Imaginary & Complex Numbers

MAC 1140 – Section 1.6 – Complex Numbers

A. Complex Numbers (Emphasize standard form) Numbers of the form: a + bi where a and b are real and i = 1

Adding complex numbers - Combine like terms and write your answer in standard form

1. (4 + 3i) + (5 - 2i) 2. (-2 - 6i) + (4 - 5i)

Subtracting complex numbers – Add the opposite

1. (4 + 3i) - (5 - 2i) 2. (-2 - 6i) - (4 - 5i)

Multiplying complex numbers: Distribute or FOIL, simplify, and write your solution in standard form. Note: Since i = 1 , i2 = -1

1. (3i) (9i) 2. (-7i) (3i) 3. 3(2 + 4i) 4. 2i (3 + 5i)

5. (5 + 2i) (4 - i) 6. (3 – 4i)2 7. (3 + 2i)(3 – 2i)

Dividing complex numbers: Multiply the numerator and denominator by the conjugate of the denominator (the same expressions with the opposite sign on the imaginary part of the complex number), then simplify results and write your answer in standard form 4  2 1. 2. i 3i

5 5i 3. 4. 3  i 2  i

Raising "i " to a power: i1  i i 2  i i  1 i 3  i 2 i  i i 4  i 2  i 2 1 i 5  i 4  i  i i 6  i 4  i 2  i 2  1 i 7  i 4 i 3  i i8  i 4  i 4 1 i 9  ______...... i 43  _____ i50  ______i115  ______

Simplify:

1. i25 = 2. i19 = 3. i216 = B. Using Imaginary/Complex numbers to simplify radical expressions

Review:  25  25  1  5i and 10  1  10  i 10 Use this property to simplify:

1. 121 2.  7 3.  98

The most common occurrence of negative numbers inside a radical is the result of using the quadratic formula to solve equations. For example: To solve:

0 = x2 – 5x + 9 or 0 = x2 – 2x + 7

We get: 5  25  36 5  11 2  4  28 2   24 x =  or x =  2 2 2 2

Which simplifies to:

5  i 11 2  2i 6 x = or x = 2 2

And converts to standard from as: 5 11 2 2i 6 x =  i or x =  1  i 6 2 2 2 2

Solve these quadratic equations and simplify your results:

1. 3x2 + 18 = 0 2. x2 = 3x – 7 3. 4x = 3 + 3x2