1.5    Complex Numbers

Complex numbers were developed with the need to solve quadratic equations. For example, the equation x2+1 = 0 has no solutions in the real numbers. Trying to solve this would give x2 = -1 and so x would have to be a square root of -1. We define the imaginary number i to be a square root of -1 so that i2 = -1. The number i is sometimes denoted by the symbol . The other square root of -1 is written -i. Now the equation x2+1 = 0 has the two solutions i and -i.

A complex number is of the form a+bi where a and b are real numbers. The number a is called the real part of the complex number and b the imaginary part. The number b is sometimes referred to as the coefficient of i.

If b = 0, then the complex number is just a real number. So every real number may be thought of as a complex number. If a is zero the numbers of the form bi are called imaginary numbers. Examples of complex numbers are -3+2i, 0.3-0.257i. Note that the coefficient of i in the last example is negative and we wrote 0.3 - 0.257i rather than 0.3 + (-0.257)i.

Like other sets of numbers, we can define arithmetic operations on complex numbers. The arithmetic of complex numbers is as follows. To add (or subtract) two complex numbers we simply add (or subtract) the real parts and add (or subtract) the imaginary parts.
Example: Find the sum and difference of -2+5i and 4-6i.

(-2+5i) + (4-6i) = -2 + 4 + 5i - 6i = 2 - i.

(-2+5i) - (4-6i) = -2 -4 + 5i + 6i = -6 + 11i.
The multiplication of two complex numbers is a little more complicated but if you remember that i2 = -1 and apply multiplication of binomials it is not difficult.

Example: Find the product of -2+5i and 4-6i.

(-2+5i)(4-6i) =
-8 + 20i +12i - 30i2=
-8 + 20i +12i + 30=
22 + 32i

Division of complex numbers is the most complicated of the four basic arithmetic operations. We want to know what complex number is represented by, (c+di)/(a+bi). That is what complex number do we get when we divide c+di by a+bi?

To describe this, we first need to define the complex conjugate. The complex conjugate of the complex number a+bi is a-bi (i.e. just take the opposite of the imaginary part). The product of a complex number with its conjugate has a very simple form, For (a+bi)(a-bi) = a2-abi+abi-b2i2 = a2+b2 which is a real number. We can use this fact to perform complex division by converting the denominator into a real number.

To divide a complex number by another complex number, multiply the numerator and the denominator by the complex conjugate of the denominator.

Example: Divide -2+5i by 4-3i.

In this exercise set, you are given two complex numbers and are to perform the indicated arithmetic operation. Your answer should be a complex number. All answers involve integers or fractions in the real and imaginary parts of the answer. Do not convert to decimal notation.

The first four questions involve addition, subtraction, multiplication and division, respectively. After that, the operation is random. Make sure that you try plenty of these problems.
Complex Arithmetic