2.5 Introduction to Complex Numbers

11/7/2012

Quick ReviewIf a number doesn’t show an exponent, it is understood that the number has an exponent of 1. Ex: 8 = 81 , x = x1 , -5 = -51 Also, any number raised to the Zero power is equal to 1Ex: 30 = 1 -40 = 1

Exponent Rule:When multiplying powers with the same base, you add the exponent.x2 • x3 = x5

y • y7 = y8

The square of any real number x is never negative, so the equation x2 = -1 has no real number solution.

To solve this x2 = -1 , mathematicians created an expanded system of numbers

using the IMAGINARY UNIT, i.

1i

12 i

Simplifying i raised to any power

1oi

iiiii 1123

ii 1

12 i

iiiii 1347

iiiii 1145

111246 iii

111224 iii

111448 iii

Do you see the pattern yet?

The pattern repeats after every 4.So you can find i raised to any power by dividing the exponent by 4 and see what the remainder is. Raise i to the remainder and determine its value.

22 :Ex i• Step 1. 22÷ 4 has a remainder of 2 • Step 2. i22 = i2

1 22 i

51 :Ex i• Step 1. 51 ÷ 4 has a remainder of 3 • Step 2. i51 = i3

ii 51

Checkpoint Find the value of

1. i 15

2. i 20

 3. i 61 

 4. i 122

i 3 = -i

i 0 = 1 

i 1 = i 

i 2 = -1

Complex Number

Is a number written in the standard form a + biwhere a is the real partand bi is the imaginary part.

State the real and imaginary part of each complex number.

9i

Real Imaginary

0 9i

0

Add/Subtract the real parts, then add/subtract the imaginary parts

Adding and Subtracting

Complex Numbers

Add Complex Numbers

Write as a complex number in standard form.

2i3(

(

+ i1(

(

–+

SOLUTION

Group real and imaginary terms.

2i3(

(

+ i1(

(

–+ = 13 + 12i i+ –

Write in standard form.

= 4 + i

Subtract Complex Numbers

Write as a complex number in standard form.

Simplify.= 5 + 0i

2i6(

(

– – 2i1(

(

–

SOLUTION

Group real and imaginary terms.

2i6(

(

= 16 2i2i +– – 2i1(

(

– – –

Write in standardform.

= 5

-1 + 2i

Checkpoint

Write the expression as a complex number in a + bi form.

Add and Subtract Complex Numbers

1. 2i4(

(

– + 3i1(

(

+ ANSWER i5 +

2. i3((

– + 4i2((

+ ANSWER 3i5 +

ANSWER 3i2 +3. 6i4(

(

+ 3i2(

(

+–

4. 4i2(

(

+ 7i2(

(

+–– ANSWER 3i4– –

Multiply Complex Numbers

Write the expression as a complex number in standard form.

a. b.1( 3i

(

+–2i 3i6(

(

+ 3i4(

(

–

SOLUTION

Multiply using distributive property.

1( 3i

(

+–2i = 2i 6i 2– +a.

1

(

(–2i 6–= + Use i 2 1.= –

6 2i––= Write in standard form.

1

:2 i

remember

Multiply Complex Numbers

Multiply using FOIL. b. 3i6(

(

+ 3i4(

(

– 24 18i– 12i+ 9i 2–=

24 6i– – 9i 2= Simplify.

24 6i– – 1

(

(–9= Use i 2 1.= –

6i33 –= Write in standard form.

Divide Complex Numbers

54 𝑖

multiply top and bottom by i

5 𝑖4 𝑖2

= = -

Complex Conjugates

Two complex numbers of the form a + bi and a - bi

Their product is a real number becauseEx: (3 + 2i)(3 – 2i) using FOIL9 – 6i + 6i - 4i 2

9 – 4i2 i2 = -19 – 4(-1) = 9 + 4 = 13Is used to write quotient of 2 complex numbers in standard form (a + bi)

SOLUTION

2i3 +

2i1 –

2i3 +

2i1 –

2i1 +

2i1 += •

Multiply the numerator and the denominator by 1 2i, the complex conjugate of 1 2i.

+–

Divide Complex Numbers

Write as a complex number in a + bi form.2i3 +

2i1 –

Multiply using FOIL. 1

2i3 6i+ + 4i 2+

2i2i+ – 4i 2–=

3 8i+ 1

(

(–4+

1 – 1

(

(–4= Simplify and use i 2 1.= –

8i+–1

5= Simplify.

51–

58 i+= Write in standard form.

Checkpoint

Write the expression as a complex number in standard form.

Multiply and Divide Complex Numbers

1. i2(

(

–3i ANSWER 6i3 +

2. ( 2i1(

+ i2((

– ANSWER 3i4 +

3.i2 +

i1 –ANSWER

2

1+2

3i

Graphing Complex Number

Real axis

Imaginary axis

Ex: Graph 3 – 2i

3

2

To plot, start at the origin, move 3 units to the right and 2 units down

3 – 2i

Ex: Name the complex number represented by the points.

A

D

C

B

Answers:A is 1 + iB is 0 + 2i = 2iC is -2 – iD is -2 + 3i

Homework

WS 2.5#1-12all, 13-27odd, 31-34all