Math 101 Intermediate Algebra
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Complex Numbers
Chapter 8, Section 7


Imaginary Numbers

The imaginary unit is and is denoted by i, and

By definition,

i2   =   -1.

For any positive real number n,


Complex Numbers

Complex numbers have the form

a + bi

where a and b are real numbers.

Examples:

  • 7 + 3i
  • 2 - 5i
  • -1 - 4i


Adding and Subtracting Complex Numbers

(a + bi) + (c + di) = (a + c) + (b + d)i

Steps to adding and subtracting complex numbers:

  1. Change all imaginary numbers to bi form.
  2. Add (or subtract) the real parts of the complex numbers.
  3. Add (or subtract) the imaginary parts of the complex numbers.
  4. Write the answer in the form a + bi.


Multiplying Complex Numbers

(a + bi)(c + di) = (ac - bd) + (ad + bc)i

Steps to multiplying complex numbers:

  1. Change all imaginary numbers to bi form.
  2. Multiply the complex numbers as you would multiply polynomials.
  3. Substitute -1 for each i2.
  4. Combine the real parts and the imaginary parts.
  5. Write the answer in the form a + bi.


Complex Conjugates

The complex conjugate of a complex number is a complex number having the same two terms with the sign inbetween changed.

Complex Number Complex Conjugate
7 + 3i 7 - 3i
3 - 2i 3 + 2i
-2 + 5i -2 - 5i
i -i


Dividing Complex Numbers

  1. Change all imaginary numbers to bi form.
  2. Write the division problem as a fraction.
  3. Rationalize the denominator by multiplying both the numerator and denominator by the conjugate of the denominator.
  4. Write the answer in the form a + bi.

Example:   Divide 7 + 3i
7 - 3i

7 + 3i
7 - 3i
= 7 + 3i
7 - 3i
· 7 + 3i
7 + 3i
  = (7 + 3i)(7 + 3i)
(7 - 3i)(7 + 3i)
  = 49 + 21i + 21i + 9i2
49 + 21i - 21i - 9i2
  = 49 + 42i + 9(-1)
49 - 9(-1)
     
  = 40 + 42i
49 + 9
  = 40 + 42i
58
  = 40
58
+ 42
58
i
  = 20
29
+ 21
29
i