Video Transcript
In this video, we’ll learn how to identify complex numbers on the Argand diagram. We’ll begin by learning what an Argand diagram actually is and how we represent complex numbers on this diagram. We’ll extend this into the geometric interpretation for addition of complex numbers and multiplication by real and imaginary numbers. Finally, we’ll learn the geometric interpretation of something called the roots of unity.
When we begin to learn about numbers, we learn that we can represent them on a one-dimensional number line. This is usually useful as it allows us to form mental strategies for addition and subtraction, in addition to providing a visual context for the idea of negative numbers. When we introduce imaginary numbers though, we add a second dimension and can begin to consider complex numbers as points in a plane. Just as a number line can allow us to gain insights into the real number set, thinking about our complex numbers as points in a plane allows us to gain insights into their properties.
We call this visual representation the Argand diagram or the Argand plane. Devised by Swiss mathematician John Argand in the early 1800s, it consists of a real axis, that’s the horizontal one, and an imaginary axis, that’s the vertical one. That means we can represent a complex number of the form 𝑥 plus 𝑦𝑖, where 𝑥 and 𝑦 are of course real numbers, by the point whose Cartesian coordinates are 𝑥, 𝑦. Let’s look at an example that uses these concepts.
If the number 𝑍 equals eight plus 𝑖 is represented on an Argand diagram by the point 𝐴, determine the Cartesian coordinates of that point.
To answer this question, we absolutely could go ahead and plot the complex number 𝑍 on the Argand diagram and then read the information from there. But that’s quite a long-winded way to go about answering this question. Instead, we remind ourselves of the definition of the Argand diagram. We know that a complex number of the form 𝑥 plus 𝑦𝑖 can be represented by a point whose Cartesian coordinates are 𝑥, 𝑦. The real part is the 𝑥-coordinate. And the imaginary part is the 𝑦-coordinate.
The real part of our complex number is eight. And we can figure the imaginary part as the coefficient of 𝑖. So in this case, the imaginary part of 𝑍 is one. This means the Cartesian coordinates of the point that represents the complex number 𝑍 on the Argand plane are eight, one.
And what about complex conjugate pairs? How might they appear on the Argand diagram?
Let’s have a look at the point that represents the complex number eight plus 𝑖 on the Argand diagram. We saw that it’s represented by a point whose Cartesian coordinates are eight, one. We can find the complex conjugate of 𝑍 by changing the sign of the imaginary part. And so the conjugate of eight plus 𝑖 is eight minus 𝑖. We therefore represent the conjugate of 𝑍 on our Argand diagram by the point whose Cartesian coordinates are eight, negative one. We can see that the point is a reflection in the real axis. And in fact, this is true for all complex numbers and their complex conjugate.
And just as we can interpret conjugate pairs on an Argand diagram, we can use the plane to interpret the addition of two complex numbers. We know that, to add two complex numbers, we add their real parts and then separately add their imaginary parts. So the sum of 𝑎 plus 𝑏𝑖 and 𝑐 plus 𝑑𝑖 is 𝑎 plus 𝑐 plus 𝑏 plus 𝑑 𝑖. We’ll plot 𝑍 one with the point whose Cartesian coordinates are 𝑎, 𝑏. 𝑍 two has Cartesian coordinates 𝑐, 𝑑. It follows that their sum has Cartesian coordinates 𝑎 plus 𝑐, 𝑏 plus 𝑑.
You might see that there’s a relationship of sorts appearing. We can actually think of complex numbers plotted in the Argand plane as vectors. And as such, we can think of complex number addition in the same way as vector addition. We can therefore think of 𝑍 one plus 𝑍 two as the resultant of the two vectors 𝑍 one and 𝑍 two. And this can be represented in the parallelogram as shown.
Just as we can represent the addition of complex numbers on an Argand plane by thinking about them as vectors, the same can be said for multiplying them by a real number. Let’s say, for example, we wanted to multiply the complex number three plus four 𝑖 by the real constant two. We represent the complex number by the point three, four on the Argand plane as shown. Multiplying a vector by two multiplies the horizontal and vertical components by two. So in this case, we represent two multiplied by the vector three plus four 𝑖 as the point six, eight. And we see that two times 𝑍 is equal to six plus eight 𝑖.
