algebra 2 book - · pdf filea. 2x1 2yi 5 6 1 4i b. 3x1 5yi 5 9 2 15i solution because the...

1.1 Write Complex Numbers (MM2N1a, MM2N1b)

1.2 Add and Subtract Complex Numbers (MM2N1b, MM2N1c, MM2N1d)

1.3 Multiply and Divide Complex Numbers (MM2N1b, MM2N1c, MM2N1d)

1.4 Use the Complex Plane (MM2N1b, MM2N1d)

1Uni t 1 Number and Operat ions

Georgia Performance Standards: MM2N1a, MM2N1b

Goal Use the imaginary unit i to write complex numbers.

Vocabulary

A number r is a square root of a number s if r 2 5 s.

The expression Ï}

s is called a radical. The symbol Ï}

is a radical sign and the number s

beneath the radical sign is the radicand of the expression.

Ï}

a b 5 Ï}

a Ï}

b Product property of radicals

Î} a}b 5

Ï}

a}

Ï}

b Quotient property of radicals

The imaginary unit i is defi ned as i 5 Ï}

21 where i 2 5 21.

By adding real numbers to real multiples of this imaginary unit, the set of complex

numbers is obtained. A complex number written in standard form is a number a 1 bi

where a and b are real numbers. The number a is the real part of the complex number,

and the number bi is the imaginary part.

If b ? 0, then a 1 bi is an imaginary number. If a 5 0 and b ? 0, then a 1 bi is a pure

imaginary number. The diagram below shows how different types of complex numbers

are related.

Two complex numbers a 1 bi and c 1 di are equal if and only if a 5 c and b 5 d. For

example, if x 1 yi 5 5 2 3i, then x 5 5 and y 5 23.

LESSON

1.1

Use properties of square roots

Simplify the expression.

a. Ï}

8 Ï}

6 5 Ï}

48 5 Ï}

16 Ï}

3 5 4 Ï}

3 Product property

b. Î} 5}36 5

Ï}

5 }

Ï}

36 5

Ï}

5 }6 Quotient property

5 2 5i2 1 3i

Complex Numbers (a 1 bi )

24i 6i

Pure Imaginary Numbers

(0 1 bi, b Þ 0)

Imaginary Numbers

(a 1 bi, b Þ 0)

Real Numbers

(a 1 0i )

219292

p 3

2 Georg ia H igh Schoo l Mathemat ics 2

Guided Practice for Example 3

Find real numbers x and y to make the equation true.

11. 6x 1 3yi 5 24 1 48i 12. 3x 2 4yi 5 221 2 32i

Equality of complex numbers


a. 2x 1 2yi 5 6 1 4i b. 3x 1 5yi 5 9 2 15i

Solution

Because the numbers are equal, their real parts must be equal and their imaginary parts

must be equal.

a. 2x 5 6 Set the real parts equal to each other.

x 5 3 Divide each side by 2 to solve for x.

2y 5 4 Set the imaginary parts equal to each other.

y 5 2 Divide each side by 2 to solve for y.

b. 3x 5 9 Set the real parts equal to each other.

x 5 3 Divide each side by 3 to solve for x.

5y 5 215 Set the imaginary parts equal to each other.

y 5 23 Divide each side by 5 to solve for y.

Write complex numbers in standard form

a. Ï}

29 5 Ï}

9(21) 5 Ï}

9 Ï}

21 5 3i

b. Ï}

25 5 Ï}

5(21) 5 Ï}

5 Ï}

21 5 iÏ}

5

c. Ï}

212 5 Ï}

12(21) 5 Ï}

12 Ï}

21 5 2iÏ}

3

d. 4 1 Ï}

216 5 4 1 Ï}

16(21) 5 4 1 Ï}

16 Ï}

21 5 4 1 4i

Guided Practice for Examples 1 and 2


1. Ï}

300 2. Ï}

98 3. Î} 19}144 4. Î} 20}

81

Write the complex number in standard form.

5. Ï}

21 6. Ï}

225 7. Ï}

22

8. Ï}

220 9. 1 1 Ï}

236 10. 7 2 Ï}

227

MM2N1a Write square roots of negative numbers in

imaginary form.

