Complex numbers

Ordered pairs (x, y) of real numbers x and y arise naturally as the coordinates of a point in the Euclidean plane, and we shall adopt the view that the plane is the set of ordered pairs of real numbers. Complex numbers arise by denoting the point (x, y) by a new symbol x + iy, and then introducing simple algebraic rules for the numbers x + iy with the assumption that i2 = −1. A complex number, then, is a number of the form x + iy, and we stress that this is no more than an alternative notation for (x, y). Thus we see that x + iy = u + iv if and only if (x, y) = (u, v); that is, if and only if x = u and y = v. Complex notation has enormous benefits, not least that while a real polynomial need not have any real roots, it always has complex roots; for example, x2 + 1 has no real roots but it has complex roots i and −i. As we identify the point (x, 0) in the plane with the real number x, so we also identify the complex number x + 0i with x. We denote the set of complex numbers by ℂ.