Imaginary numbers.

What is the deal? What are they?

Why are they freaky… or, wait, are they freaky?

Some basics to get out the way:

Say we have a number.

Like 4.

We can square that number (or multiply it by itself):

4x4 and get 16.

We can also take a negative 4, multiply it by itself

-4 x-4 and get… 16.

Two different problems giving the same answer because a negative number multiplied by a negative number gives us a positive number.

Stay calm. This is just like in English. An even multiple of negatives gives a positive outcome.

“I don’t not like you.” means, “I like you”.

Yes?

(Okay, so there might be some passive aggressive angst going on there, but you get my essential meaning.)

Now, let’s reverse our thinking. (Mathematics likes to work like that, backward and forwards):

If 4x4=16 and -4x-4=16, then there has to be a reverse button that says what number times itself equals 16.

This is our “square root”. The square root of 16 is either 4 or -4.

Now, what if we wanted to know what number times itself equals a -16?

Well, that’s tricky, since we’ve already established that a negative times a negative equals a positive.

We can’t pull a ‘real-life’ example that works. There is a temptation to say,

“That doesn’t work. Full stop. Walk away. Let’s go think about a lovely holiday.”

Mmm… a holiday… wouldn’t it be nice to be able to just appear in Paris, with that internet buddy you’ve never met in real life, and to be able to eat ALL THE PASTRIES. Yes. That would be nice. I’m going to suspend my reality and just daydream a bit.

Oh! What if I did that with math?

What if I just pretended that there was a number times itself that equaled a negative number.

I could call it *i*, or my imaginary number. It will be the square root of a negative 1.

Why -1? Because 1 is special. I can multiply 1 by any number and end up with that number. This means I can multiply any number by my special *i* and it can be imaginary too! (The magic wand of maths land!)

Why is this a good thing?

Well, if I suspend my disbelief (just like with movies or TV or a good juicy novel), I can think about things (here mathematical things) that may not exist is ‘real life’. Just because something doesn’t exist in the touchable world doesn’t mean I can’t learn from it. Being able to manipulate equations by using my imaginary number *i* allows me to hold onto information that would otherwise be lost. Imaginary numbers are a tool for the "what if's" of the mathematical world.

There is one more thing I should mention.

If I take my *i* and multiply it by itself ( *i *x* i *) or *i* squared, I will get my -1 back.

It’s like closing a book, and being back in the ‘real’ world. Or, it’s like getting a cleaning deposit back after leaving a vacation rental (I hear that can actually happen).

Are imaginary numbers scary?

Well, is it scary to think about things that don’t exist in a way we can touch?

I guess the honest answer is ‘sometimes’. We tend to think of math (and science) as being serious things made of “Real” concepts, but sometimes both fields are grounded not in proven fact or touchable reality, but in purely mental constructs. There is a blurring of the lines between Real and not-Real.

This happens in all facets of our lives. The internet is filled with Real and not-Real things. One of the advantages of imaginary numbers is that they come with a label. That wee italic “i” states, 'Imaginary'. Wouldn’t it be nice if all things from the not-Real realm came with a handy Imaginary tag?