First off, take the Cartesian Plane. We'll label the x-axis with the integers and the y-axis with the imaginary numbers.
To plot a point, just move the constant to the left/right and the coefficient up/down. So, for 2 + 3i, you would move two to the right and three up. For 4 - 8i, you would go four right and eight down.
Let's say you had to add these two numbers together. First off, you could do it algebraically, which isn't that cool, and you get 6 - 5i. However, you could also add them on the imaginary plane.
Let's plot the points:
How about we just see what happens when we plot 6 - 5i. We'll connect all of the points to make a little quadrilateral.
What do you notice? The points have formed a parallelogram, with the opposite sides equal in length and parallel. At this point, we should go over how to figure out the lengths.
To figure out the lengths, we use the Pythagorean Theorem. Remember back to the problems of the week? We saw that a^2 + b^2 = c^2? We will be proving that in a later post, believe it or not. Anyways, we will be doing exactly that, literally! The form for a complex number is a + bi, so we can just plug those in and solve for the length c.
In the case of 2 + 3i, we will do 2^2 + 3^2 = c^2.
2^2 + 3^2 = c^2
4 + 9 = c^2
13 = c^2
3.606 ≈ c
To keep things more simple, we will use two different points; 8 + 6i and -3 + 4i. In this case, the lengths are 10 and 5.
Of course, we could add them together pretty easily with algebra or geometry to get 5 + 10i. However, let's figure out the angles of the lines. To do it, we use trigonometry, which we also used for the problem of the weeks.
To briefly review/explain, we will take the b term (which is the opposite side) and divide it by the c term, or hypotenuse, to get the sine of the angle. To retrieve the angle, press the sin^-1 button on your calculator.
In this case, here are our angles:
8 + 6i: 37° (approx.)
-3 + 4i: 127° (approx.)
Now, we will learn how to multiply them together. All you have to do is two easy steps: multiply the lengths and add the angles. For this one, we multiply 5 and 10 and add 37 and 127.
5 x 10 = 50
37 + 127 = 164
If you do some trigonometry, you will get an approximate answer of 48 + 14i, which happens to be the correct answer.
You can also easily multiply complex numbers by real numbers; just multiply the length by that number. If you think about it, this corresponds to the original method. These are just a handful of the many cool things you can do with the imaginary plane. To be honest, I really love algebra and arithmetic, but I'm not one of those people who is all over geometry (I'm not Archimedes, the famous Greek mathematician who died doing geometry in the sand), but this little method is just so cool.