Complex multiplier

Have you ever played with toy blocks that had different shapes and colors? Imagine if you could put two of these blocks together and get a new block with a different shape and color!

That's a bit like what a complex multiplier does. It takes two special blocks called complex numbers and combines them to make a new complex number.

A complex number is like a special kind of toy block that has two parts: a real part and an imaginary part. The real part is a regular number that you can count with, like 4 or 7. The imaginary part is a bit weird, because it uses a special number called i.

The number i is like a magic block that makes the imaginary part of a complex number work. So if we're talking about a complex number like 3 + 2i, the real part is 3 and the imaginary part is 2i.

When we use a complex multiplier, we take two complex numbers and multiply them together. But we have to remember some rules about how the imaginary part works.

If we have two complex numbers, like 3 + 2i and 4 - 5i, we can put them together like this:

(3 + 2i) × (4 - 5i)

To multiply these together, we need to use something called FOIL. That stands for

First (multiply the first terms): 3 × 4 = 12
Outside (multiply the outside terms): 3 × -5i = -15i
Inside (multiply the inside terms): 2i × 4 = 8i
Last (multiply the last terms): 2i × -5i = -10i²

Now we have to remember another rule: i² is equal to -1. So we can rewrite the last term as:

-10i² = -10(-1) = 10

Now we can add up all our terms to get the final answer:

12 - 15i + 8i + 10 = 22 - 7i

So the complex multiplier combined the two complex numbers and gave us a new one: 22 - 7i.

In the end, a complex multiplier helps us combine two complex numbers into a new one, by using magic blocks and some special rules.