Complex Numbers

   There are a number of ways to define the field of complex numbers, C.
One way is to define it as an extension field of the real numbers, C = R[i] where i is a root of the real polynomial x² + 1.

 Sums and Products

    Definition. The field of complex numbers, C, is the set of ordered pairs of real numbers: C = {(x, y) | x, y ∈ R}, where addition is defined as

\[({x_1},\;{y_1})\; + \;\left( {{x_2},{y_2}} \right)\; = \;({x_1} + \;{x_2},\;{y_1}\; + \;{y_2})\]

and multiplication is defined as

\[\left( {{x_1},\;{y_1}} \right)\left( {{x_2},{y_2}} \right)\; = \;\left( {{x_1}{x_2}\; - \;{y_1}{y_2},{y_1}{x_2}\; + \;{x_1}{y_2}} \right)\]

   Note. As you might expect, we denote z = (x, y) ∈ C as z = x+iy. Geometrically, C is the same as the Cartesian plane, R² (however they are different algebraically; we do not multiply elements of R² together, say. . . though if we take R² as a vector space, then we can add elements). So we can associate any element of C with a point in R². When doing so, we call the x-axis the “real axis” and call the y-axis the “imaginary axis.” We have:

For z ∈ C, we denote x = Re(z) (the " real part of z ") and y = Im(z) (the " imaginary part of z ").

Notice that z = Re(z) + i Im(z), where i = (0, 1).

Note. With (0, 1) denoted as i, we have by the definition of multiplication that
i² = (0, 1)(0, 1) = ((0)(0) - (1)(1), (1)(0) + (0)(1)) = (-1, 0) = -1 + i0 = -1.