So we saw last time that we can take a complex number and put it in a 2 dimensional plane called the complex plane, where its horizontal distance from the origin is given by its real part, and the vertical distance from the origin is given by its imaginary part. We can thus think of the real and imaginary parts as the Cartesian coordinates of that point.

It turns out that there is another way to represent a complex number, but rather than using the real and imaginary parts to specify it, we will use two other pieces of information.

If I tell you that a complex number is a distance $|z|$ away from the origin in the complex plane, then this leaves you with a whole circle of possibilities. All the points on the circle of radius $|z|$ about the origin are the same distance from the origin. But if I also give you an angle subtended between the x-axis and the line joining the complex number and the origin, read anti-clockwise from the x-axis, this will completely pin down the point in the complex plane. Let’s say I tell you that there’s a complex number of modulus 2, and angle $\frac{\pi}{4}$ anti-clockwise from the x-axis, this is clearly given by:

We already know that the magnitude of the complex number $|z|$ is sometimes called the modulus. The angle subtended between the x-axis and the line between the number and the origin is called the argument of the complex number, and is denoted by $arg(z)$. Notice that once $|z|$ and $arg(z)$ are given, the point in the complex plane is completely specified. It is just as good as giving the values of the real part and the imaginary parts. Of course there is a way to calculate one from the other.

Given the pair $(|z|,arg(z))$, we can see that the real part of the complex number is $|z|\cos(arg(z))$ and the imaginary part is given by $|z|\sin(arg(z))$. For instance, in the example above we have:

$z=a+bi=|z|\cos(arg(z))+|z|\sin(arg(z))i=2\cos(\frac{\pi}{4})+2\sin(\frac{\pi}{4})i=\frac{2}{\sqrt{2}}+\frac{2}{\sqrt{2}}i=\sqrt{2}+\sqrt{2}i$

If we had the modulus and argument given by $(3,2.5)$ then this would correspond to the complex number:

$z=3\cos(2.5)+3\sin(2.5)i\approx -2.40343 + 1.79542 i$

Why do we care about having another way to write out complex numbers?

Well, it turns out that it’s very useful when we want to multiply two complex numbers together. Adding together complex numbers in the so called Cartesian form ($a+bi$) is very easy as we simply add together the real parts and imaginary parts separately:

$(a_1+b_1i)+(a_2+b_2i)=a_1+a_2+(b_1+b_2)i$

However, multiplying complex numbers together in the Cartesian form turns out not to be so convenient.

$(a_1+b_1i)(a_2+b_2i)=a_1a_2-b_1b_2+(a_1b_2+a_2 b_1)i$

How about if we write everything in terms of their modulus and argument? If we have two complex numbers where the moduli are $r_1$ and $r_2$ and their arguments are $\theta_1$ and $\theta_2$, then we have:

$z_1 z_2=(r_1\cos(\theta_1)+r_1\sin(\theta_1)i)(r_2\cos(\theta_2)+r_2\sin(\theta_2)i)=r_1 r_2(\cos(\theta_1)\cos(\theta_2)-\sin(\theta_1)\sin(\theta_2)+(\cos(\theta_1)\sin(\theta_2)+\sin(\theta_1)\cos(\theta_2))i)$

Well, this is a mess, but actually we can use our double angle formulae to see that this is just:

$z_1 z_2=r_1 r_2(\cos(\theta_1+\theta_2)+\sin(\theta_1+\theta_2)i)$

Which is amazing! So we see that when we multiply two complex numbers together then the result is a complex number where its magnitude is the product of the magnitudes of the two original numbers, and its argument is simply the sum of the original arguments – moduli multiply, arguments add! Much nicer than the whole $ac-bd$ mess.

This now means that when you multiply two numbers together which are given in the complex plane, you can easily see what the result is going to be. You simply multiply the magnitudes together to give the magnitude of the new number and you add the angles together to get the angle of the new number:

ok, so multiplying numbers together in the so-called modulus argument form (where we write explicitly $r(\cos(\theta)+i\sin(\theta_1))$) turns out to be really simple, and we can see how to go from the modulus-argument form to the Cartesian form just by evaluating the $\sin$ and $\cos$ and multiplying them by $r$. How about going the other way? Surely if I give you a number in Cartesian form I should be able to calculate the value of the modulus and argument.

Well, yes and no! Certainly we already know how to calculate the modulus of a complex number given the Cartesian form ($r=\sqrt{a^2+b^2}$), but how about how to calculate the argument? Well, it turns out that it’s not unique. Of course we can measure the angle between the line joining the complex number and the origin, with the x-axis, but it turns out that we can always add an integer multiple of $2\pi$ onto this angle and we will end up in the same place – ie. we can go around a full circle and end up where we started. It turns out that the argument of a given complex number isn’t unique and is actually given by a set.

For instance if I give you the number $z=2+2i$, it is certainly true that $\theta=\frac{\pi}{4}$ is an argument for this, but in fact it is more accurate to say that:

$arg(z)=\{\frac{\pi}{4}+2\pi n; n\in {\mathbb Z}\}$

In fact we have to be a bit careful when calculating the argument of a complex number from the cartesian form, because the fomula depends on whether the number is in the left or right half of the complex plane.

For a number of the form $z=a+bi$, the following holds: If the number is in the right part of the complex plane (ie. $a$ is positive) then:

$r=\sqrt{a^2+b^2}\, , \,\,\, \theta=\arctan\frac{b}{a}+2n\pi\, |\, n\in {\mathbb Z}$

$a=r\cos\theta\, , \,\,\, b=r\sin\theta$

but if $a$ is negative (ie. the number is in the left half of the complex plane) then the angle is given by

$r=\sqrt{a^2+b^2}\, , \,\,\, \theta=\pi+\arctan\frac{b}{a}+2n\pi\, |\, n\in {\mathbb Z}$

$a=r\cos\theta\, , \,\,\, b=r\sin\theta$

OK, so let’s summarise what we’ve learnt here:

• There are two ways to specify a complex number. Either we write it in Cartesian form as $a+bi$ or we write it in modulus argument form as $r(\cos\theta+i\sin\theta)$ where $r=|z|$ and $\theta\in \arg(z)$.
• Adding together complex numbers is easy in Cartesian form
• Multiplying together complex numbers is easy in modulus argument form
• We can go back and forth between the two forms, but the argument is not a unique number, but is only defined up to addition of integer factors of $2\pi$.
 How clear is this post?