Well, without wishing to appear rude, DrRocket's explanation had me tugging my beard for a while. Of course he is correct, but perhaps a little opaque.
Try this instead. Consider the polynomial . One says that the "zeros" of this polynomial are those values of for which is true.
First note that there are implied integer coefficients in this polynomial - in this case 1.
So a number is defined to be an algebraic number iff it is a zero of some polynomial with integer coefficients
Now there is a theorem, called the Fundamental Theorem of Algebra, that states that any polynomial of degree has at most zeros and moreover that . (The proof is hard!)
So we seek at least one algebraic number such that . In fact there are 2, which is the usual case for degree 2 polynomials, as indeed for those of higher degree.
So one defines the objects as the 2 zeros for our polynomial. Since these quite obviously not real numbers (in the usual sense of the word) our is called the "imaginary unit". (Notice that )
Further, one says that, given the field of real numbers, there is an extension of this field by our imaginary unit such that .
Now it is not hard to show (using the field axioms on ) that is also a field, whose elements must be of the form
And so (finally!!) one says that he complex numbers are the algebraic completion of the reals, which quite simply means that there is no polynomial whose zeros cannot be found in or its subfield (usually both)
OMG, I had intended to add clarity to this thread - I now see I have done the opposite.