MM2N1b Write complex numbers in the form a 1 bi.


LESSON

3.65Exercise Set A

LESSON

1.1MM2N1a Write square roots of negative numbers in

imaginary form.



1. Ï}

500 2. Ï}

108 3. Ï}

242 4. Ï}

128

5. Ï}

125 6. Ï}

343 7. Î} 25}169 8. Î} 16}

36

9. Î} 23}121 10. Î} 6}

9 11. Î} 27}

4 12. Î} 32}

25

Match the complex number with the correct standard form.

13. 10 1 Ï}

216 A. 4i

14. Ï}

216 B. 10 1 4i

15. 10 2 Ï}

216 C. 10 2 4i


16. Ï}

24 17. Ï}

264 18. Ï}

2100

19. Ï}

232 20. Ï}

217 21. Ï}

275

22. Ï}

245 23. Î}2 108}49 24. Î}2 68}

81

25. 3 1 Ï}

22 26. 5 2 Ï}

221 27. 9 1 Ï}

281

28. 22 1 Ï}

218 29. 24 2 Ï}

49 30. 8 2 Ï}

248

31. Ï}

211 1 6 32. 2Ï}

225 1 12 33. Ï}

144 2 11

Error Analysis Describe and correct the error in writing the complex number

in standard form.

34.5 2 Ï

}

29 5 5 2 1Ï}

29 2

5 5 2 (23)

5 5 1 3

5 8

35. 3 1 Ï

}

28 5 3 1 Ï}

8(21)

5 3 1 Ï}

8 Ï}

21

5 3 1 2i Ï}

2

5 5i Ï}

2


36. 2x 2 4yi 5 8 1 12i 37. 5x 1 yi 5 10 1 i 38. 23x 1 2yi 5 15 2 4i

39. 22x 1 6yi 5 14 2 24i 40. 4x 1 3i 5 16 2 yi 41. 11 2 5yi 5 x 1 10i

42. x 1 7yi 5 26 1 21i 43. 24x 2 4yi 5 4 1 48i 44. 22 2 8i 5 11x 1 4yi

45. The complex numbers 22xi 1 8 and 6 1 4yi are equal. Find the values of x and y.

XX


LESSON

3.65Exercise Set B

LESSON

1.1MM2N1a Write square roots of negative numbers in

imaginary form.



1. Ï}

52 2. Ï}

99 3. Ï}

175

4. Ï}

392 5. Ï}

375 6. Ï}

270

7. Î} 76}225 8. Î} 200}

49 9. Î} 405}

4

10. Î} 75}256 11. Î} 216}

625 12. Î} 140}

196

Match the complex number with the correct standard form.

13. 24 1 Ï}

250 A. 24 2 2i Ï}

19

14. 2Ï}

225 B. 25i

15. 24 2 Ï}

276 C. 24 1 5i Ï}

2


16. Ï}

2121 17. Ï}

2169 18. Ï}

2225

19. Ï}

278 20. Î}2 63}121 21. Î}2 208}

25

22. Ï}

2180 23. Ï}

2(2128) 24. Ï}

2168

25. 8 1 Ï}

28 26. 215 1 Ï}

228 27. 1 2 Ï}

2400

28. 221 1 Ï}

252 29. 4 2 Ï}

17 2 49 30. 18 2 Ï}

97 2 48

31. Ï}

2161 2 4 32. 2Î}2 125}9 2 15 33. Ï

}

2 120}169 1 29


34. x 2 3yi 5 5 1 15i 35. 25x 2 yi 5 20 2 9i

36. 26x 1 7yi 5 18 1 28i 37. 24x 1 5yi 5 22 2 35i

38. 4x 2 7i 5 22 2 2yi 39. 3}4 2 2yi 5 2x 1 5i

40. 5x 2 7yi 5 2 7}10 2 56i 41. 22x 2

1}3 yi 5 26 1 4i

42. 27 2 8i 5213x 1 3yi 43. 214 1 12i 5 7x 2 6yi

44. The complex numbers 2}5 xi 1 3 and 24 2

1}4 yi are equal. Find the values of x and y.


