[/latex]
(Between the second and third parts of that line of algebra, we used the facts that the square root of 36 is 6 and the square root of -1 is [latex]i[/latex])
So, [latex]x^2-2x+10=0[/latex] has no real solutions (as we know), but it does have two complex solutions, [latex]-1+3i[/latex] and [latex]-1-3i[/latex].
Now, please do not think that complex numbers only occur when trying to solve quadratic equations having no real solutions. They crop up all over the place. Quadratic equations just suited us as a ‘way in’ to the concept. And if we try to think of [latex]t=-1\pm3i[/latex] as the solutions to when the projectile is 50 m above its starting point, that won’t help us very much – we already know that it never reached that height, and it isn’t very easy to visualise [latex]-1\pm3i[/latex] seconds of time, to say the least! However, there are myriad examples of when complex solutions are useful in science, and that is what Sections 12 to 14 are all about. For the rest of this page, we’ll show you some useful properties of complex numbers, now that you know what they are.
Representing complex numbers: the complex plane
As we have said before, the complex number [latex]3+4i[/latex] has a real part (3) and an imaginary part (4). The real part ‘lives’ on the normal number line. But the imaginary part doesn’t. So where shall we put it? If we draw an axis at right angles to the number line to represent the imaginary part of a complex number, then we have just formed the ‘complex plane’, sometimes called an ‘Argand diagram’.
It has to be at right angles, otherwise it will have some ‘component’ along the real axis (see Section 11), and that won’t do because by definition we know it isn’t real.
Then the coordinates (3, 4) on the complex plane represent the complex number 3 + 4i.
If you are familiar with the idea of vectors being formed of two perpendicular components (see Section 11), then you might see similarities to the complex plane. [latex]3[/latex] and [latex]4i[/latex] can be thought of as the real and imaginary ‘components’ (parts) of the complex number [latex]3+4i[/latex]. Then the complex number [latex]3+4i[/latex] has what is called a ‘modulus’ equal to [latex]\sqrt{3^2+4^2}=5[/latex], much like finding the ‘magnitude’ of a vector from its components.
In general, the modulus, [latex]r[/latex], of a complex number, [latex]x+iy[/latex], is given by [latex]r =\sqrt{x^2+y^2}[/latex]
You might not then be surprised to find that we can also write the complex number [latex]3+4i[/latex] in a different form, using the modulus of the complex number and the angle it forms with the real axis on the complex plane to express it in polar co-ordinates. We will see this in the video below.
To summarise the video:
A complex number, [latex]z=x+iy[/latex], with modulus [latex]r=\sqrt{x^2+y^2}[/latex], can be expressed in polar coordinates. Its real and imaginary parts are given by:
[latex]x=r\cos{\theta}[/latex]
and
[latex]y=r\sin{\theta}[/latex]
where [latex]\theta[/latex] is the angle (going anticlockwise) from the real axis to the complex number. These relationships follow directly from the definition of the trigonometric ratios, sine and cosine; if you do not find this instinctively obvious, you might like to look at Section 11, where we use the same mathematics to resolve a vector into its components.
Then we have the following, where the ‘Cartesian’ form of a complex number on the left is equated to its ‘polar’ form on the right:
[latex]\displaystyle x+iy=r\cos{\theta}+ir\sin{\theta}[/latex]
or, factorising out the [latex]r[/latex]:
[latex]\displaystyle x+iy=r(\cos{\theta}+i\sin{\theta})[/latex]
This equivalence turns out to be absolute gold-dust to scientists, and is pretty much the reason why scientists like complex numbers so much. It might not be obvious to you that this will revolutionise your science, so we’ll spend Section 12 explaining why it is so important. There is a clue in the fact that the real part of the complex number involves a cosine and the imaginary part involves a sine. It makes complex numbers really useful in fields involving oscillatory or rotating quantities in mechanical or electrical systems.
Complex conjugates
The complex conjugate of a complex number [latex]x+iy[/latex] is defined as [latex]x-iy[/latex] (we just swap the sign of the imaginary part).
So why would it be a good idea to invent one of those? Well, one reason that the complex conjugate is useful is because of what happens when you multiply a complex number by its own complex conjugate:
[latex]\displaystyle zz^*=(x+iy)(x-iy)=x^2+y^2=r^2[/latex]
When we expand the brackets [latex](x+iy)(x-iy)[/latex], the [latex]ixy[/latex] term and the [latex]-ixy[/latex] term cancel, leaving the real number [latex]x^2+y^2[/latex], which we know from Pythagoras is equal to the square of the modulus. So [latex]zz^*[/latex] is a real number and is equal to the square of the modulus of [latex]z[/latex].
A more fundamental reason for the importance of complex conjugates is given in Section 22.5 of Volume 1 of the Feynman Lectures in Physics, which you can find here.
A reminder why we bothered doing this
We haven’t yet seen an instance of complex numbers being useful to scientists. After all, when we tried and failed to solve the quadratic equation concerning the time at which the ball is 50 m above the starting position, we were satisfied with the answer “it doesn’t reach” 50 m high. We didn’t care about the solution [latex]t=-1 \pm 3i[/latex].
So when are complex numbers useful to scientists? Well, this page has been long enough that it might be worth reiterating the magic trick sketched out at the top. 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 in the real world.
- The universe seems to know that physical quantities that are quantified by real numbers can be more simply dealt with by using complex numbers.
We know the description in these bullets is fairly sketchy, so we’ll explore in greater detail in Section 12. It’s so important that we’ll devote two sections, 13 and 14, to examples of it…
Trying to make peace with the mathematicians this page has offended…
We are not trying to start a war with mathematicians. And, in fact, there are lots of great maths resources that explain complex numbers really well. One of our favourites is the series on imaginary numbers from Welch Labs, the first of which can be found here.
In their determination to make complex numbers ‘seem real’, many of these resources take a similar route. They, like us above, discuss the discovery/invention of fractional, negative, and irrational numbers. They then say that generalising to complex numbers is just as natural as any of the generalisations of numbers that have gone before. This may well be true for mathematicians, and they are certainly at liberty to take that view – mathematicians don’t exist for the sake of scientists.
However, we are unashamedly aiming this site at people at the science, rather than maths, end of the argument. Scientists tend to think of numbers in less abstract terms than mathematicians – see our blog post ‘What does 4380 mean?’ For a scientist, numbers tend to quantify actual quantities, and from that angle, perhaps complex numbers really are different from real numbers. Scientists love complex numbers as a tool, and love mathematics because it provides a language and structure of reasoning that leads to powerful, testable predictions. Those testable predictions lead to experiments in which measurements are made (to test the predictions). And those measurements lead to real numbers describing quantities. That predicting and testing by experiment is central to what science is.
Link to Section 11 – Vectors
Because the maths of the complex plane bears resemblances to dealing with the horizontal and vertical components of a vector, and you might like to think about the connection.
Link to Section 12 – Euler’s formula
Where complex numbers will really come into their own and you will find all scientists’ favourite formula in mathematics (well, maybe not all scientists…)
Link to Section 13 – Circular motion and phasors
Via Euler’s equation we will find that any oscillatory phenomenon can be described as a rotating arrow, and is simplified by being described in terms of complex numbers
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