Complex Numbers Explained
When we think about complex numbers, we often think about performing algebra with this weird i term and it all seems a bit arbitrary and easily forgettable. In actuality what we’re doing is tangible and can be visualized.
So get ready.
After this post you’ll probably never think of complex numbers the same again…and yeah, that’s a good thing.
Note: If you’re blanking on what imaginary numbers are and how they operate on a complex plane check out this post.
Plotting Complex Numbers
Complex numbers are the sum of a real and an imaginary number, represented as a + bi. Using the complex plane, we can plot complex numbers similar to how we plot a coordinate on the Cartesian plane.
Here are a few examples:
3 + 2i
1 – 4i
-3 + 3.5i
Just draw a point at the intersection of the real part, found on the horizontal axis, and the imaginary part, found on the vertical axis.
Adding & Subtracting Complex Numbers
This is by far the easiest, most intuitive operation. Adding/subtracting real numbers translates the point right/left on the real axis, and adding/subtracting imaginary numbers translates the point up/down on the imaginary axis.
Arithmetically, this works out the same as combining like terms in algebra.
For example, if we subtract 1 – 4i from 3 + 2i, we simply compute the real difference:
3 – 1 = 2,
and the imaginary difference:
2i – (-4i) = 2i + 4i = 6i.
This is the same as plotting the point 3+2i and translating it left 1 unit and up 4 units. The resulting point is the answer: 2+6i.
We can also think about these points as vectors.
First distribute the minus sign so we have the addition: (3+2i) + (-1+4i).
Next plot the two points with line segments shooting out from the origin.
To add these points, simply stack one on top of the other. Since addition is commutative, it doesn’t matter which way we stack them.
This may seem like overkill, but here’s the thing: understanding the vector representation is going to make multiplying and dividing complex numbers so much easier.
Multiplying Complex Numbers
This operation is a little less obvious and leaves us wondering:
What does it mean to multiply two complex numbers together?
In general, we know multiplying by a real number scales the value, and we learned in the last post that multiplying by i rotates a value by 90˚ counter clockwise, but how about this?
To get a better grasp, let’s distribute the first binomial through the second.
Alright, now we can perform addition by stacking the vectors after we’ve performed the transformations. Let’s try it out.
First we have (3+2i)(1), which is (3+2i) scaled by 1.
Next we have (3+2i)(-4i). We have two things happening here: scaling and rotating.
First let’s scale it by 4 by multiplying (4)(3+2i) to get (12 + 8i).
We also need to multiply by -i. Recall multiplying by -i is a 90˚ clockwise rotation.
Note: This matches the algebra had we subbed in i = √-1:
The final step is to perform addition by stacking the vectors.
Our final answer is 11 – 10i.
Now you might be thinking,
“Brett, why can’t we just solve this with algebra??”
And it’s true, we can solve this using algebra. In fact, it’s the most efficient way to solve the problem (although it lacks the insight you get from graphing).
I’d be a lousy mathematician if I didn’t show you both ways. So for all my algebra-loving friends out there, here’s how to expand and simplify the above problem:
Dividing Complex Numbers
Let’s divide (3+2i)/(1–4i).
At this point you might think you can just divide the real parts and the imaginary parts…but not so fast.
Just like in algebra, we have to divide the denominator into both terms of the numerator, which leaves us with the same issue:
What does dividing by a complex number really mean?
Truthfully, it’s confusing and there isn’t a great explanation for it. Wouldn’t it be nice if we could get rid of the imaginary number in the denominator??
Good news → That’s exactly what we’re going to do!
The Complex Conjugate
The key to solving this problem is figuring out how to change the denominator into a plain ole real number.
The simplest way to do this is to use the complex conjugate.
To find the complex conjugate, simply flip the sign on the imaginary part. For example, the complex conjugate of (1–4i) is (1+4i).
When I multiply these together I get 17:
Of course, I can’t just multiply the denominator by (1+4i). Like any fraction, if I want to multiply the denominator by a value I must also multiply the numerator by that value.
Now this makes sense. We have two complex numbers being multiplied in the numerator, which we know how to handle from the previous section, and we are scaling the whole thing by 1/17.
You can solve this with a graph or take the algebra shortcut:
That wasn’t so bad, was it?
What I love about seeing problems solved in multiple ways is you get the opportunity to really get to know a concept and fully understand it in a way you wouldn’t be able to had you not seen both methods. Not only are you more likely to stumble across that coveted aha! moment, you now have way more tools in your arsenal for when you need to solve tougher problems.
Thanks for reading!