Georgia Performance Standards: MM2N1b, MM2N1c, MM2N1d

Goal Add and subtract complex numbers.

Vocabulary

To add (or subtract) two complex numbers, add (or subtract) their real parts and their

imaginary parts separately.

Sum of complex numbers:

(a 1 bi) 1 (c 1 di) 5 (a 1 c) 1 (b 1 d)i

Difference of complex numbers:

(a 1 bi) 2 (c 1 di) 5 (a 2 c) 1 (b 2 d)i

LESSON

1.2


Write the expression as a complex number in standard form.

1. (5 2 2i) 1 (2 1 7i) 2. (21 1 i) 1 (6 2 3i) 3. (3 1 2i) 1 (25 2 i)

Add complex numbers


a. (7 2 i) 1 (5 1 3i)

b. (3 1 2i) 1 (7 1 6i)

c. (25 1 3i) 1 (22 2 8i)

Solution

a. (7 2 i) 1 (5 1 3i) 5 (7 1 5) 1 (21 1 3)i Complex addition

5 12 1 2i Write in standard form.

b. (3 1 2i) 1 (7 1 6i) 5 (3 1 7) 1 (2 1 6)i Complex addition


c. (25 1 3i) 1 (22 2 8i) 5 (25 2 2) 1 (3 2 8)i Complex addition


Subtract complex numbers


a. (6 2 5i) 2 (1 1 2i)

b. (4 1 2i) 2 (5 2 i)

c. (26 1 5i) 2 (23 1 9i)


Guided Practice for Examples 2 and 3


4. (9 2 4i) 2 (22 1 3i) 5. (6 1 2i) 2 (5 2 4i) 6. (23 1 7i) 2 (23 2 i)


7. 8 2 (2 1 4i) 1 3i 8. 2 2 (7 1 2i) 2 4i 9. 27i 1 (9 1 i) 1 6

Add and subtract complex numbers


a. 11 2 (8 1 7i) 1 3i

b. 9 2 (10 1 2i) 2 5i

c. 5i 1 (24 1 4i) 1 17

Solution

a. 11 2 (8 1 7i) 1 3i 5 [(11 2 8) 2 7i] 1 3i Complex subtraction

5 (3 2 7i) 1 3i Simplify.

5 3 1 (27 1 3)i Complex addition


b. 9 2 (10 1 2i) 2 5i 5 [(9 2 10) 2 2i] 2 5i Complex subtraction

5 (21 2 2i) 2 5i Simplify.

5 21 1 (22 2 5)i Complex subtraction


c. 5i 1 (24 1 4i) 1 17 5 [24 1 (5 1 4)i] 1 17 Complex addition

5 (24 1 9i) 1 17 Simplify.

5 (24 1 17) 1 9i Complex addition



MM2N1c Add, subtract, multiply, and divide complex numbers.

MM2N1d Simplify expressions involving complex numbers.

Subtract complex numbers (continued)

Solution

a. (6 2 5i) 2 (1 1 2i) 5 (6 2 1) 1 (25 2 2)i Complex subtraction


b. (4 1 2i) 2 (5 2 i) 5 (4 2 5) 1 (2 1 1)i Complex subtraction


c. (26 1 5i) 2 (23 1 9i) 5 (26 1 3) 1 (5 2 9)i Complex subtraction



Match the expression with the correct standard form.

1. (3 1 5i) 2 (22 1 5i) A. 13 1 i

2. (5 1 2i) 1 (8 2 i) B. 24 1 13i

3. (2 1 i) 1 (22 2 3i) C. 22i

4. (3 1 4i) 2 (7 2 9i) D. 5

5. Multiple Choice What is the standard form of the expression (3 1 2i) 2 (8 1 3i)?

A. 24 B. 25 1 7i C. 25 2 i D. 5 1 i


6. (23 1 3i) 1 (2 2 i) 7. (8 1 2i) 1 (2 2 3i)

