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Complex Number

  • Definition: By a complex number we shall mean an ordered pair of real numbers which we denote by (x_1, x_2). The first member, x_1, is called the real part of the complex number; the second member, x_2 is called the imaginary part. Two complex numbers x=(x_1, x_2) and y=(y_1, y_2) are called equal, and we write x = y, if, and only if, x_1 = y_1 and x_2 = y_2. We define the sum x+y and the product by the equations x+y = (x_1 + y_1, x_2 + y_2), xy = ((x_1)(y_1) - (x_2)(y_2), (x_1)(y_2) + (x_2)(y_1)).

    Source: Mathematical Analysis, second edition by Tom M. Apostol

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