Last time, we left off with a real math problem: what is the connection between complex multiplication and the complex plane?
To get to the bottom of this, we'll use the 4 examples we mentioned last time. For each example, we'll plot the two numbers are multiplying together. We'll also compute the result algebraically and add it to each plot.
Our job now is to look for patterns. Back in part 5, we learned that 'i' had something to do with rotation on the complex plane.
So a good thing to keep track of here will be the angle our complex numbers make with the real axis.
We can determine our angles using a little trigonometry, specifically the arctangent function.
Now, let's look for a connection between our three angles.
After a little pondering, we see that the angle of the result is exactly equal to the angles of the numbers are multiplying added together.
This is the first half of the connection we're looking for: when multiplying on the complex plane, the angle of our result is equal to the sum of the angles of the numbers we're multiplying.
Let's now have a closer look at our first 2 examples. Notice that the angles are identical, but the resulting complex numbers are not.
This means that just keeping track of angles alone is not enough to sufficiently describe complex multiplication in the complex plane. There is something else going on.
So what is the difference between these examples?
It looks like multiplying by 2i has pushed our results further from the origin than multiplying by i. A good follow-up question is, "How much further?"
We can measure the distance between the origin and our complex numbers by forming right triangles and using the Pythagorean theorem.
Just as before, let's compute our measurement for each example and look for patterns.
After some more pondering, we see that if we multiply our distances, we obtain the distance from the origin of the result.
We now have the complete picture. When we multiply complex numbers on the complex plane, their angles from the real axis add and their distance from the origin multiply. This is the connection we were looking for between complex multiplication and the complex plane.
We now have completely separate but completely equivalent interpretations of complex multiplication.
To multiply two complex numbers together, we can follow the rules of algebra, or we can find each numbers distance from the origin and angle to the real axis on the complex plane and multiply and add each.
And what's really cool here is that although these approaches look and are totally different, they do the same exact thing.
What we're seeing here is the same underlying process from two separate vantage points.
I really like this idea because it reminds me that there's more to math than what we see on the page. There are deeper truths embedded in our universe, and math is one way of expressing them.
Now that we've made our discovery, let's formalize our results a bit.
We found that the quantities we should keep track of when multiplying complex numbers in the complex plane are the distance from the origin and the angle from the real axis.
These quantities turn out to be so important that we use them as another way to write complex numbers.
Instead of writing complex numbers as the sum of their real and imaginary parts, we instead write them as their distance from the origin and the angle they make with the real axis.
This is called polar form, and the distance from the origin gets a special name 'magnitude.'
Multiplying complex numbers in polar form is super easy. We just multiply the magnitudes and add the angles.
Division is pretty simple too, especially compared to dividing in rectangular form. To divide in polar form, we divide the magnitudes and subtract our angles.
Next time, we'll show that this discovery is not only cool, but useful. We'll use the complex plane to make hard algebra problems easier, faster, and more intuitive.