Today I discussed what it looks like when we multiply complex numbers on the plane. In this entry, I’m going to give some more examples of how that works.

We already know how to multiply complex numbers algebraically. For example, what is the product of \(z_1 = (4 – 2i)\) and \(z_2 = (3 + i)\)? First we use the distributive law: \(z_3 = (4 – 2i)(3 + i) = 12 + 4i – 6i – 2i^2\). We replace \(i^2\) with \(-1\) then simplify this to \(z_3 = 14 – 2i\).

Let’s look at these three numbers on the complex number plane.

On the standard complex plane, there doesn’t seem to be much relationship between the two multiplicands and their product. The product is farther away, which we would expect, but it’s not clear why it’s located where it is.

First, let’s look at the absolute values of each point.

- \(|z_1| = |4-2i| = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5}\)
- \(|z_2| = |3+i| = \sqrt{9 + 1} = \sqrt{10}\)
- \(|z_3| = |13-2i| = \sqrt{196+4} = \sqrt{200} = 10\sqrt{2} \)

The product of the absolute values of the multiplicands is the absolute value of the product. This is the first property of the product of complex numbers.

This explains why \(z_3\) has the distance that it does, but not why it’s located where it is. For this, we need to look at how much each point is rotated from the positive real axis (the x-ray).

We can create three right triangles by connecting each point to the x-ray and to the point \(0\). For instance, looking at the triangle formed by \(z_2\), it has legs of \(3\) and \(1\) and a hypotenuse of \(\sqrt{10}\). This means the angle at the lower left corner is \(\tan^{-1}{1/3} \approx 18.43^o\).

Using the inverse tangents, we can calculate the angle of rotation for each point. Here’s an important detail: If we’re rotating counterclockwise, we’ll call it a positive angle; if we’re rotation clockwise (as with \(z_1\) and \(z_3\)), we’ll call it a negative angle.

Here are the respective angles of rotation, approximately:

- \(z_1: -26.57^o\)
- \(z_2: 18.43^o\)
- \(z_3: -8.13^o\)

What do we get if we add the angles of rotation for \(z_1\) and \(z_2\)? It’s pretty close to what we got for \(z_3\); the difference is because we rounded the values.

So here’s the second property of the product of complex numbers: The angle of rotation for the product will be the sum of the angles of rotations of the multiplicands.

Let’s do another example. Take \(z_1 = (-1 + i)\) and \(z_2 = (1 – i)\). Then \(z_3 = -1 + i + i – i^2 = 2i\). Here are the respective absolute values:

- \(|z_1| = \sqrt{2}\)
- \(|z_2| = \sqrt{2}\)
- \(|z_3| = 2\)

and the angles of rotation:

- \(z_1: 135^o\)
- \(z_2: -45^o\)
- \(z_3: 90^o\)

… which follows the pattern we’ve established: The absolute values are multiplied, the angles of rotation are added.

### Key consequences

In general, this is a fun observation that might help you understand what multiplication of complex numbers means. But it’s a very powerful observation to explain two key mathematical truths.

The first and more important of these is that *the product of two negative real numbers is positive*. Because all **negative** real numbers are on the negative portion of the real number line, they have an angle of rotation of \(180^o\) from the x-ray. When we multiply two negative real numbers, the angle of rotation of the product will be \(360^o\), that is, the product will be on the x-ray.

The second is that *the product of a non-zero complex number and its conjugate will always be a positive real number*. The conjugate of a complex number is formed by keeping the real part the same and taking the opposite of the imaginary part.

Let’s look at \(z_1 = 1+2i\) and its conjugate \(z_2 = 1-2i\).

If we create our triangles, we see two congruent triangles (two right triangles with congruent legs). Hence the angle of rotation for \(z_2\) is the clockwise equivalent of \(z_1\)’s, and the angle of rotation for their product with be \(0^o\). A non-zero complex number with an angle of rotation of \(0^o\) is a positive real number.

### Advanced section! (Here be dragons)

Incidentally, there is a different way of giving complex numbers. Rather than giving a real part and an imaginary part, we could instead state the absolute value and the angle of rotation. These are called polar coordinates. Our class calculator (the TI-84) even has a setting that lets us work with them.

If you set the calculator to \(a + bi\), you will be working with complex numbers in the way we’ve discussed in class. This is the most common way to work with complex numbers. However, if you set the calculator to \(re^{\theta i}\), then enter \(\sqrt{-1}\), you’ll get \(1e^{90i}\).

The number before \(e\) is the absolute value of the complex number; “r” stands for “radius”, because in this system you’re working with the radius of the circle that the point is on, and the angle of rotation of the number.

You probably haven’t met \(e\) before. This is a constant called Euler’s number that we’ll discuss later in this course.

The number between \(e\) and \(i\) is the angle of rotation. Depending on your settings, it can be given in either degrees or radians.

Please don’t set your calculator to the \(re^{\theta i}\) setting. It will confuse other users greatly.

Now, why does this work? Recall that when we take the product of complex numbers, the absolute value is the product of the multiplicands, while the angles are the sum. Let’s look at \(re^{\theta i}\) for each of our complex number multiplicands.

- \(z_1 = r_1e^{\theta_1 i}\)
- \(z_2 = r_2e^{\theta_2 i}\)
- \(z_3 = z_1\cdot z_2 = r_1e^{\theta_1 i} r_2e^{\theta_2 i}\)

What is the rule for multiplying when we have the same number (\(e\)) to different powers? We ** add** the powers! So \(z_3 = r_1r_2 e^{(\theta_1 + \theta_2) i}\). This is exactly what we want: Multiply the absolute values (r) and add the angles of rotation (\(\theta\)).