8. (1 2 i) 1 (1 2 i) 9. (5 1 4i) 2 (4 1 2i)

10. (7 1 3i) 2 (1 1 7i) 11. (26 1 2i) 1 (2 2 i)

12. (14 1 19i) 1 (216 2 11i) 13. (28 1 4i) 1 (21 1 13i)

14. (21 1 19i) 2 (4 2 i) 15. (218 2 11i) 1 (214 1 17i)

16. (220 1 4i) 2 16i 17. 6 2 (211 1 9i)

18. 18 1 (22 1 18i) 19. (1 2 17i) 1 (12 2 i)

20. 214 1 (217 1 7i) 2 20i 21. 14 2 (6 2 12i) 1 8i

22. 27i 1 (26 2 19i) 1 5 23. 3 1 (12 1 15i) 2 7i

24. 28i 1 (9 1 16i) 2 16 25. 218i 1 (28 2 4i) 2 6

26. 9 1 (210 1 5i) 1 14i 27. 212 2 (7 2 5i) 1 4i

Error Analysis Describe and correct the error in writing the expression as

a complex number in standard form.

28. (2 1 4i) 2 (21 1 9i) 5 (2 2 1) 1 (4 2 9)i

5 1 2 5i

29.6 2 (3 1 4i) 1 (24i) 5 6 2 (3 1 4i 2 4i)

5 6 2 3

5 3

LESSON

3.65Exercise Set A

LESSON

1.2MM2N1b Write complex numbers in the form a 1 bi.



X

X



1. 6 2 (23 1 8i) 1 5i A. 224 1 3i

2. 28 1 (218 2 20i) 1 12i B. 4 1 14i

3. 10i 1 (6 1 4i) 2 2 C. 226 2 8i

4. 2i 2 (14 2 4i) 2 10 D. 9 2 3i

5. Multiple Choice What is the standard form of the expression 10 2 (26 1 5i) 2 18i?

A. 216 B. 216 2 13i C. 216 2 23i D. 16 1 23i


6. (26 2 29i) 1 (8 2 12i) 7. (8 1 11i) 1 (25 1 28i)

8. (16 2 26i) 1 (23 1 27i) 9. (214 2 6i) 2 (28 1 13i)

10. (17 2 21i) 1 (16 1 2i) 11. (9 1 20i) 2 (14 1 11i)

12. (28 1 9i) 2 (217 2 9i) 13. (25 2 i) 1 (222 1 22i)

14. (27 1 13i) 2 (4 1 6i) 15. (25 1 i) 2 (221 2 23i)

16. (230 1 4i) 2 17i 17. 13 2 (29 1 13i)

18. 24 1 (29 2 8i) 19. (1.5 2 4.2i) 1 (1.2 2 2.6i)

20. (6.3 1 1.5i) 2 (2.5 2 7.1i) 21. (22.7 1 7.8i) 1 (0.6 1 5.3i)

22. 227i 1 (227 2 26i) 1 9 23. 13 2 (23 1 7i) 2 25i

24. 210i 1 (217 1 16i) 2 23 25. 12i 1 (1 2 4i) 2 25

26. 20 1 (223 2 8i) 1 7i 27. 213 2 (25 2 3i) 1 25i

28. Multiple Representations Every complex number corresponds

to a point in the complex plane. (You will learn more about

the complex plane in Lesson 1.4.) The complex plane has a

horizontal axis called the real axis and a vertical axis called the

imaginary axis. The graph shows how you can geometrically

add two complex numbers (in this case, 3 1 i and 2 1 4i)

to fi nd their sum (in this case, 5 1 5i). Use the defi nition of

complex addition to fi nd each of the following sums and then

fi nd the sums by drawing a graph.

imaginary

real1

i

5 1 5i

4i

3

2i

a. (3 1 i) 1 (1 1 3i) b. (25 1 2i) 1 (6 2 3i)

c. (4 2 4i) 1 (23 2 i) d. (8 1 2i) 1 (24 2 6i)

LESSON

3.65Exercise Set B

LESSON





LESSON

1.2

Problem Circuit components such as resistors, inductors, and capacitors all

oppose the fl ow of current. This opposition is called resistance for resistors and

reactance for inductors and capacitors. A circuit’s total opposition to current fl ow

is impedance. All of these quantities are measured in ohms (V).

