2 The polar form of a complex number

We have seen, above, that the complex number $z=a+ßb$ can be represented by a line pointing out from the origin and ending at a point with Cartesian coordinates $\left(a,b\right)$ .

To locate the point P we introduce polar coordinates $\left(r,\theta \right)$ where $r$ is the positive distance from $0$ and $\theta$ is the angle measured from the positive $x$ -axis, as shown in Figure 5. From the properties of the right-angled triangle there is an obvious relation between $\left(a,b\right)$ and $\left(r,\theta \right)$ :

$a=rcos\theta b=rsin\theta$

or equivalently,

$r=\sqrt{a^2+b^2}tan\theta =\frac{b}{a}.$

This leads to an alternative way of writing a complex number:

The angle $\theta$ is called the argument of $z$ and written, for short, arg $\left(z\right)$ . The non-negative real number $r$ is the modulus of $z$ . We normally consider $\theta$ measured in radians to lie in the interval $-\pi <\theta \le \pi$ although any value $\theta +2k\pi$ for integer $k$ will be equivalent to $\theta$ . The angle $\theta$ may be expressed in radians or degrees.

Example 4

Find the polar coordinate form of

1. $z=3+4i$
2. $z=-3-i$

Solution

1. Here

   $r=\left|z\right|=\sqrt{3^2+4^2}=\sqrt{25}=5\theta =\text{arg}\left(z\right)={tan}^{-1}\left(\frac{4}{3}\right)=53.13^{\circ }$

   so that $z=5\left(cos53.13^{\circ }+isin53.13^{\circ }\right)$

2. Here

   $r=\left|z\right|=\sqrt{{\left(-3\right)}^2+{\left(-1\right)}^2}=\sqrt{10}\approx 3.16\theta =\text{arg}\left(z\right)={tan}^{-1}\frac{\left(-1\right)}{\left(-3\right)}$

   It is natural to assume that ${tan}^{-1}\frac{\left(-1\right)}{\left(-3\right)}={tan}^{-1}\left(\frac{1}{3}\right)$ . Using this value on your calculator (unless it is very sophisticated) you will obtain a value of about $18.43^{\circ }$ for ${tan}^{-1}\left(\frac{1}{3}\right)$ . This is incorrect since if we use the Argand diagram to plot $z=-3-i$ we get:

   Figure 6

   

   The angle $\theta$ is clearly $-180^{\circ }+18.43^{\circ }=-161.57^{\circ }$ .

   This example warns us to take care when determining arg $\left(z\right)$ purely using algebra. You will always find it helpful to construct the Argand diagram to locate the particular quadrant into which your complex number is pointing. Your calculator cannot do this for you .

   Finally, in this example, $z=3.16\left(cos198.43^{\circ }+isin198.43^{\circ }\right)$ .

Task!

Find the polar coordinate form of the complex numbers

1. $z=-i$
2. $z=3-4i$

Answer

Answer

Remember, to get the correct angle, draw your complex number on an Argand diagram.

2.1 Multiplication and division using polar coordinates

The reader will perhaps be wondering why we have bothered to introduce the polar form of a complex number. After all, the calculation of arg $\left(z\right)$ is not particularly straightforward. However, as we have said, the polar form of a complex number is a much more convenient vehicle to use for multiplication and division of complex numbers. To see why, let us consider two complex numbers in polar form:

$z=r\left(cos\theta +isin\theta \right)w=t\left(cos\varphi +isin\varphi \right)$

Then the product $zw$ is calculated in the usual way

in which we have used the standard trigonometric identities

$cos\left(\theta +\varphi \right)\equiv cos\theta cos\varphi -sin\theta sin\varphi sin\left(\theta +\varphi \right)\equiv sin\theta cos\varphi +cos\theta sin\varphi .$

We see that in calculating the product that the moduli $r$ and $t$ multiply together whilst the arguments arg $\left(z\right)=\theta$ and arg $\left(w\right)=\varphi$ add together.

Task!

If $z=r\left(cos\theta +isin\theta \right)$ and $w=t\left(cos\varphi +isin\varphi \right)$ find the polar expression for $\frac{z}{w}.$

Answer

We see that in calculating the quotient that the moduli $r$ and $t$ divide whilst the arguments arg $\left(z\right)=\theta$ and arg $\left(w\right)=\varphi$ subtract .

We conclude that addition and subtraction are most easily carried out in Cartesian form whereas multiplication and division are most easily carried out in polar form.

2.2 Complex numbers and rotations

We have seen that, when multiplying one complex number by another, the moduli multiply together and the arguments add together. If, in particular, $w$ is a complex number with a modulus $t$

$w=t\left(cos\varphi +isin\varphi \right)\left(\text{i.e.}r=t\right)$

and if $z$ is a complex number with modulus 1

$z=\left(cos\theta +isin\theta \right)$   (i.e. $r=1$ )

then multiplying $w$ by $z$ gives

$wz=t\left(cos\left(\theta +\varphi \right)+isin\left(\theta +\varphi \right)\right)$ (using Key Point 7)

We see that the effect of multiplying $w$ by $z$ is to rotate the line representing the complex number $w$ anti-clockwise through an angle $\theta$ which is arg $\left(z\right)$ , and preserving the length. See Figure 7.

Figure 7

This result would certainly be difficult to obtain had we continued to use the Cartesian form.

Since, in terms of the polar form of a complex number

$-1=1\left(cos180^{\circ }+isin180^{\circ }\right)$

we see that multiplying a number by $-1$ produces a rotation through $180^{\circ }$ . In particular multiplying a number by $-1$ and then by $\left(-1\right)$ again (i.e. $\left(-1\right)\left(-1\right)$ ) rotates the number through $180^{\circ }$ twice, totalling $360^{\circ }$ , which is equivalent to leaving the number unchanged. Hence the introduction of complex numbers has ‘explained’ the accepted (though not obvious) result $\left(-1\right)\left(-1\right)=+1$ .

Exercises

1. Display, on an Argand diagram, the complex numbers $1-i$ , $1+3i$ and $-1+2i$ .
2. Find the polar form of
   1. $1-i$ ,
   2. $1+3i$
   3. $2i-1$ . Hence calculate $\frac{\left(1+3i\right)}{\left(-1+2i\right)}$
3. On an Argand diagram draw the complex number $1+2i$ . By changing to polar form examine the effect of multiplying $1+2i$ by, in turn, $i$ , $i^2$ , $i^3$ , $i^4$ . Represent these new complex numbers on an Argand diagram.
4. By utilising the Argand diagram convince yourself that $\left|z+w\right|\le \left|z\right|+\left|w\right|$ for any two complex numbers $z,w$ . This is known as the triangle inequality .

Answer