1 The argand diagram

In Section 10.1 we met a complex number $z=x+iy$ in which $x,y$ are real numbers and

$i^2=-1$ . We learned how to combine complex numbers together using the usual operations of addition, subtraction, multiplication and division. In this Section we examine a useful geometrical description of complex numbers.

Since a complex number is specified by two real numbers $x,y$ it is natural to represent a complex number by a vector in a plane. We take the usual $Oxy$ plane in which the ‘horizontal’ axis is the $x$ -axis and the ‘vertical’ axis is the $y$ -axis.

Figure 3

Thus the complex number $z=2+3i$ would be represented by a line starting from the origin and ending at the point with coordinates $\left(2,3\right)$ and $w=-1+i$ is represented by the line starting from the origin and ending at the point with coordinates $\left(-1,1\right)$ . See Figure 3. When the $Oxy$ plane is used in this way it is called an Argand diagram . With this interpretation the modulus of $z$ , that is $\left|z\right|$ is the length of the line which represents $z$ .

Note: An alternative interpretation is to consider the complex number $a+ib$ to be represented by the point $\left(a,b\right)$ rather than the line from $0$ to $\left(a,b\right)$ .

Task!

Given that $z=1+i$ , $w=i$ , represent the three complex numbers $z,w$ and $2z-3w-1$ on an Argand diagram.

Answer

If we have two complex numbers $z=a+ib$ , $w=c+id$ then, as we already know

$z+w=\left(a+c\right)+i\left(b+d\right)$

that is, the real parts add together and the imaginary parts add together. But this is precisely what occurs with the addition of two vectors . If $\underset{̲}{p}$ and $\underset{̲}{q}$ are 2-dimensional vectors then:

$\underset{̲}{p}=a\underset{̲}{i}+b\underset{̲}{j}\underset{̲}{q}=c\underset{̲}{i}+d\underset{̲}{j}$

where $\underset{̲}{i}$ and $\underset{̲}{j}$ are unit vectors in the $x$ - and $y$ -directions respectively. So, using vector addition:

$\underset{̲}{p}+\underset{̲}{q}=\left(a+c\right)\underset{̲}{i}+\left(b+d\right)\underset{̲}{j}$

Figure 4

We conclude from this that addition (and hence subtraction) of complex numbers is essentially equivalent to addition (subtraction) of two-dimensional vectors. (See Figure 4.) Because of this, complex numbers (when represented on an Argand diagram) are slidable — as long as you keep their length and direction the same, you can position them anywhere on an Argand diagram.

We see that the Cartesian form of a complex number: $z=a+ib$ is a particularly suitable form for addition (or subtraction) of complex numbers. However, when we come to consider multiplication and division of complex numbers, the Cartesian description is not the most convenient form that is available to us. A much more convenient form is the polar form which we now introduce.