Complex Numbers

Complex numbers can be thought of as an extension of the set of real numbers to include the imaginary numbers. These numbers must obey the laws, such as the distributive law, which are already in place. This they do with two exceptions, the fact that the "sum" a + bi must be left uncombined, and the law i² = -1, which runs counter to the rule that the product of two numbers of like sign is positive.

Arithmetic with complex numbers is much like the "arithmetic" of binomials such as 5x + 7 with an important exception. When such a binomial is squared, the term 25x² appears, and it doesn't go away. When a + bi is squared, the i² in the term b²i² does go away. It becomes -b². These are the rules:

• Equality: To be equal two complex numbers must have equal real parts and equal imaginary parts. That is a + bi = c + di if and only if a = c and b = d.
• Addition: To add two complex numbers, add the real parts and the imaginary parts separately. The sum of a + bi and c + di is (a + c) + (b + d)i. The sum (3 + 5i) + (8 - 7i) is 11 - 2i.
• Subtraction: To subtract a complex number, subtract the real part from the real part and the imaginary part from the imaginary part. The difference (a + bi) - (c + di) is (a - c) + (b - d)i; (6 + 4i) - (3 - 2i) is 3 + 6i.
• Zero: To equal zero, a complex number must have both its real part and its imaginary part equal to zero: a + bi = 0 if and only if a = 0 and b = 0.
• Opposites: To form the opposite of a complex number, take the opposite of each part: -(a + bi) = -a + (-b)i. The opposite of 6 - 2i is -6 + 2i.
• Multiplication: To form the product of two complex numbers multiply each part of one number by each part of the other: (a + bi)(c + di) = ac + adi + bci + bdi², or (ac - bd) + (ad + bc)i. The product (5 - 2i)(4 - 3i) is 14 - 23i.
• Conjugates: Two numbers whose imaginary parts are opposites are called "complex conjugates." These complex numbers a + bi and a - bi are conjugates. Pairs of complex conjugates have many applications because the product of two complex conjugates is real: (6 - 12i)(6 + 12i) = 36 - 144i², or 180.
• Division: Division of complex numbers is an example. Except for division by zero, the set of complex numbers is closed with respect to division: If a + bi is not zero, then (c + di)/(a + bi) is a complex number. To divide Figure 1. Illustration by Hans & Cassidy. Courtesy of Gale Group. c + di by a + bi, multiply them both by the conjugate a - bi, which eliminates the need to divide by a complex number. For example

While the foregoing rules suffice for ordinary complex-number arithmetic, they must often be coupled with ingenuity for non-routine problems. An example of this can be seen in the problem of computing a square root of 3 - 4i.

One starts by assuming that the square root is a complex number a + bi. Then 3 - 4i is the square of a + bi, or a² - b² + 2abi.

For two complex numbers to be equal, their real and imaginary parts must be equal

Solving these equations for a yields four roots, namely 2, -2, i, and -i. Discarding the imaginary roots and solving for b gives 2 - i or -2 + i as the square roots of 3 - 4i. These roots seem strange, but their squares are in fact 3 - 4i.