Component

and symbol

Resistor Inductor Capacitor

Resistance

or reactanceR L C

Impedance R Li 2CiAlternating current source

4

5

6

The table shows the relationship between a component’s resistance or reactance and its contribution to impedance. A series circuit is also shown with the resistance or reactance of each component labeled.

The impedance for a series circuit is the sum of the impedances for the individual components. Find the impedance of the circuit shown above.

What do you know? The resistance or reactance of each component in the circuit and how it is related to impedance

What do you want to fi nd out? The impedance of the circuit

Use what you know to fi nd the impedance of the circuit by adding the contribution to impedance of each component.

The resistor has a resistance of 6 ohms, so its impedance is 6 ohms. The inductor has a reactance of 5 ohms, so its impedance is 5i ohms. The capacitor has a reactance of 4 ohms, so its impedance is 24i ohms.

Impedance of circuit 5 6 1 5i 1 (24i) Add the individual impedances.

5 6 1 i Simplify.

The impedance of the circuit is 6 1 i ohms.

Double-check to make sure that you converted the resistance or reactance of each component to the correct impedance.

1. What If? Suppose the series circuit above has a resistor with a resistance of 10 ohms, an inductor with a reactance of 7 ohms, and a capacitor with a reactance of 5 ohms. What is the impedance of the circuit?

2. Circuits A series circuit has a resistor with a resistance of 14 ohms, an inductor with a reactance of 8 ohms, and a capacitor with a reactance of 8 ohms. What is the impedance of the circuit?





Georgia Performance Standards: MM2N1b, MM2N1c, MM2N1d

Goal Multiply and divide complex numbers.

Vocabulary

To multiply two complex numbers, use the distributive property or the FOIL method

just as you do when multiplying real numbers or algebraic expressions.

Two complex numbers of the form a 1 bi and a 2 bi are called complex conjugates.

The product of complex conjugates is always a real number. For example,

(2 1 4i)(2 2 4i) 5 4 2 8i 1 8i 1 16 5 20. You can use this fact to divide complex

numbers. When a quotient has an imaginary number in the denominator, rewrite the

denominator as a real number so you can express the quotient in standard form.

LESSON

1.3

Multiply complex numbers


a. 3i(27 1 i) b. (8 2 2i)(26 1 5i)

c. (23 1 i)(8 1 5i) d. (4 1 3i)(4 2 3i)

Solution

a. 3i(27 1 i) 5 221i 1 3i 2 Distributive property

5 221i 1 3(21) Use i 2 5 21.

5 221i 2 3 Simplify.


b. (8 2 2i)(26 1 5i) 5 248 1 40i 1 12i 2 10i 2 Multiply using FOIL.

5 248 1 52i 2 10(21) Simplify and use i 2 5 21.

5 248 1 52i 1 10 Simplify.


c. (23 1 i)(8 1 5i) 5 224 2 15i 1 8i 1 5i 2 Multiply using FOIL.

5 224 2 7i 1 5(21) Simplify and use i 2 5 21.

5 224 2 7i 2 5 Simplify.


d. (4 1 3i)(4 2 3i) 5 16 2 12i 1 12i 2 9i 2 Multiply using FOIL.

5 16 2 9(21) Simplify and use i 2 5 21.

5 25 Simplify.



1. 22i(1 1 4i) 2. (25 1 i)(2 2 7i) 3. (3 2 2i)(25 2 9i)


Divide complex numbers


a. 8 1 3i}1 2 2i

b. 5 2 2i}3 1 8i

c. 4 1 3i}5i

Solution

a. 8 1 3i}1 2 2i

5 8 1 3i}1 2 2i

3 1 1 2i}1 1 2i

Multiply numerator and denominator by 1 1 2i.

5 8 1 16i 1 3i 1 6i 2

}} 1 1 2i 2 2i 2 4i 2

Multiply using FOIL.

