it's a very useful abstraction which helps solve real world problems.

And you can usefully manipulate it by assigning "i" to the quantity.

People started out with only positive integers. everything was fine.

you could count and add and multiply these

Then eventually after much messing about people came up with the notions of an "equation" and of "algebra" in their quest to abstract and solve some real problems and puzzles.

This is where many of the real headaches with numbers started

people began to come up with equations which gave absurd solutions.

4x +1=1 (x must be nothing)

adding "0" as an acceptable number allowed this class of equations to have a solution (x=zero)

The greek Diophantus came up with an equation 4x+20=0 which was considered absurd

because it demanded that "negative numbers" were needed to solve this.

This class of equation could be solved if you just allowed for the existence in the abstract of these "negative numbers". So we extended our set of known numbers

to include negative numbers

However this was not enough either.

Some absolute bastard came up with the notion of ax=b

which implies x=b/a (the rational numbers)

again we extended our set of known numbers to include numbers

that were solutions for this class of equations

Then yet another complete dickhead opened his trap and asked

about the solution of x= the squareroot of 2.

This was not a number that could be represented as a ratio of integers or anything else we had thus far. So we came up with (complicated!) definitions for the real numbers. Numbers with non terminating non recurring decimals that cannot be represented properly by integer ratios.

This class of numbers included such numbers as pi, e, and root 2

As if this was not enough, some other total shithead decided it was a good idea to ask about equations of the form x=square root -1. Fuck that guy!

So again we had to further generalise our set of known classes of numbers into the form a+ib

in order to manipulate the set of numbers which include this quantity i (root -1).

Numbers in this form were referred to as "complex numbers"

However magically this is where the process of extending our set of possible numbers to allow for solutions to algebraic equations seems to stop.

We now can solve pretty much any algebraic equation

that any annoying dipshit asshole decides to come up with.

Anyway, to summarise, complex numbers of the form a+ib (where i=square root of -1)

were part of a natural and very logical progression that proved mathematically necessary to

allow for the existence of solutions for various classes of algebraic equations that

annoying little fuckers insisted on asking questions about!

A nice potted intro lecture here from the master himself Richard Feynman

goes slightly further and links these complex numbers back to geometry

which is ultimately what leads to it's use in the calculation of phase information

in time varying electric signals