>>13388280My bad. The reals are constructed analytically, which is an entirely separate branch of math from algebra. Analysis deals with sequences approaching a value, which is required to put values like pi and e on the number line.
However, algebra deals with the roots of polynomials, for example what value of "x" that satisfies the relation . This is not possible in the rational numbers, but it is possible in the reals, since you can produce a sequence of values that approach the square root of 2, for example by the Babylonian method.
Back when algebra was getting on its feet, it was noticed that polynomials like had no solution, or did not have a sequence which approached a value. Eventually, when cubic polynomials were examined in detail, it became clear that an "imaginary" value was required to express the root in the cubic formula, even if the root itself was real. Thus, the complex numbers were born. Later geometric reasoning arranged such numbers onto a plane.
It turns out that extending the real numbers with complex numbers is enough to express the root of any polynomial (with real coefficients). This is called the "algebraic completion" of the reals. There's still a bit of a lie here, since quintic and higher polynomials cannot be guaranteed to have expressible roots, and some people don't believe in the "completion" of the reals, but it's more or less satisfactory.
If you're interested in other roots of unity, you can try studying the cyclotomic polynomials, which have to do with factoring the polynomial . They begin:
(Ran out of room for the fifth)
