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Video: The Complex Plane

Lauren McNaughten

Learn how to represent complex numbers on the complex plane using an Argand diagram with real and imaginary axes. We also describe polar coordinates and how to calculate the length (modulus) and the direction (argument) of the line in the diagram.

14:58

Video Transcript

In this video, we are going to look at how to represent complex numbers on the complex plane. Now this is also sometimes referred to as “an Argand diagram.” And it’s very similar in structure to the Cartesian coordinate grid that you’re most likely already be familiar with. But the axes take on a slightly different meaning when we’re looking at representing complex numbers.

So here is the complex plane. And at first glance, it probably looks exactly like the Cartesian coordinate grid that you’re already used to, but the key difference is if you look at the labelling of the axes, so you see that I’ve labelled the horizontal axis as Re, which means real, and vertical axis as Im, which means imaginary. And so what these two axes do is they represent the real and imaginary parts of a complex number. And so what the Argand diagram does is it enables us to represent not just real numbers on a one-dimensional number line, but also complex numbers on a two-dimensional grid by considering the real and the imaginary parts separately.

So let’s look at representing some complex numbers on this grid. So the first one 𝑍 one is the complex number three plus four 𝑖. So as a real part of three, so I’ll move along to three on the real axis and imaginary part of four, so then I’ll move up to four on the imaginary axis. And therefore this complex number 𝑍 one would be represented by a cross or a dot in this position here, corresponding to a real part of three and an imaginary part of four.

The second complex number 𝑍 two is the complex of a negative two 𝑖. So its real part is zero. And if I go to negative two 𝑖 on the imaginary axis, I would be representing that complex number 𝑍 two down here. The third one has a real part of negative two, so negative two on the real axis and then positive five 𝑖. I would be representing 𝑍 three about here. And finally, 𝑍 four is just the complex number seven, which is in fact just a real number. So it would be represented on the real axis here. So every complex number that we could write can be represented somewhere in this complex plane by considering its position relative to these two axes with the horizontal representing the real part and the vertical representing the imaginary part.

Now when we’re using the complex plane to represent complex numbers, there is actually an alternative form that we may often consider useful. Now the complex number 𝑍 equals two plus four 𝑖, this is currently what we’re thinking of as rectangular form, where we’re considering the position relative to the real and the imaginary axes. But there is an alternative form that we can consider. This alternative form is what’s known as the polar form of a complex number or polar coordinates when we’re plotting it on the complex plane. And what we do is we consider the length and the direction of the vector of the line that joins that complex number back to the origin — so this line that I’ve marked in here.

So there are two pieces of information here. First of all, the length of that vector which is sometimes referred to as the modulus or the magnitude and is represented in various different ways. Sometimes it’s represented using vertical lines to denote modulus; sometimes it’s represented using the letter 𝑟. But in both cases what it means is the length of that line joining the complex number back to the origin.

The other piece of information that we’re interested in is the angle that this vector makes with the positive real axis; so that would be this angle here. And we often use the letter 𝜃 to represent that or it can sometimes be written as arg 𝑍 because the name that we give to this angle is known as the argument of a complex number. So two things that we can calculate, we can work out the length of the line and we can work out the direction of the line. And those are referred to as the modulus and the argument of the complex number, which is why polar form is also sometimes referred to as modulus-argument form.

Now let’s look at how we would calculate both of those things for this example. So let’s look first of all at how we would calculate the modulus. And if you imagine just sketching in a little triangle here, so what we’re going to do is we’re actually going to use an application of Pythagoras’s theorem in order to work out the length of that line, which is the hypotenuse of this right-angled triangle. So this right-angled triangle has sides of two and four, which are the real and imaginary parts of the complex number. And therefore the length of the hypotenuse using Pythagoras is going to be the square root of two squared plus four squared. So that’s going to be the square root of four plus sixteen, which is the square root of twenty. And if I want to simplify that surd, that will become two root five.

So calculating the modulus is just a direct application of Pythagoras’s theorem. If I want to calculate the argument, it’s going to be using a little bit of trigonometry. So in this right-angled triangle, remember we have sides of four and two. And if I want to calculate this angle here, well using those two sides I will be using the opposite side and I will be using the adjacent. So using trigonometry, that tells me that I will be using tan as the trig ratio that I’m interested in. So tan of this angle, tan of 𝜃 is opposite over adjacent. Tan will be four over two, which tells me that 𝜃 will be tan inverse of two.

Now we can use a calculator to evaluate that. And working in degrees or radians is acceptable, although radians is more usual. That will give an answer of sixty-three point four degrees in degrees or one point one one radians, both to three significant figures. So a reminder of what have I done, I’ve used Pythagoras in order to work out the modulus of that complex number. And I’ve used trigonometry in order to work out the argument — the angle that it makes with the real axis.

So let’s look at how to write a general complex number into its polar form. So I have the complex number 𝑍 which is 𝑎 plus 𝑏𝑖. I’ve worked out its modulus 𝑟 and its argument 𝜃 using the methods described in the previous slide. And I want to see what its polar form will be.

So what I want to do is look at this right-angled triangle and look in a bit more detail at the different trig ratios that I’d play here. So first of all if I think about cosine, if I think about cos of 𝜃. Cos remember is adjacent divided by hypotenuse. So in this right-angled triangle, that’s going to be 𝑎 divided by 𝑟. And if I do one step of rearrangement there, I get the relationship 𝑎 is equal to 𝑟 cos 𝜃, just by multiplying up by 𝑟. If I think about the sine ratio in this triangle, sine remember is opposite divided by hypotenuse. So I’ll have sin 𝜃 is 𝑏 over 𝑟. And again one step of rearrangement gives me the relationship 𝑏 equals 𝑟 sin 𝜃.

