Operations on Complex Numbers

To add or subtract complex numbers (which are entities of the form $a + bi$) you do the appropriate thing to the real parts (the $a$'s) and the imaginary parts (the $b$'s) separately.

For example, we have

$( 4 + 3i) - (7 - i) = (4 - 7) + (3 - -1)i = -3 + 4i$

To multiply two complex numbers, you multiply out the terms in the two factors (using the linearity of multiplication (aka the distributive law) and use the fact that $i^2$ is $-1$.

For example, we get

$\begin{aligned} (4 + 3i) * (7 - i) &= 4 * 7 + 4 * (- i) + 3i * 7 + 3i * (-i)\\ &= 28 - 4i + 21i - 3i^2\\ &= 28 + 3 + 17i\\ &= 31 + 17i \end{aligned}$

Division is slightly trickier, because we want our answer to have the form $a + bi$ and not that of a ratio of such things (though $a$ and $b$ can be ratios).

To get this we use the wonderful fact that any complex number multiplied by its complex conjugate (what you get by reversing the sign of its $b$) is a real number.

In symbols, this reads $(a + bi) * (a - bi) = (a^2 + b^2)$.

How come?

Multiply it out using the distributive law and see.

What good is this?

We rewrite this equation as $a + bi = \frac{a^2 + b^2}{a - bi}$, which tells us that multiplying by $(a + bi)$ is the same as multiplying by the real number $(a^2 + b^2)$, and dividing by $(a - bi)$.

This means dividing by $(a + bi)$ is the reciprocal operation to that, which is multiplying by $(a - bi)$ and also dividing by the real number $(a^2 + b^2)$.

So dividing by a complex number, say $(3 + 2i)$ is the same as multiplying by $(3 - 2i)$ and dividing the result by $3^2 + 2^2$ which is $9 + 4$ or $13$.

So, for example, $\frac{7-i}{(3+2i)}$ is $\frac{(7-i)(3-2i)}{13}$ which is $\frac{(21-2)-(14+3)i)}{13}$ or $\frac{19-17i}{13}$.

We thus have rules for adding, subtracting, multiplying and dividing complex numbers.

By the way, the quantity $a^2 + b^2$ is called the square of the magnitude of the complex number $a + bi$.

Geometric Representations of Complex Numbers

A complex number, ($a + ib$ with $a$ and $b$ real numbers) can be represented by a point in a plane, with $x$ coordinate $a$ and $y$ coordinate $b$.

This defines what is called the "complex plane". It differs from an ordinary plane only in the fact that we know how to multiply and divide complex numbers to get another complex number, something we do not generally know how to do for points in a plane.

This picture suggests that there is another way to describe a complex number. Instead of using its real and imaginary parts, which are its $x$ and $y$ coordinates to describe it. We can use the distance from its point in the complex plane to the origin $(0,0)$, and the angle formed by a line segment from the origin to that point, and the positive half of the $x$ axis. The distance to the origin is usually denoted as $r$, that angle is usually called $θ$ (theta). $θ$ is called the "phase" and sometimes the "argument"" of the complex number. The distance to the origin is called its "magnitude" and also its "absolute value".

How are these parameters, $r$ and $\theta$, related to $a$ and $b$?

We use the Euclidean definition of distance, for which the Pythagorean theorem holds. This tells us

$r^2 = a^2 + b^2 \enspace \text{and so} \enspace r = \sqrt{a^2 + b^2}$

As for $\theta$, we use the standard trigonometric definitions of sines and cosines. The sine of an angle is defined to be the ratio of its y-coordinate $b$ to length $r$, and the cosine is the ratio of its x-coordinate $a$ to $r$. Thus $\theta$ is an angle whose sine is $\frac{y}{r}$, and whose cosine is $\frac{x}{r}$.

This gives us the relations

$a = r\cos\theta \enspace \text{and} \enspace b = r\sin\theta$

What good is this?

Lots of good as we shall eventually see. But right now we can notice the following curious fact:

In terms of $a$ and $b$, called the real and imaginary parts of the complex number, addition and subtraction are easy to describe, (add or subtract each part separately as if the other didn’t exist: $(a+bi) + (c+di) = (a+c) + (b+d)i$, but multiplication and division are a bit ugly.

In terms of $r$ and $θ$, the magnitude and phase of a complex number, the opposite is true. That is, multiplication and division are easy to describe, while addition and subtraction are a bit ugly.

How so?

Well, you can multiply two complex numbers together by multiplying their magnitudes together, and adding their phases. You divide by dividing the magnitudes correspondingly, and subtracting the phase of the denominator from that of the numerator.

Explicitly, we have the product of the complex number with magnitude $r_1$ and phase $\theta_1$ with the complex number with magnitude and phase $r_2$ and $\theta_2$, is the complex number with magnitude $r_1*r_2$, and phase $\theta_1 + \theta_2$.

(The rules for adding and subtracting in terms of magnitude and phase can be deduced from the rules in terms of real and imaginary parts, but are not particularly illuminating, because they are messy.)

You can see all this on the following mathlet. You can move the complex numbers, $w$ and $z$ around by clicking the left mouse button on the appropriate head and holding it down as you move. It allows you to examine the behavior of sums products differences and ratios of complex numbers as you change them. To see what you can do with this mathlet, click on "+ about" in its upper right hand corner.