High School: Number and Quantity
The Complex Number System HSN-CN.B.5
5. Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation. For example, because has a modulus 2 and argument 120°.
As the name suggests, complex numbers can occasionally get a little, well, complex. But relax. We won't drop complex bombs on you right now.
As we know, the x-coordinate of a complex number represents its real part, and the y-coordinate represents its imaginary part. So adding and subtracting complex numbers is pretty much combining like terms. Work first with the real component, a. Then, move on to the b, the imaginary component. Put them together what do you get? Well, uh, the answer.
For instance, let's say we have to graph the point made by adding 3 + 2i and 6 – i. We should add the a values first. Going over 3, then over 6 more, and we get to 9. So the a value is 9.
Now the b values. First, we go up 2 units (because of 2i), but then back down one for that -i term. That puts us at i. That means our point is at 9 units to the right and one i unit up. That's because 3 + 2i + 6 – i = 9 + i.
Easier than balancing a walrus on your head, right? Hopefully. Ready for that complex part? Neither are we.
Multiplication of imaginary numbers in a + bi form is easy. Students can use FOIL as though the i were an x or some other variable. But when we switch to polar coordinates, things get a little more… challenging.
Here's the basic rule to find the product of two complex numbers in polar form:
- Multiply the radii.
- Add the angles.
So, to find the product of (4, 30°) and (7, 20°), we just multiply 4 and 7 for the radial coordinate, and add 30° and 20° for the angular coordinate. Our product is (28, 50°). That's not hard at all, right?
Finally, we should cover how to find the reciprocal of a complex number. If it's in a + bi form, it's just algebra. Put it under 1, then multiply the top and bottom by the conjugate of the bottom.
What we mean is that the reciprocal of a + bi is . Since we can't have imaginary numbers in denominators, we have to multiply by the conjugate, which will give us , or .
Sometimes, it will involve using FOIL or the double distributive property on either the top or the bottom (or both). It's just algebra, but it can sometimes be a substantial dose of it.
Students should know that in polar form, the reciprocal of the number (r, θ) is 1 over the r value and the negative angle. For instance, the reciprocal of (2, 30°) is (½, -30°). More generally, the reciprocal of (r, θ) is (1⁄r, -θ).
That's just one of the reasons these numbers are called complex!