Interpreting complex numbers as vectors on the Argand plane allows us to interpret multiplication by a real number 𝑐 as a dilation or an enlargement scale factor of 𝑐 about the origin. This even extends into the idea of multiplying by a negative number. And alternatively, that can be interpreted as a rotation about the origin by 𝜋 radians, followed by dilation by the scale factor modulus of 𝑐. But how do we represent multiplying a complex number by a purely imaginary number on the Argand plane?
Four complex numbers 𝑧 one, 𝑧 two, 𝑧 three, and 𝑧 four are shown on the Argand diagram. Part 1) Find the image of the points 𝑧 one, 𝑧 two, 𝑧 three, and 𝑧 four under a transformation that maps 𝑧 to 𝑖𝑧. Part 2) By plotting these points on an Argand diagram, or otherwise, give a geometric interpretation of the transformation.
We’re looking to find the transformation that maps 𝑧 to 𝑖𝑧. To do this, we’re going to first need to find the complex numbers 𝑧 one, 𝑧 two, 𝑧 three, and 𝑧 four. Remember, the horizontal axis represents the real part of a complex number. And the vertical axis represents the imaginary part. 𝑧 one has Cartesian coordinates three, zero. So in complex number form, it’s three plus zero 𝑖, which is just three. 𝑧 two is two plus three 𝑖. 𝑧 three is negative two minus one 𝑖. 𝑧 four has Cartesian coordinates zero, negative one. So as a complex number, it’s negative 𝑖.
Next, we’re going to multiply each of these numbers by 𝑖, remembering of course that 𝑖 squared equals negative one. This means that 𝑖𝑧 one is three 𝑖. 𝑧 two is two 𝑖 plus three 𝑖 squared. And since 𝑖 squared is negative one, that’s negative three plus two 𝑖. And in the same way, 𝑖𝑧 three is one minus two 𝑖. And 𝑖𝑧 four is one. We now need to plot these points on the Argand diagram.
We can see that 𝑖𝑧 one has Cartesian coordinates zero, three. That’s here. 𝑖𝑧 two has Cartesian coordinates negative three, two. That’s here. 𝑖𝑧 three is here. And 𝑖𝑧 four is here. We can see that 𝑧 one has moved a quarter of a turn here. 𝑧 two has moved a quarter of a turn, as had 𝑧 three and 𝑧 four. And we can see that the transformation that maps 𝑧 to 𝑖𝑧 is a rotation about the origin in a counterclockwise direction by 𝜋 by two radians.
Now, it follows that since multiplying a complex number by 𝑖 leads to a rotation, multiplying a complex number by some real multiple of 𝑖 will lead to a rotation, followed by dilation as we saw earlier. And whilst we aren’t quite ready to represent multiplication of two complex numbers using an Argand diagram, we can look at the geometric interpretation of something called the roots of unity.
1) Find all the solutions to 𝑧 to the power of six equals one. 2) By plotting the solutions on an Argand diagram, or otherwise, describe the geometric properties of the solutions of 𝑧 to the power of six equals one.
We could solve this equation by finding the sixth root of both sides. However, we know that there are going to be six solutions to this equation. So we need to consider an alternative method. Instead, we rearrange by subtracting one from both sides. And we see that 𝑧 to the power of six minus one equals zero. This is actually a special case of the difference of two squares, meaning we can write the expression on the left-hand side as 𝑧 cubed minus one multiplied by 𝑧 cubed plus one. And now, we have two numbers whose product is zero. This can only be the case if 𝑖, the number itself, is equal to zero.
Let’s start with saying that 𝑧 cubed minus one is equal to zero. We can observe that one of the solutions to this equation is one since one cubed minus one is indeed zero. This means that 𝑧 minus one must be a factor of 𝑧 cubed minus one. We could use polynomial long division to find the other factor. Or we could say that this means that 𝑧 cubed minus one is equal to 𝑧 minus one multiplied by some quadratic. And then, we can equate coefficients of 𝑧. Distributing the brackets, and we see that 𝑎𝑧 cubed plus 𝑏 minus 𝑎 𝑧 squared plus 𝑐 minus 𝑏 𝑧 minus 𝑐 equals 𝑧 cubed minus one.