5 8 1 19i 1 6(21)

}}1 2 4(21)

Simplify and use i 2 5 1.

5 2 1 19i}5

Simplify.

5 2}5 1

19}5 i Write in standard form.

b. 5 2 2i}3 1 8i

5 5 2 2i}3 1 8i

3 3 2 8i}3 2 8i

Multiply numerator and denominator by 3 2 8i.

5 15 2 40i 2 6i 1 16i 2

}} 9 2 24i 1 24i 2 64i 2

Multiply using FOIL.

5 15 2 46i 1 16(21)

}} 9 2 64(21)


5 21 2 46i}

73 Simplify.

5 2 1}73 2


c. 4 1 3i}5i

5 4 1 3i}5i

3 1 25i}25i 2 Conjugate of 5i is 25i.

5 220i2 15i 2}225i 2

Distributive property

5 220i2 15(21)

}}225(21)


5 15 2 20i}25

5 3}5 2




4. 5}1 1 i

5. 5 1 2 i}3 2 2i

6. 2 2 6i}4i





LESSON

3.65Exercise Set A

LESSON




1. What is the complex conjugate of a 2 bi?


2. (28 2 2i)(1 1 2i) A. 210 1 20i

3. 10i(2 1 i) B. 24 2 18i

4. (1 2 4i)2 C. 215 2 8i


5. 26(8 2 2i) 6. 11i(5 1 8i)

7. 6i(23 2 7i) 8. (1 2 6i)(2 1 3i)

9. (2 1 4i)(2 2 6i) 10. (24 2 10i)(6 2 5i)

11. (1 2 i)(1 1 2i) 12. (9 1 2i)(4 1 11i)

13. (28 1 9i)(27 2 i) 14. (5 2 i)(22 1 12i)

15. (27 1 3i)(4 1 6i) 16. (25 1 i)(25 2 i)

17. (2 1 8i)(22 1 i) 18. (27 2 5i)(2 2 4i)

19. (23 1 4i)7i 20. (2 1 3i)2i

21. (22 2 5i)2 22. (1 2 4i)(2 2 6i)

23. 2(3 1 i)(2 2 7i) 24. (22 1 3i)2

25. 2i(5 1 i)(22 1 7i) 26. 3i(4 2 2i)(1 2 4i)

27. 4}5 2 2i

28. 5}3 2 2i

29. 2 2 i}3 1 4i

30. 22 1 4i}6 1 3i

31. 7 2 4i}6i

32. 2 1 9i}27i

33. 24 2 5i}9 1 i

34. 7 1 4i}22 2 5i

35. 1 1 2i}

Ï}

2 1 i36.

1 2 3i}

Ï}

3 2 i

37. 5 1 6i}22 2 3i

1 4 2 i 38. 3}2 2 4i

2 (3 1 2i)

39. Using the fact that i 2 5 21, make a table that shows the powers of i from i 1 to i 8 in

the fi rst row and the simplifi ed forms of these powers in the second row. Describe

the pattern you observe in the table. Verify that the pattern continues by evaluating

the next four powers of i.


LESSON

3.65Exercise Set B

LESSON





1. 6 1 2i}5i

A. 25 1 12i

2. 7i(2 2 i) B. 2}5 2

6}5 i

3. (22 2 3i)2 C. 7 1 14i


4. (6 2 3i)(4 1 2i) 5. (4 2 i)(2 1 5i)

6. (2 2 3i)(7 1 3i) 7. (25 1 2i)(4 2 10i)

8. (28 1 4i)(25 2 7i) 9. (5 2 5i)(25 1 5i)

10. (6 1 3i)(4 2 6i) 11. (23 1 4i)(24 1 3i)

12. (2 1 5i)(22 2 9i) 13. (7 2 4i)(6 2 8i)

14. (27 1 5i)2i 15. 27i(5 2 6i)

16. (3 2 2i)2 17. (2 2 2i)(3 2 5i)

18. 3i(2 2 i)(4 1 2i) 19. (2 1 i)(3 2 i)(2 2 3i)

20. 2(2 1 i)(6 2 5i) 1 (24 1 2i) 21. 2i(1 1 i)2(2 1 i)

22. 7}3 1 i

23. 8}5 2 2i

24. 3 1 i}22 2 5i

25. 6 2 4i}2 2 i

26. 26 1 i}2 1 4i

27. 4 2 6i}22 1 2i

28. 4 2 i}6i

29. 29 1 2i}23i

30. 1 1 i Ï

}

2 }

Ï}

2 2 i31.

