Two complex numbers x = a + i b and y = c + i d are multiplied as follows: x y | = | (a + i b)(c + i d) | = | a c + i b c + i a d - b d | = | (a c - b d) + i(a d + b c). In component form, (x, y)(x', y') = (x x' - y y', x y' + y x') (Krantz 1999, p. 1). The special case of a complex number multiplied by a scalar a is then given by a(x, y) = (a, 0)(x, y) = (a x, a y).
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