In the section before last, on drawing complex numbers, I mentioned the "modulus" and "argument" of a complex number. Here's a recap.
The modulus is the length of the vector from the origin to the position of the complex number in the Argand diagram. We can find the modulus using Pythagoras' theorem. For the complex number 3+4i, for example, the modulus is 5, since 32+42=25.
The argument is the angle that the vector makes with the horizontal x-axis. So the complex number 3+3i has argument 45o, since the line from the origin to the point (3,3) is at 45o to the horizontal axis.
These terms modulus and argument may be familiar to you from using polar coordinates. If you are not familiar with polar coordinates, you may find it useful to look in the MathHelp notebook on Coordinates and Graphs.
When using polar coordinates, instead of the Cartesian coordinates x and y, we define the position of a point in terms of its distance from the origin and its angle from the x-axis. Here's a diagram to illustrate this idea.
The position of the point A can be described as (2,2) in the Cartesian coordinates x and y. ALternatively its position can be described as (r,q), where r is the distance from A to the origin and q is the angle shown.
In this case we can work out that r must be about 2.8 (the square root of 22+22) and that q must be 45o.
We calculate the value of q by looking at the ratio of the y value and the x value. By its definition, tan(q) is equal to the length of the side opposite the angle q, divided by the length of the side adjacent to the angle q.
In other words, tan(q)=y/x. In this case y=2 and x=2, so y/x=1. That means q=45o, since tan(45o)=1.
To find out what x and y are, if we know r and q,
we can use the definitions of sine and cosine to give the following
formulae:
x=rcos(q)
y=rsin(q).
OK so that was a quick revision of polar coordinates. What does it have to do with complex numbers?