# Common Core Standards: Math See All Teacher Resources

#### The Standards

# 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 + 2*i* 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 2*i*), 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 + 2*i* + 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!