Equating coefficients of 𝑧 cubed, we see that 𝑎 is equal to one. And that’s because the coefficient of 𝑧 cubed on the right-hand side is just one. The coefficient of 𝑧 squared on the right-hand side is zero. So we see that when we equate coefficients of 𝑧 squared, we get 𝑏 minus 𝑎 equals zero. 𝑎 is of course one. So 𝑏 minus one is zero, which means that 𝑏 must be equal to one. We’re going to skip equating coefficients of 𝑧 to the power of one and go straight to equating constants or coefficients of 𝑧 to the power of zero.
We see that negative 𝑐 equals negative one, which means that 𝑐 is equal to one. And this means that 𝑧 cubed minus one is equal to 𝑧 minus one multiplied by 𝑧 squared plus 𝑧 plus one. We then solve 𝑧 squared plus 𝑧 plus one equals zero by either using the quadratic formula or completing the square.
If we use the quadratic formula, we see that 𝑧 is equal to negative one plus or minus the square root of one squared minus four times one times one, all over two times one. That’s negative one plus or minus the square root of negative three over two. We’ll split this up and write it as negative one-half plus or minus the square root of negative three over two. And since the square root of negative one is 𝑖, our solutions for 𝑧 become negative a half plus or minus the square root of three over two 𝑖.
We’ll repeat this process for 𝑧 cubed plus one is equal to zero. This time, we can spot that one of the solutions to this equation is 𝑧 equals negative one. And that’s because negative one cubed plus one is equal to zero. This time, that means that 𝑧 plus one must be a factor of 𝑧 cubed plus one. And we can say that we can write 𝑧 cubed plus one as 𝑧 plus one multiplied by some quadratic in 𝑧.
This time, distributing the brackets, and we see that 𝑎𝑧 cubed plus 𝑎 plus 𝑏 𝑧 squared plus 𝑏 plus 𝑐 𝑧 plus 𝑐 equals 𝑧 cubed plus one. And this time, when we equate coefficients, we get that 𝑎 is equal to one. 𝑏 is equal to negative one. And 𝑐 is equal to one. So 𝑧 cubed plus one is equal to 𝑧 plus one multiplied by 𝑧 squared minus 𝑧 plus one. This time, we solve 𝑧 squared minus 𝑧 plus one equals zero, once again using the quadratic formula or possibly completing the square. And when we do, we can see that 𝑧 is equal to one-half plus or minus the square root of three over two 𝑖.
And we see that we now have the six solutions to the equation 𝑧 to the power of six equals one that we were looking for. And if we want to, we could check these solutions by substituting them back into the equation 𝑧 to the power of six equals one and checking that our answers make sense.
For part 2), we’re going to plot these points on an Argand diagram. 𝑧 equals one and 𝑧 equals negative one are fairly straightforward. We have the point a half, root three over two representing the solution one-half plus root three over two 𝑖. And we have negative a half root three over two, representing the solution negative a half plus root three over two 𝑖. We can plot the other two solutions as shown. And what about the geometric properties? Well, we can see that these complex numbers are evenly spaced about the origin. In fact, the solutions are the vertices of a regular hexagon inscribed with a unit circle whose centre is the origin.
In this video, we’ve seen that we can represent a complex number 𝑥 plus 𝑦𝑖 on the Argand plane by the point whose Cartesian coordinates are 𝑥, 𝑦. We’ve also seen that there are many geometric interpretations to operations with complex numbers. The addition of complex numbers can be represented by a translation with a vector 𝑎𝑏. We saw that complex conjugate pairs are reflections of one another in the real axis. We learned that multiplication by a real number is a dilation with a centre at the origin whose scale factor is that real number. And we saw that multiplication by 𝑖 is a rotation about the origin counterclockwise by 𝜋 by two radians.
