Complex Numbers

Any number of the form a + bi, where a andb are real numbers and i is an imaginary number whose square equals -1.

If a and b are real numbers, the number a+bi is a complex number, and it is said to be written in standard form. If b=0, the number a + bi= a is a real number. If b doesn’t equal 0, the number a + bi is an imaginary number.

Examples:        3+5i

                        6+4i

Adding and Subtracting Complex Numbers

(3-i) + (2+3i)= 3-i+2+3i                Remove Parenthesis

                  = 3+2-i+3i               Group Like terms

                  = 5+2i                     Write in standard form

Multiplying Complex Numbers

(i)(3i)= 3i²                                             Multiply

            =(3)(-1)              i²=-1

            =-3                   Simplify

(2-i)(4+3i)= 8+6i-4i-3i²     Product of binomials

            = 8+6i-4i-3(-1)    i²=-1

            = 8+3+6i-4i        Group like terms

            =11+2i

Complex Conjugate

(a+bi)(a-bi)

Dividing Complex Numbers

2+3i/4-2i= 2-3i(4+2i)/4-2i(4+2i)     Multiply numerator and denominator by conjugate

            = 8+4i+12i+6i²/16-4i²       Expand

            = 8-6+16i/16+4              i²=-1

            =(2+16i)/20                    Simplify

            = 1/10 + 4/5i                  Write in standard form

Applications

Vertical axis of graph is imaginary axis

Horizontal axis of graph is real axis