Complex numbers: multiplication

Multiplication done algebraically.

Complex multiplication is a more difficult operation to understand from either an algebraic or a geometric point of view. Let’s do it algebraically first, and let’s take specific complex numbers to multiply, say 3 + 2i and 1 + 4i. Each has two terms, so when we multiply them, we’ll get four terms:

(3 + 2i)(1 + 4i) = 3 + 12i + 2i + 8i².

Now the 12i + 2i simplifies to 14i, of course. What about the 8i²? Remember we introduced i as an abbreviation for √–1, the square root of –1. In other words, i is something whose square is –1. Thus, 8i² equals –8. Therefore, the product (3 + 2i)(1 + 4i) equals –5 + 14i.

If you generalize this example, you’ll get the general rule for multiplication

Remember that (xu – yv), the real part of the product, is the product of the real parts minus the product of the imaginary parts, but (xv + yu), the imaginary part of the product, is the sum of the two products of one real part and the other imaginary part.

Let’s look at some special cases of multiplication.

Multiplying a complex number by a real number

In the above formula for multiplication, if v is zero, then you get a formula for multiplying a complex number x + yi and a real number u together:

(x + yi) u = xu + yu i.

In other words, you just multiply both parts of the complex number by the real number. For example, 2 times 3 + i is just 6 + 2i. Geometrically, when you double a complex number, just double the distance from the origin, 0. Similarly, when you multiply a complex number z by 1/2, the result will be half way between 0 and z. You can think of multiplication by 2 as a transformation which stretches the complex plane C by a factor of 2 away from 0; and multiplication by 1/2 as a transformation which squeezes C toward 0.

Multiplication and absolute value.

Even though we’ve only done one case for multiplication, it’s enough to suggest that the absolute value of zw (i.e., distance from 0 to zw) might be the absolute value of z times the absolute value of w. It was when w was the real number u just above. In fact, this is true in general:

The verification of this identity is an exercise in algebra. In order to prove it, we’ll prove it’s true for the squares so we don’t have to deal with square roots. We’ll show |zw|² = |z|²|w|². Let z be x + yi, and let w be u + vi. Then, according to the formula for multiplication, zw equals (xu – yv) + (xv + yu)i. Recall from the section on absolute values that

|z|² = x² + y²

Similarly, we have

|w|² = u² + v²

and, since zw = (xu – yv) + (xv + yu)i,

|wz|² = (xu – yv)² + (xv + yu)²

So, in order to show |zw|² = |z|²|w|², all you have to do is show that

(xu – yv)² + (xv + yu)² = (x² + y²) (u² + v²)

and that’s a straightforward exercize in algebra.

Powers of i.

For our next special case of multiplication, consider the various powers of the imaginary unit i. We started with the assumption that i² = –1. What about i³? It’s just i² times i, and that’s –1 times i. Therefore, i³ = –i. That’s interesting: the cube of i is its own negation.

Next consider i⁴. That’s the square of i², that is, the square of –1. So i⁴ = 1. In other words, i is a fourth root of 1. You can show that –i is another fourth root of 1. And since both –1 and 1 are square roots of 1, we now know all four fourth roots of 1, namely, 1, i, –1, and –i. This observation connects to the Fundamental Theorem of Algebra since the equation z⁴ = 1 is a fourth-degree equation so must have exactly four roots.

Higher powers of i are easy to find now that we know i⁴ = 1. For example, i⁵ is i times i⁴, and that’s just i. You can reduce the power of i by 4 and not change the result. For another example, i¹¹ = i⁷ = i³ = –i.

How about negative powers of i? What is the reciprocal of i, that is, i^–1? For the same reason that you can subtract 4 from a power of i and not change the result, you can also add 4 to the power of i. That means i^–1 = i³ = –i. Thus, the reciprocal of i is –i. Imagine–a number whose reciprocal is its own negation! Of course, it’s easy to check that i times –i is 1, so, of course, i and –i are reciprocals.

Roots of unity.

The various roots of 1 are called roots of unity. In general, by the Fundamental Theorem of Algebra, the number of n-th roots of unity is n, since there are n roots of the n-th degree equation z^u – 1 = 0. The square roots of unity are 1 and –1. The fourth roots are ±1, ±i, as noted earlier in the section on absolute value. Also, in that section, it was mentioned that ±√2/2 ± i√2/2 were square roots of i and –i, and now with the formula for multiplication, that’s easy to verify. Therefore, the eight eight-roots of unity are ±1, ±i, and ±√2/2 ± i√2/2. Notice how these eight roots of unity are equally spaced around the unit circle.

We can use geometry to find some other roots of unity, in particular the cube roots and sixth roots of unity. But let’s wait a little bit for them.

Multiplying a complex number by i.

In our goal toward finding a geometric interpretation of complex multiplication, let's consider next multiplying an arbitrary complex number z = x + yi by i.

z i = (x + yi) i = –y + xi.

Let's interpret this statement geometrically. The point z in C is located x units to the right of the imaginary axis and y units above the real axis. The point z i is located y units to the left, and x units above. What has happened is that multiplying by i has rotated to point z 90° counterclockwise around the origin to the point z i. Stated more briefly, multiplication by i gives a 90° counterclockwise rotation about 0.

You can analyze what multiplication by –i does in the same way. You'll find that multiplication by –i gives a 90° clockwise rotation about 0. When we don't specify counterclockwise or clockwise when referring to rotations or angles, we'll follow the standard convention that counterclockwise is intended. Then we can say that multiplication by –i gives a –90° rotation about 0, or if you prefer, a 270° rotation about 0.

A geometric interpretation of multiplication.

To completely justify what we're about to see, trigonometry is needed, and that is done in an optional section. For now, we'll see the results without the justification. We've seen two special cases of multiplication, one by reals which leads to scaling, the other by i which leads to rotation. The general case is a combination of scaling and rotation.

Let z and w be points in the complex plane C. Draw the lines from 0 to z, and 0 to w. The lengths of these lines are the absolute values |z| and |w|, respectively. We already know the length of the line from 0 to zw is going to be the absolute value |zw| which equals |z| |w|. (In the diagram, |z| is about 1.6, and |w| is about 2.1, so |zw| should be about 3.4. Note that the unit circle is shaded in.) What we don't know is the direction of the line from 0 to zw.

The answer is that “angles add”. We'll determine the direction of the line from 0 to z by a certain angle, called the argument of z, sometimes denoted arg(z). This is the angle whose vertex is 0, the first side is the positive real axis, and the second side is the line from 0 to z. The other point w has angle arg(w). Then the product zw will have an angle which is the sum of the angles arg(z) + arg(w). (In the diagram, arg(z) is about 20°, and arg(w) is about 45°, so arg(zw) should be about 65°.)

In summary, we have two equations which determine where zw is located in C: