(6 A) ­ Complex Numbers.notebook February 18, 2015Bell Work

4) Graph y = -x2 - 4x + 5

Vertex:

AOS:

y - int:

Compare to Parent:D: R:

Factor:

1) 4x2 + 26x + 12 2) 25x2 - 1

Solve by Factoring

3) x2 = -6x + 16

(6 A) ­ Complex Numbers.notebook February 18, 2015

1.6 Complex Numbers

a + bi standard form of a complex number

real partimaginary part

i - imaginary number

i 2

= ____

i = - 1

-1 

- a = -1 *a = -1 * a = i a√ √ √ √

1+ 3i

3i +1

Real Life Context

Complex Numbers are useful in representing a phenomenon that has two parts varying at the same time, for example an alternating current. Also, radio waves, sound waves and microwaves have to travel through different media to get to their final destination. There are many instances where, for example, engineers, doctors, scientists, vehicle designers and others who use electromagnetic signals need to know how strong a signal is when it reaches its destination. The two parts in this context are: the rotation of the signal and its strength. The following are examples of this phenomenon:

• A microphone signal passing through an amplifier

• A mobile phone signal travelling from the mast to a phone a couple of miles away

• A sound wave passing through the bones in the ear

• An ultrasound signal reflected from a fetus in the womb

• The song of a whale passing through miles of ocean water

(6 A) ­ Complex Numbers.notebook February 18, 2015

Complex Numbers are also used in:

• The prediction of eclipses

• Computer game design

• Computer generated images in the film industry

• The resonance of structures (bridges, etc.)

• Analyzing the flow of air around the wings of a plane in aircraft design

i 2 = √-1 * √ -1 = -1

i 3 = i 2 * i =√-1* √ -1 * i = -i

-1 * i = -i

(6 A) ­ Complex Numbers.notebook February 18, 2015

i =

i 2 =

i 3 =

i 4 =

i 5 =

i 6 =

i 7 =

i 8 =

This pattern of powers, signs, 1's, and i's is a cycle:

Taking the Square Root of a Negative Number

a. √-5 b. √-9 c. √-20

(6 A) ­ Complex Numbers.notebook February 18, 2015

Simplify

a. √-21 b. √-144 c. (-5i )2

Adding and Subtracting Complex Numbers

Write the expression as a complex number in standard form.

(-1+2i) +(3+3i)

add the real parts togetheradd the imaginary parts together

think of them as like terms

(6 A) ­ Complex Numbers.notebook February 18, 2015

Write the expression as a complex number in standard form.

(2-3i) - (3-7i)

Write the expression as a complex number in standard form.

2i +(3 +i) + (2-3i)

(6 A) ­ Complex Numbers.notebook February 18, 2015

Multiplying Complex Numbers

a. -i(3 + i)

b. (2 + 3i)(-6 - 2i)

Write the expression as a complex number in standard form.

Write the expression as a complex number in standard form.

b. (3 - 4i)(3 + 4i)

a. (1 + 2i)(1 - 2i)

(6 A) ­ Complex Numbers.notebook February 18, 2015

Dividing Complex Numbers

Write the quotient of the complex number

2 - 7i1 + i

_______

Dividing Complex Numbers

3 + 11i -1 - 2i_________

Write the quotient of the complex number

(6 A) ­ Complex Numbers.notebook February 18, 2015

Assignment:p. 45 # 4-10 even, 14 - 24 even, 25,

30, 32