1 1 i Ï}

3 }

Ï}

3 1 i

32. 2 1 3i}4 2 i

3 1 2 3i}4 1 i

33. 4 1 7i}3 1 2i

3 1 2 i}3 2 2i

34. (8 1 2i)2 (3 2 2i) }}

(26 1 7i) 1 (4 2 5i)35.

(6 2 2i)1 (3 2 7i) }}

(2 1 6i)2 (8 2 2i)

36. (9 2 4i)1 (3 1 8i)

}} 4i(6 2 3i)

37. 1 2 5i}3 2 5i

2 (4 2 5i)

38. Using the fact that i 2 5 21, fi nd the product (i 4)(i 3)(i 2)(i).

39. Show that the complex conjugate of the product of two complex numbers a11 b

1i

and a21 b

2i is the product of their complex conjugates.


Use a graphing calculator to write the expression in standard form.

1. (6 1 2i) 2 (2 2 3i) 2. (24 1 5i)(1 2 4i) 3. 2 1 i}3 2 i

Write (24 2 8i ) 1 (5 2 3i ) as a complex

(-4-8i)+(5-3i) 1-11i

number in standard form.

the expression.

the ENTER key.

Use after Lesson 1.3

How can you use a graphing calculator to

Normal SCI ENGFLOAT 0123456789Radian DegreeFUNC PAR POL SEQCONNECTED DOTSEQUENTIAL SIMULREAL a+bi re^Øi

FULL HORIZ G-T

perform operations with complex numbers?

Change the mode of the calculator to handle complex

numbers by pressing MODE and highlighting a 1 bi.

To insert the imaginary unit i in an expression, press 2nd [i].




Write the expression 7 1 4i}2 2 3i

as a complex number in standard form.

the expression. the MATH menu and

choose Frac to display the answer using

fractions. Press the ENTER key.

(7+4i)/(2-3i) (7+4i)/(2-3i) Frac2/13+29/13i


How can you fi nd the absolute value of a complex number?

The absolute value of a real number is the distance between the number

and zero on a number line. Likewise, the absolute value of a complex

number is the distance between the number and the origin in the complex

plane. The complex plane has a horizontal axis called the real axis and a

vertical axis called the imaginary axis.

Find the absolute value of the complex number.

1. 6 1 8i 2. 23 1 6i 3. 25 2 12i

4. Reasoning Plot the complex number 3 1 i in the complex plane. Plot

the conjugate of the number. Find the absolute values of both numbers.

Will the absolute values of a complex number and its conjugate always

be equal? Explain.

Find the absolute value of 4 1 2i.

Plot the complex number in the complex Notice that a right triangle can be

plane. Start at the origin, move 4 units formed with legs of 4 units and 2 units.

to the right and 2 units up. The hypotenuse is the distance from the

origin to the complex number.

imaginary

real1

i

4 1 2i

imaginary

real1

i

4 1 2i

Use the Pythagorean Theorem to fi nd the distance between the complex number

and the origin.

Distance 5 Ï}

42 1 22 5 Ï}

16 1 4 5 Ï}

20 5 2 Ï}

5

Because the absolute value of a complex number is the distance between the number

and the origin, 4 1 2i 5 2 Ï}

5 .

Use before Lesson 1.4




Georgia Performance Standards: MM2N1b, MM2N1d

Goal Use the complex plane and fi nd the absolute value of a

complex number.

Vocabulary

Just as every real number corresponds to a point on the real number line, every complex

number corresponds to a point in the complex plane. The complex plane has a horizontal

axis called the real axis and a vertical axis called the imaginary axis.

