So, last time we discovered that numbers are maybe not quite as real as we thought that they were, and that we can have numbers which don’t obviously correspond to something in the real world (though we’ll discover later that they are a way to jump between islands of reality).
In the resources on Vula you will find some great notes on complex numbers, so I want this to be an additional resource, and not an alternative. This means that sometimes we will look at things from a slightly different perspective than in the resource book.
Let’s start off discussing a bit more about the complex plane.
When you learnt about integers, one of the first things that you learnt to do was to put them in order. 3 came after 2 and 7 came after 6. You could put them all in a line. When you learnt about the negative numbers, it was quite clear that this line which had previously started with zero simply went backwards in the other direction, and you could count backwards to whatever large negative integer you wanted.
Then you learnt about halves, and it was clear where these should go on the number line, then there were more and more numbers which had very obvious places on this line. Even the irrationals, of which there are not only an infinite number, but an uncountable infinity (as opposed to the countable infinity of the rationals), had a place on the number line – every one of them.
OK, so how about our new number, , this number which is basically ?. Where does this go on our number line? In fact that’s not clear at all. How about ? Is that bigger than in the same way that ? Well, it would seem natural that you could maybe order these numbers in this way, but things are a bit more complicated as we also have numbers like . Is that somewhere near , or , or perhaps on our number line?
Well, as it happens, the number line just isn’t flexible enough when we have added this new type of number into our repertoire. We could cope with the negative numbers, the rationals, and the irrationals, but suddenly we are at a loss. However, we can see that whereas for the real numbers there is a single dial you can turn to turn up and down the value of a real number. Now we have two dials. Now we can change the real part, and the imaginary part separately of our number, and can specify a complex number by two real numbers: The real part, and the imaginary part.
Remember that, quite strangely, the imaginary part of a complex number is real. This is because the imaginary part is defined as . It’s just the coefficient of .
OK, so now we see that we need two dimensions to specify any complex number. We can think that rather than having a line, we have a plane, with two coordinates on it, one of the coordinates is the real part, and we normally put it as the horizontal axis on a page, and the other is the imaginary part which we normally put as the vertical axis on a page.
There’s something subtle but important about this. You can think of every point in the complex plane as being given by the coordinate of the point. This is just a set of coordinates of the form . In this respect there is nothing about the complex nature of the plane. It’s just a two dimensional plane with points all indicated by two real numbers. It is only when we start to perform operations on points in the complex plane that it’s going to be any different from a regular two dimensional plane, but until then we don’t have to think about its complex nature.
Given a complex number we can see the real and imaginary parts and and put this at position in the complex plane. This is described as the ordered pair formalism of complex numbers where is an ordered pair of real numbers which completely specifies the complex number.
We posted it previously, but here let’s post the complex plane with a load of complex numbers labelled on it:
Note that while the points are labelled by the complex numbers in the form , to put them in their place on the plane, all you need are the values of and , which here are called and .
In class we played around then with adding complex numbers together and found that the rules were pretty simple. If we have two complex numbers and , then to add them together, you simply add together the real parts, and the imaginary parts separately to get .
In multiplying two complex numbers together we simply multiplied out the brackets and remembered that and that in the end it was best to get the real parts and the imaginary parts together, so:
When we looked at dividing complex numbers by each other we used a trick. Given a division of the form:
If we multiply and divide by the expression is magically simplified into a form with the real part and the imaginary parts separated, as we normally like:
We came up with a special name for this funny changing of the sign of the imaginary part of a complex number. If we have a complex number then we denote and say that is the complex conjugate, or simply conjugate of .
In the resources you will find a list of the properties multiplication and addition of complex numbers, showing how they are commutative and associative under addition and multiplication as well as being distributive. Make sure that you can prove that all of these are true by writing out the complex numbers in their real and imaginary parts separately and remembering that .
We also saw in class how complex conjugation works on products and sums of complex numbers. ie. the fact that and . Again, make sure that you can follow all of the properties of complex conjugation. These sorts of operations are going to be your basic staple diet for complex numbers.
If you are reading through these notes and are not studying MAM1000, please leave a note in the comments and I will write up these details more thoroughly.
OK, let’s go back to the complex plane and ask again a question about ordering.
If you have two real numbers and I ask you which is bigger it is about the most intuitive thing in the world. In fact, if you read anything in the field of numerical cognition (and I suggest you do, for instance The Number Sense, by Dehaene is a great text),
you will find that understanding the magnitude of numbers is one of the first things we do. We have a very intuitive sense of relative size. Given 5 dots and 3 dots we know from a very early age what we mean by more dots.
But there is no such thing as having dots, so is that more or less than having dots? Well, the answer is that on the complex numbers themselves, there is no sensible way to order them. We can’t write them in a one dimensional list going from smallest to largest, which is why we had to imagine them living in a two dimensional plane.
However, we have seen something rather special already. When we had our division problem we saw that if we took and multiplied it by its conjugate , we ended up with . But we know that and are real numbers, so we have something which is positive.
In fact this value has a very clear description in the complex plane. If we have a point in the complex plane given by , then is the distance along the real axis and is the distance along the imaginary axis. Then the value is simply the square of the distance from the origin to the point.
You don’t have to worry about the fact that these are complex numbers. We are simply asking – when we plot the position of the number in this two dimensional plane, what is the distance from the origin. We call this the magnitude, or modulus, or absolute value of and label it by . So we have that for :
So now if you are asked which of two complex numbers are bigger, you can’t answer that, but you can order them by their magnitude, if you so wish. While you can’t say that you can very easily say that: .
In fact we’ve skipped one little visualisation in the complex plane. We defined the complex conjugate simply to be the original complex number with the sign of the imaginary part changed. For instance:
, and .
In the complex plane this has a very simply picture. It is simply the reflection of a complex number about the x-axis:
But now we have something rather intriguing, which is perhaps the first thing which isn’t obvious about the arithmetic operations of complex numbers when we think about them in the complex plane. If you take and and you multiply them together, then the answer you get is going to be a positive real number and its length is going to be the square of the length of . Notice that the magnitude of and are the same. Just reflecting the number across the x-axis does not change its size.
OK, that’ll do for now. We’re soon going to see that there’s a whole new way to describe complex numbers using their positions in the complex plane, but for now I suggest that you practice adding, multiplying, taking the conjugate of, and finding the magnitude of as many complex numbers as you can.