Warm-Up: December 13, 2011 Solve for x:

022)3

753)2

04)1

2

2

x

xx

x

Complex Numbers

Section 2.1

Imaginary Unit, i

1

12

i

i

Imaginary Numbers Multiples of i are imaginary numbers. Square roots of negative numbers are

imaginary.

ii

5220

39

Complex Numbers Any number that can be written in the form

“a+bi” is a complex number. This is called the standard form.

In “a+bi” both a and b are real numbers a is called the real part b is called the imaginary part A complex number is simplified iff it is written

in standard form, “a+bi”

Operations with Complex Numbers Addition: Combine like terms

Add the real parts, add the imaginary parts Subtraction: Combine like terms

Subtract the real parts, subtract imaginary parts Don’t forget to distribute the negative

Multiplication: Use distributive property or FOIL Replace i2 with -1

Always write final answer in standard form, “a+bi”

Example 1 (like HW #1-8)

iib

iia

3724)

532)

You-Try #1 (like HW #1-8)

iib

iia

3187)

2495)

Example 2 (like HW #9-20)

iib

iia

423)

523)

You-Try #2 (like HW #9-20)

iib

iia

3251)

642)

Complex Conjugates The complex conjugate of “a+bi” is “a-bi” Switch the sign of the imaginary part The product of a complex number and its

complex conjugate is:

22 babiabia

Dividing Complex Numbers1. Multiply the numerator and denominator by

the complex conjugate of the denominator2. FOIL the numerator and simplify3. Denominator becomes a2+b2

4. Write the result in standard form

Example 3 (like HW #21-28)

i

i

24

53

Warm-Up: December 14, 2011

i

i

3

72

Homework Questions?

You-Try #3 (like HW #21-28)

i

i

35

32

Principal Square Root of Negative Numbers

bib Product and quotient properties of square

roots do not apply to square roots of negative numbers.

Take out the i before multiplying or dividing square roots

Example 4 (like HW #29-44) Perform the indicated operations and write the

result in standard form

2412)

2218385)

b

a

You-Try #4 (like HW #29-44) Perform the indicated operations and write the

result in standard form

7618)

1242732)

b

a

Quadratic Formula It works to find complex solutions of quadratic

equations, not just real ones.

a

acbbx

2

42

Example 5 (like HW #45-50) Solve for x. Express solutions in standard

form. 643 2 xx

You-Try #5 (like HW #45-50) Solve for x. Express solutions in standard

form. 01722 xx

Assignment Page 252 #1-49 Every Other Odd