5.7 Complex Numbers

12/17/2012

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 given any powers

1oi

iiiii 1123

ii 1

12 i

iiiii 1347

iiiii 1145

111246 iii

111224 iii

111448 iiiDo 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. Based on that remainder, you can determine it’s value.

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

1 22 i

50 :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

Properties of Square Root of Negative Number

rrr 11

rir

rriiririri 2

rrrandiiiSince 1 2

rrri or 1 2

1

12

i

i

Example 1 Solve a Quadratic Equation

Solve the equation.

=7x 2 49–a. b. =3x 2 5– 29–

SOLUTION

Write original equation. =7x 2 49–a.

Divide each side by 7.=x 2 7–

Take the square root of each side.=x +– 7–

Write in terms of i.=x +– 7i

Example 1 Solve a Quadratic Equation

Write original equation. b. =3x 2 29–5–

Add 5 to each side.=3x 2 24–

Divide each side by 3.=x 2 8–

Write in terms of i.=x +– 8i

Take the square root of each side. =x +– 8–

Simplify the radical. =x +– 2i2

22248

Checkpoint

Solve the equation.

Solve a Quadratic Equation

1. =x 2 3– ANSWER 3,i 3i–

2. =x 2 7–

3. =x 2 20–

4. =x 2 3 2+ –

5. =y 2 4– 12–

ANSWER 7,i 7i–

ANSWER 5,2 52–i i

ANSWER 5,i 5i–

ANSWER 2,2 22–i i

Complex Number

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

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

Adding and Subtracting

Complex Numbers

Example 2 Add Complex Numbers

Write as a complex number in standard form.

2i3(

(

+ i1(

(

–+

SOLUTION

Group real and imaginary terms.

2i3(

(

+ i1(

(

–+ = 13 + 12 i+ –

Write in standard form.

= 4 + i

i

Example 3 Subtract Complex Numbers

Write as a complex number in standard form.

Simplify.= 5 + 0i

2i6(

(

– – 2i1(

(

–

SOLUTION

Group real and imaginary terms.

2i6(

(

= 16 22i i+– – 2i1(

(

– – –

Write in standardform.

= 5

-1 + 2i

Checkpoint

Write the expression as a complex number in standard form.

Add and Subtract Complex Numbers

6. 2i4(

(

– + 3i1(

(

+ ANSWER i5 +

7. i3((

– + 4i2((

+ ANSWER 3i5 +

ANSWER 3i2 +8. 6i4(

(

+ 3i2(

(

+–

9. 4i2(

(

+ 7i2(

(

+–– ANSWER 3i4– –

Checkpoint

Write the expression as a complex number in standard form.

Add and Subtract Complex Numbers

11. 2i1(

(

– + 5i4(

(

+ ANSWER 3i5 +

ANSWER 3i3 +12. i2((

– ((

– 4i1– –

Example 4 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

Example 4 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.

Complex Conjugates

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

Their product is a real number because(3 + 2i)(3 – 2i) using FOIL9 – 6i + 6i -4i2

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.

+–

Example 5 Divide Complex Numbers

Write as a complex number in standard 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.

a + bi

Checkpoint

Write the expression as a complex number in standard form.

Multiply and Divide Complex Numbers

13. i2(

(

–3i ANSWER 6i3 +

14. ( 2i1(

+ i2((

– ANSWER 3i4 +

15.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

5.7 p.264 #17-20, 27/29, 33-35, 40, 43, 45, 46, 52-

54, 64-71

Checkpoint

Solve the equation.

Solve a Quadratic Equation

1. =x 2 3–

2. =x 2 7–

3. =x 2 20–

4. =x 2 3 2+ –

5. =y 2 4– 12–

Checkpoint

Write the expression as a complex number in standard form.

Add and Subtract Complex Numbers

6. 2i4(

(

– + 3i1(

(

+

7. i3((

– + 4i2((

+

8. 6i4(

(

+ 3i2(

(

+–

9. 4i2(

(

+ 7i2(

(

+––

Checkpoint

Write the expression as a complex number in standard form.

Add and Subtract Complex Numbers

10. 2i1(

(

– + 5i4(

(

+

11. i2(

(

– (

(

– 4i1– –

Write the expression as a complex number in standard form.

12. i2(

(

–3i

13. ( 2i1

(

+ i2(

(

–

14.i2 +

i1 –