Maths Times Infinity!

So this is more a 4 unit topic area but whatever, anybody can learn it, it's very interesting and universal.

You may have heard of real numbers before. Well they're just all the normal numbers you're used to. Anything you can come up with: #\pi, 8, -2, 5.22, -\frac{9}{7}, e, \sqrt{5}, 0, 3253249876, \dots#

But notice how when you square a number, you always get a positive result (or zero, perfectionists). Doesn't matter if it's big or small, positive or negative.

Let's make up an imaginary number just for fun, that squares to give a negative number. Probably kind of silly, won't get us anywhere, right? WRONG. Anyway, whatever, let's just play around and see what happens.

From now on, #i# is a special magical mystical special number that squares to give #-1#. Specifically. So, #i^2 = -1#. You might say #i = \sqrt{-1}# but let's just focus on the fact that when you square it, you get minus one.

What can we do with this new number? Well, let's try some algebra. What's #i+i#? Let's call that #2i#. What about #5 \times i#. That can just be #5i#. So far so good.

Then what's #(3i)^2#? Well, I guess it should be #3^2 \times i^2 = 9 \times -1 = -9#. Okay, kind of weird but I guess that's okay. Looks like we found the square root of minus nine as well. In fact, we could get any negative square root.

#(\sqrt{r}i)^2 = -r# so we can sort of say that #\sqrt{-r} = \sqrt{r}i#.

Boring. What else can we do? What about #(1+i)^2#? Hmmmm, tricky. I know! Let's just expand it. Treat #i# like any ordinary variable except we know it has this special made up property that #i^2 = -1#.

#(1+i)^2 = 1 + 2i - 1 = 2i#. Hey, we got a result! In fact, if we then try #(\frac{1+i}{\sqrt{2}})^2#, we get the same except it's divided by 2 (because of the #\sqrt{2}# part), so the result is #i#. Cool, so the "square root" of #i# is #\frac{1+i}{\sqrt{2}}#. How confusing and meaningless! Yay!

Let's try #(3+2i)(5-i) = 15 + 10i - 3i + 2 = 17 + 7i#. Hey that's not so bad. Looks like we always get a real part and an imaginary part. Turns out this is true.

Okay, how about #i^7#, what's that Mr. smarty-pants? Good question. Well #i^2 = -1#. So cubing both sides gives #i^6 = -1#. Times both sides by #i# to get #i^7 = -i#. Done. Funny how everything seems to simplify so nicely! That #i^2 = -1# property is pretty powerful!

In a sense, addition and multiplication are such a big part of maths that basically everything revolves around them, everything uses them. Because #i^2 = -1#, we can simplify larger powers of #i# and every function we'll ever use will always give a result in the form #a + bi#, where #a# and #b# are real.

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