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Mathematical Journeys: My Imaginary Friend

There's no such thing as the square root of a negative number. Right?

Since squaring a number is defined as multiplying it by itself, and multiplying a negative times a negative gives a positive, all squares should be positive. Right?

So any number you want to take the square root of should be positive to begin with. Right?

So what if it's not?

What do you do if you're chugging through a problem and suddenly find yourself confronted with

x = √(-9)

It seems like to finish this problem we'll need to take the square root of a negative number – but we can't, so what do we do? Drop the sign and hope nobody notices? Mark it as 'undefined' like dividing by zero? Give up? Cry?

Well, actually, we don't have to do any of that, because we've got an imaginary friend to help us.

Meet i.

i is a mathematical constant, whose sole definition is that i2 = -1. Or, in other words, i = √(-1). i is an imaginary number – people used to think taking the square root of a negative number was impossible, so they called such results imaginary. i is known as the “imaginary unit” or the “unit” imaginary number, and he functions very similarly to the number 1 in the realm of real numbers.

Because he's imaginary, i can be a bit difficult to wrap your head around. Just remember that he's a constant, like 3 or 12 or even π. Unlike other special constants like pi or e, though, we have no real way to articulate his value. We can say that pi is roughly equivalent to 3.14..., and that e is roughly 2.718..., but what is i? i is just i. i is the square root of negative one, and that's the only way we can really describe it, since he's not a real number. We just have to accept that “the square root of negative one” would theoretically have a concrete value, and assign it a special symbol like we did with the other special constants.

So let's put our imaginary friend i to work on our earlier problem of

x = √(-9)

i functions much as 1 does for real numbers, so we can rewrite that equation as

x = √(9 * -1)

Which we can split up into

x = √(9) * √(-1)

And now we know how to deal with that second term – we just use our imaginary friend! √(-1) = i, so

x = 3 * i

Which we can rewrite as simply x = 3i.

Imaginary numbers all include the symbol i, since they're all essentially multiples of the imaginary unit. So when math hands you an impossibility, just grab your imaginary friend and jump into the realm of the unreal!