Do you believe in √(-1)?

[pbrower2a]

i is an answer; one can debate the relevance of the answer.

We have a hierarchy of genuineness of numbers. First we have the natural numbers that one counts with. Add the number "zero" for completeness through the inclusion of the concept of "nothingness" or a void.

The ancient Egyptians and Greeks hardly accepted zero as a number, perhaps suggesting that "nothingness" had no significance. Neither did they have the mirror image of the whole numbers, the negative integers, which with the whole numbers and zero comprise the integers, the "complete" numbers. 

The Egyptians found fractions -- "broken numbers" useful, as did the Greeks. Add to such numbers as 3/8  and 2/5  such inverses as 8/3 and 5/2 and one has all the broken, if sensible "rational" numbers -- if one allows 1 as a divisor. Zero is of course prohibited as a divisor.

Trouble arose with numbers that could not be seen as fractions -- like the square roots of 2 or 23, or the cube roots of 4 or 25. They could only be approximated with fractions, but they were solutions. The Egyptians knew well of the 3-4-5 right triangle and perhaps 5-12-13, 7-24-25, 8-15-17, 9-40-41, and 11-60-61 ... but the great Euclid could prove that the diagonal of a right triangle with rational sides could never be expressed precisely as a fraction. Such numbers didn't seam quite 'reasonable', so they were called 'irrational'. The weird constant π created its own problems.

Some cultures discovered zero -- our use of zero derives from Indian practice. They had the concept of the void and nothingness and had a number to express it. The Arabs found it useful, and we learned it from the Arabs.

In the Renaissance, people started trying to solve equations that supposedly had no answers -- like x^4 =1. "1" and "-1" are solutions, but so are "i" and "-i"; after all, i^4 =1. Numbers involving i were thus "imaginary" even if three-dimensional mathematics is  a sane proposition. But they were algebraic, establishing them as solutions of polynomial equations.  One could do about everything but establish order among them.

Numbers like π and the frequently-used e (base of the natural logarithms, and heavily in use) could later be proved impossible to find polynomial equations with π or e (among other numbers) from which one can find a root. Such numbers are called transcendental -- as they "go beyond" computation as they are un-algebraic.