1 what you will learn lots of vocabulary! a new type of number! how to add, subtract and multiply...

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What you will learn

Lots of vocabulary! A new type of number! How to add, subtract and multiply this

new type of number How to graph this new type of number How to simplify expressions involving

this new type of number

Objective: 5.4 Complex Numbers

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Lions, Tigers and Complex Numbers…oh my!

What is:

Mathematicians created an “imaginary unit” to help us do math with negative square roots.

i =

i is called the imaginary unit.

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Simplifying with “i”

Example:

How do we simplify

What is i2?

Any guesses for 5i2?

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You Try! Simplify:

1.

2.

3. 12i2

7

20


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Solving a Quadratic Equation Solve 3x2 + 10 = -26


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You Try! Solve 2x2 + 26 = -10


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More Vocabulary We are adding to our “types” of numbers. We

had whole, integer, rational, irrational and real. We are adding “complex” numbers. A complex number is of the form a + bi where a

and b are real numbers. This is called standard form.

“a” is the real part. “bi” is the imaginary part. If you have both an “a” term and a “bi” term,

the number is an imaginary number. If you have only a “bi” term, the number is a

pure imaginary number.


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“Types” of Numbers – Expanded!

Whole 1, 2, 3

Integer -3, -2, 0

Rational ¾, ½, -1/2

Real Numbers

Irrational

,2

Imaginary Numbers

2+3i, 5-5i

Pure Imaginary Numbers

-4i, 6i

Complex Numbers


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Graphing Complex Numbers We graph complex numbers in the

“complex plane”.

Real axis

Imaginary axis


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Graphing continued… Plot the complex numbers in the complex

plane.

2 – 3i -3 + 2i 4i

Real axis

Imaginary axis

2i

4i

-2i

-4i

2 4-4


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Adding and Subtracting Complex Numbers

Like adding and subtracting “like terms”1. ( 4 – i) + (3 + 2i)

2. (7 – 5i) – (1 – 5i)

3. 6 – ( -2 + 9i) + (-8 + 4i)


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You Try! Write the expression as a complex

number in standard form.1. ( -1 + 2i ) + (3 + 3i)

2. (2 – 3i) – (3 – 7i)

3. 2i – (3 + i) + (2 – 3i)


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Multiplying Complex Numbers You will use the distributive property or

FOIL.Write the expression as a complex number

in standard form:1. 5i(-2 + i)

2. (7 – 4i)(-1 + 2i)


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You Try

Write the expression as a complex number in standard form:

1. -3i(3 + 2i)

2. (3 + 2i)(5 - 2i)


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Complex Conjugates What happens when we multiply:

(6 + 3i)(6 – 3i)

(6 + 3i)(6 – 3i) are called complex conjugates.


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Simplifying “Division” Problems We can use complex conjugates to simplify

complex numbers of the form:

i

i

21

35


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You Try Write the quotient in standard form.

i

i

21

113


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Homework page 277, 18-34 even, 38-42 even, 48-

52 even, 56-60 even, 80-85 all

1 what you will learn lots of vocabulary! a new type of number! how to add, subtract and multiply...

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