The final step then is to go back to this complex number up here and to replace 𝑎 and 𝑏 with their new values in terms of 𝑟 and 𝜃. So I have 𝑍 is equal to 𝑎 plus 𝑏𝑖, but I can replace 𝑎 with our cos 𝜃 and 𝑏 with our sin 𝜃. And so that’s what I’ve done here. And then the final step is just to take out a common factor of 𝑟, which leads to 𝑍 equals 𝑟 brackets cos 𝜃 plus 𝑖sin 𝜃. And this is known as the polar form of a complex number, where you work out the modulus 𝑟 and the argument 𝜃. And then you can write it in this alternative form 𝑍 equals 𝑟 brackets cos 𝜃 plus 𝑖sin 𝜃 instead of its form 𝑍 equals 𝑎 plus 𝑏𝑖 in terms of those pure real and imaginary parts.

So we’ve got our general form — our general polar form. If we return to the previous example which was two plus four 𝑖, we’d already calculated the modulus was two root five and the argument in radians was one point one one. So if I now want to write this complex number in its polar form instead of its rectangular form, I just need to replace our own 𝜃 with the calculated values. So I will have 𝑍 equals two root five lots of cos one point one one plus 𝑖sin one point one one. And then I’ve converted that complex number from its rectangular form into its polar form.

Let’s just look at how to generalize that polar form for the complex number 𝑎 plus 𝑏𝑖. So if you recall, we use Pythagoras in order to work out the modulus. So 𝑟 will be equal to the square root of 𝑎 squared plus 𝑏 squared by using the Pythagorean theorem. And then we use trigonometry in order to work out the argument. So we use the fact that tan 𝜃 is opposite over adjacent 𝑏 over 𝑎, which gave 𝜃 is inverse tan of 𝑏 over 𝑎. And so that gives us some general formulae that we can use in order to work out the modulus and argument of a complex number. Now this first one for 𝑟, this will always work no matter where the complex number is in the plane. But the one for the argument does need a little bit more consideration because if 𝑏 or 𝑎 are negative and therefore the complex number is not in the first quadrant of the plane, we need to think a little bit more carefully about how we actually calculate the argument.

So here’s an example on the screen, where we have a complex number that would be in a different quadrant. So we have negative three plus four 𝑖 and it’s represented over here in the second quadrant. Now if I go to the process using these general results, then I get 𝑟 equals two root five and there were no problems there. If I work out 𝜃 using tan inverse of 𝑏 over 𝑎, I get negative fifty-three point one or negative nine point two seven radians because I’m doing tan inverse of four over negative three. Now remember 𝜃 is supposed to represent the angle that this line makes with their positive real axis. So it’s this angle that I’ve marked in purple. And that can’t be negative fifty-three point one or negative nought point nine two seven radians; it should be an obtuse angle, somewhere between ninety degrees and a hundred and eighty degrees or somewhere between 𝜋 over two and 𝜋. So just using the formula doesn’t give us the actual argument that we are looking for.

However, it’s still helpful. What I need to do is use the fact that the angle between the positive real axis all the way around to the negative real axis — that angle there — is 𝜋 radians or a hundred and eighty degrees. And what I find is if I add a hundred and eighty degrees or add 𝜋 radians, then I do get a value that is in the range we’re looking for. I get a hundred and twenty-six point nine degrees or I get two point two one radians, which both do represent an obtuse angle. So if the complex number isn’t in that first quadrant, then I do have to think more carefully about how to use this formula. I can use it as a starting point, but I may have to add or subtract 𝜋 or add or subtract from 𝜋 in order to work out the argument correctly and I can do that by using the Argand diagram.

One final point on this argument is that we have what’s known as the principal argument for a complex number. And it’s always between negative 𝜋 and 𝜋. So for complex numbers that are above the real axis in quadrants one and two, we are measuring in an anticlockwise direction from the positive real axis, giving an answer anywhere between zero or 𝜋 radians. For complex numbers that are below the positive real axis, we are measuring in a clockwise direction, giving arguments of anything from zero to negative 𝜋 depending on where they are placed in the complex plane. So the argument that we’re looking for should always be in this range negative 𝜋 to 𝜋. And depending which of the four quadrants you are in, you will get values in these different ranges, as shown on the screen at the moment.

As discussed, we can use that formula tan inverse of 𝑏 over 𝑎 to work out an initial value for 𝜃. But then we need to adjust by adding or subtracting 𝜋 or adding and subtracting from 𝜋 in order to ensure the argument we get is in this predefined range.

Final thing to consider is converting back from the polar form into the rectangular form. So here I have the complex number ten and then cos two point two one four plus 𝑖sin two point two one four. And I want to convert it back to rectangular form. So I can do this with the aid of a calculator, just evaluating cos and sin of two point two one four and multiply the result by ten. So if I do that, I see that it gives me negative five point nine nine seven six plus eight point zero zero one one seven 𝑖. And it is in fact just the complex number negative six plus eight 𝑖. So there you have it, a summary of what the complex plane is, how you can represent complex numbers on it, and the two different forms: polar form and rectangular form of a complex number.