Section 1.4:  Complex Numbers

The last set of numbers that we need to add to our earlier sets is the complex numbers, which occur because of the use of imaginary numbers. We had to add them because someone wanted to solve the simple equation:
x2 + 1 = 0

or
x2 = -1


This couldn’t be solved with real numbers because the square of a real number is never negative.   So mathematicians had to add a new type of number, the imaginary number.  (Since we already had real numbers, …)

Definition:  The imaginary unit i is defined as

Using i, the square root of any negative number may be expressed as a real multiple of i.


We get Complex Numbers by adding real numbers and multiples of the imaginary unit together.
Complex number = real part + imaginary part
                           = a + bi

If b = 0, we have a real number
If a = 0, we have an imaginary number

A complex number is simplified, if it is written in the standard form a + bi.

Equality of complex numbers – Complex numbers are equal if and only if their real parts are equal AND their imaginary parts are equal.



Operations with Complex Numbers

To add or subtract complex numbers, add or subtract their real parts AND add or subtract their imaginary parts.



Example 1.4.1


Multiplying Complex Numbers

When multiplying, treat i like a variable. At the end, reduce the power of i using the following relations:

All powers of i greater than 4 can be reduced to a power between 1 and 4.


Example 1.4.2

Example 1.4.3


Example 1.4.4


If you noticed, in the last example, I was multiplying conjugates. In this case, we call them complex conjugates. The result of multiplying complex conjugates is ALWAYS a real number.

Complex Conjugates and Division

As we noted in the last example, we can multiply complex numbers and get real numbers.

Definition:  The complex conjugate of the number a + bi is a – bi and the complex conjugate of a – bi is a + bi. The multiplication of complex conjugates gives a real number.

Complex conjugates are used to divide complex numbers.  By multiplying the top and bottom by the complex conjugate of the denominator (i.e., multiplying the expression by 1), we get a real number in the denominator.  The real denominator is then used to divide the real parts of the numerator.
Example 1.4.5


Roots of Negative Numbers

For any positive real number b,


We usually write the i in front of the radical to minimize confusion.  It’s hard to tell if the i is under the radical or not.  (Also you can put a tail on the radical.)

When performing operations with square roots of negative numbers, first express all square roots in terms of i. Then perform the operation.
Example 1.4.6
Write the expression in the form a + bi, where a and b are real numbers.

Example 1.4.7
Write the expression in the form a + bi, where a and b are real numbers.

Example 1.4.8
Write the expression in the form a + bi, where a and b are real numbers.