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Complex Numbers A complex number is a number that can be expressed in the form a bi, where a and b are real numbers, and i is a solution of the equation x2 =-1, which is called an imaginary number because there is no real number that satisfies this equation. For the complex number abi, a is called the real part, and b is called the imaginary part. Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers, and are fundamental in many aspects of our description of the natural world. Complex Number Operations Complex numbers are added by separately adding the real and imaginary parts of the summands. That is to say, assuming the complex numbers (a bi) and (c di): (a+bi) + (c + di) = (a + c) + (b + d) Subtraction is defined similarly: (a + bi)-(c+di) = (a-c) + (b-d)i The multiplication of two complex numbers is defined by the following formula: (a + bi)(c + di) = (ac-bd) + (bc + ad)i The division of two complex numbers is defined in terms of complex multiplication, which is described above, and real division. When at least one of c and d is non-zero, we have: be - ad i. c+did