Believe it or not, imaginary numbers aren't these mystical things
that don't exist except in some obscure mathematician's
head. They are real
things that you can find real life
examples of. So my goal in this write-up is to explain what they are without resorting to weirdness like "the imaginary number is the square root of negative one".
So let's start easy. Suppose I ask you to show me the number three. Now, you can't show me "three", but you can certainly show me three of
something, like three apples
, three dollars
, or three steps forward
. So numbers exist in the real world as a count or a measure of something.
Let's try something a little harder: show me negative three. Well... you can't really show me three negative apples, but you can give
me negative three dollars by taking
three dollars from me, and you can
show me negative three steps forward
by walking three steps backwards
. A negative number
is a positive number done backwards
; to do something a negative number of times is to do the opposite a positive number of times.
Why do we have negative numbers? Because sometimes it's more convenient to say "minus three" than to say "do the opposite three times." For example, suppose that you're filling out a form and there's a blank which says, "How much money have you given to charities this year?" And let's say that you are actually a cradle-robber who steals candy from impoverished babies
. Rather than wasting your time scratching off the words "given to" and writing in "taken from", you can instead lazily write in the negative of the dollar amount of the candy you stole and mean the same thing.
So if negative numbers are just the opposite of what you would do with a positive number, what's an imaginary number? An "imaginary number" is when you do something countable, but it was neither what you were expecting to count, nor the opposite of what you were expecting to count. So say you wanted to count how many steps a girl
was going to walk forward, but then she surprises you by walking five steps left
instead. Since she did not travel either forwards or backwards, neither a positive nor a negative number will work here. At this point you have two options. First, you could scratch out the word "forwards" on your notepad and replace it with "left". Or... if you are too lazy, you could instead write down that she walked an imaginary
five steps forward. So just as negative numbers are a convenient way of indicating we mean the opposite of whatever positive means, imaginary numbers are a convenient way of indicating we mean the "left" of what positive meant.
Now, here's a cute trick
: If walking left five steps is walking forward an imaginary five steps, what would it mean to walk an imaginary number of steps left? Well, since in this case imaginary already means "left", that means you are walking five steps left of left, or five steps backwards! So if you do the "imaginary" thing an "imaginary" number of times, you get the opposite of what you are doing! That this works is nothing more magical than saying when you do the opposite of the opposite of what you meant to do, then you are really just doing what you meant to do in the first place. Just like many actions have natural opposites
, so do some actions have a natural "imaginary".
When we want to say using fancy mathematical equations that the opposite of the opposite is the original, we say that (-1) * (-1) = 1, or equivalently (-1)2
=1. Likewise, when we want to say that the imaginary of an imaginary is the opposite, we can say that i * i = i2
= -1. There is no mathematical trickery
here; the only trickery lay in finding an "imaginary" way of doing something, which you cannot always do, just like you can't always find an "opposite". It is as puzzling to figure out what it means to give you an imaginary amount of money as it is to figure out what how one can physically hold a negative number of apples.
So in conclusion, the imaginary number, i
, is more than just some crazy symbol with the property that i2=-1
; it is rather real
concept that has practical
meaning behind it, just like positive
and negative numbers