Complex number arithmetic tutorial
© Stu Savory, 2006.
Naturally, I once wrote "you can think of the real numbers as points along a line".
Today I want to talk about complex numbers, which you can think of as points in a plane.
Complex numbers have a real number part and a socalled "imaginary" part.
The real part is shown as the position along a horizontal axis.
The imaginary part is shown as the position along a vertical axis. The unit on the
vertical axis is called 'i' and is the square root of minus one.
I'll show you why in just a minute. I've drawn two examples of complex numbers here,
point A is at 3+4i and point B is at 4+3i. In polar coordinates both are at radius 5 from
the origin (remember Pythagoras theorem with 32+42=52 ?),
however their respective angular coordinates are θA and θB.
BTW, θ is pronounced theta. All clear so far, everybody ?
In this example the angle θA=arctan(4/3) and the
angle θB=arctan(3/4), as you see.
Values inside the arctan expression are the Imaginary part divided by the Real part.
.
Now let's look at doing arithmetic with complex numbers. We use shifts for addition
and twirls for multiplication (your school may have taught this differently,
so what?).
In this illustration we start at the origin with the complex number K.
Adding 1 to K just shifts the starting point of the vector right by one unit, the other end of the vector is now at (1+K).
Adding i (the square root of -1) to K just shifts the starting point of the
vector up by one unit, the other end of the vector is now at (i+K).
So if we want to add the complex number K to the complex number Z, it is just shifting the
vector end-point from Z to (Z+K). This is done by just adding the real and imaginary
components respectively.
So (3+4i) + (5+9i) = (8+13i). That clear so far?
Now let's take a twirl through the multiplication of complex numbers :-)
Let's say we want to multiply the complex number Q by -1. So we can just put a minus sign
in front of both the real (R) and imaginary(I) components of Q, thus we get -1*Q= (-R, -I).
In polar coordinates Q=(r,θQ), and we can get from Q to -Q just by
twirling around through 180° along the path which I have marked B in my diagram.
Now let's look at path A. It takes us from 1 to -1 via i. We see that if we turn
twice through a right angle we turn a total of 180°.
θi * θi = θ-1. i.e.
i*i=-1, so i is the square root of -1 as I stated above. All clear now?
Now lets do the complex multiplication by algebra :
(A+Bi) * (C+Di) = AC + BCi + ADi * BDi2 = ((AC-BD)) + (BC+AD)i),
so (3+4i)*(4+3i) = 0 + 25i or, in polar coordinates (r,θ)=(25,90°)
The radii have been multiplied as real numbers are
(R1 shifted R2 times) and the angles θ1,
θ2 have been added (twirled). That is all there is to complex multiplication.
Any dancers amongst you will have already known how to twirl in their shifts, I guess ;-)
Now go visit my blog please :-)
