Complex numbers
No real number squares to a negative, so x^2 + 9 = 0 has no real solution. What happens if we allow a number whose square is -1?
Where you are 5 inside
What this area is about
The complex numbers add one new number, i, with i^2 = -1. Every complex number can be written a + bi with a and b real, and the real numbers sit inside as the case b = 0. With them, every quadratic equation ax^2 + bx + c = 0 with real coefficients and a \ne 0 has complex solutions, and the old arithmetic still works.
How these fit together
The definition. Imaginary and complex numbers introduces i, writes \displaystyle \sqrt{-a} = i\sqrt{a} for a > 0, and names the real and imaginary parts.
Arithmetic. Adding, subtracting and multiplying complex numbers treats a + bi like a binomial and replaces i^2 with -1. Conjugates and dividing complex numbers multiplies above and below by the conjugate a - bi, which makes the denominator real.
A picture. The complex plane, modulus and argument plots a + bi at the point (a, b), with size \displaystyle |z| = \sqrt{a^2 + b^2} and a direction angle.
A result. Complex roots come in pairs: if a polynomial has real coefficients and z is a root, so is its conjugate \overline{z}.
A path through it
Standard form first, then adding, subtracting and multiplying; dividing and the complex plane follow, in either order. The pairing of roots comes last, after the discriminant, which itself uses this area.
Finish here and you are ready for the fundamental theorem of algebra and for polar form.