It's the only way to get to the complexes without doing anything spooky! Defining i to be "the square root of minus one" is about as sane as defining it to be the square root of the colour blue.
The first people who thought about it followed that approach, and were rightly scared stiff and confused by it. Even Euler made trivial mistakes.
Argand came up with the right way of thinking about it:
Take all the tuples (x,y). (Where x and y are just integers). Define on them addition and multiplication rules.
Oh look! There's a big system of these things, and embedded within it is a sub-system which works exactly the same as the integers and their rules.
Since they're exactly the same for all practical purposes, we may as well forget about the difference and say that we'll write (a,0) as a and (0,b) as ib, and (a,b) as a+ib
And although non of the pairs in that subsystem square to be (-1,0), also known as -1, there are two things that do! So now we know that both (0,1) and (0, -1), otherwise known as 1i and -1i, or i and -i, are square roots of -1.
No slight of hand or magical thinking necessary.
As it happens, we can do the same thing with the reals, to form something that embeds the reals. But the reals really are dark and mysterious and need to be brought about by a kind of magic.
http://www.math.toronto.edu/mathnet/answers/imagexist.html
I'll gratuitously paste the relevant part here, but the summary is that to multiply two tuples, you simply take the cross product, as you would a binomial.
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Remember that any collection of objects for which there is a definition of what the objects are and when two objects are equal, there is a rule for how to add two objects, there is a rule for how to multiply two objects, and these rules obey familiar arithmetic laws like commutativity, associativity, and distributivity, is, by definition, a number system.
These properties are all satisfied by complex numbers.
We have a definition of when two complex numbers are to be considered equal: they are equal if and only if they are the same pair of real numbers.
We have a rule for adding two complex numbers (which, remember, are nothing more than pairs of real numbers):
(a,b) + (c,d) = (a+c, b+d)
(a,b)(c,d) = (ac-bd, ad+bc)
Therefore, complex numbers form a number system.
Within this number system, is there an object which, when squared, gives -1? Yes. It is the pair (0,1). When you square it using the above rule of multiplication, you get
(0,1)(0,1) = ( (0)(0) - (1)(1), (0)(1)+(1)(0) ) = (-1,0).
However, complex numbers of the form (a,0) behave identically to the way ordinary real numbers a behave. They add and multiply in exactly the same way that ordinary real numbers do:
(a,0) + (b,0) = (a+b,0)
(a,0)(b,0) = (ab,0)
http://betterexplained.com/articles/intuitive-arithmetic-wit...
The first link was posted here a while ago, I think.
I really like Euler's Formula, but feel many explanations focus on the raw bits (showing the series are equal). The physics explanation "90-degree rotation" is a good one. Here's how I explained it to my mom:
"The formula is saying there's two ways to get to a place on a circle. One is to go across (cos) and up (sin), and the other is to go out (1.0) and rotate (x radians)."
This is a beautiful result which ties together radians, exponential growth, and complex numbers -- I have a post drafted that I want to share on this :).