The absolute value of a complex number z 5 a 1 bi, denoted by z , is a nonnegative

real number defi ned as z 5 Ï}

a 2 1 b 2 . This is the distance between z and the origin in

the complex plane.

LESSON

1.4

Plot complex numbers

Plot the complex numbers in the same complex plane.

a. 3 2 2i b. 22 1 4i

c. 3i d. 24 2 3i

e. 6 1 5i f. 26 1 6i

Solution

a. To plot 3 2 2i, start at the origin, move imaginary

real1

i

6 1 5i

3i

3 2 2i

24 2 3i

26 1 6i

22 1 4i

3 units right, and then move 2 units down.

b. To plot 22 1 4i, start at the origin, move

2 units left, and then move 4 units up.

c. To plot 3i, start at the origin and move

3 units up.

d. To plot 24 2 3i, start at the origin, move

4 units left, and then move 3 units down.

e. To plot 6 1 5i, start at the origin, move

6 units right, and then move 5 units up.

f. To plot 26 1 6i, start at the origin, move

6 units left, and then move 6 units up.


Plot the complex numbers in the same complex plane.

1. 1 1 4i 2. 2 2 2i 3. 23 1 3i

4. 25i 5. 25 2 6i 6. 6 2 4i


Find absolute values of complex numbers


a. 4 2 i

b. 23 1 4i

c. 3i

d. 2 1 6i

Solution

a. 4 2 i 5 Ï}

42 1 (21)2 Defi nition of absolute value

5 Ï}

16 1 1 Evaluate powers.

5 Ï}

17 Simplify.

b. 23 1 4i 5 Ï}

(23)2 1 42 Defi nition of absolute value

5 Ï}


5 Ï}

25 Simplify.

5 5 Evaluate square root.

c. 3i 5 Ï}

02 1 32 Defi nition of absolute value

5 Ï}


5 Ï}

9 Simplify.

5 3 Evaluate square root.

d. 2 1 6i 5 Ï}

22 1 62 Defi nition of absolute value

5 Ï}


5 Ï}

40 Simplify.

5 2 Ï}

10 Evaluate square root.



7. 15 1 8i 8. 7 2 24i

9. 25 1 6i 10. 9i

11. 22 2 4i 12. 4 2 9i




Match the complex number with the correct letter on the graph.

1. 2 1 i 2. 23 2 6i 3. 1 2 4i 4. 25

5. 4 1 5i 6. 26 1 3i 7. 6 1 2i 8. 7 2 5i

imaginary

real1

iE

D

H

C

B

A

G

F


9. 5 2 2i 10. 5 1 12i

11. 23 2 4i 12. 2 1 3i

13. 2 2 6i 14. 6 2 5i

15. 1 1 2i 16. 9 1 40i

17. 5i 18. 22 1 12i

19. 4 1 6i 20. 25 2 i

21. 22 1 i 22. 27 2 5i

23. 23 2 11i 24. 2i

25. 22 2 5i 26. 1 2 4i

27. 2 2 7i 28. 22 1 3i

29. 5 1 4i 30. 4 2 2i

31. Ï}

5 1 2i 32. Ï}

11 1 5i

33. 2 2 i Ï}

21 34. 2Ï}

23 1 i Ï}

26

35. What is the absolute value of a 2 bi?

36. Is every complex number an imaginary number? Explain.

LESSON

3.65Exercise Set A

LESSON




LESSON

3.65Exercise Set B

LESSON



Match the complex number with the correct letter on the graph.

1. 6 1 4i 2. 3 2 2i 3. 22 1 6i 4. 2i Ï}

26

5. Ï}

17 2 5i 6. 26 2 i 7. Ï}

29 1 2i 8. 2Ï}

42 2 2i Ï}

7

imaginary

real1

i

E

D

H

C

B

A

G

F


9. 24 2 10i 10. 4 2 i 11. 23i

12. 29 1 12i 13. 25 2 7i 14. 5 2 5i

15. 4 2 6i 16. 211 1 60i 17. 22 2 9i

18. 7 1 4i 19. 22 1 2i 20. 5 2 6i

21. 4 1 2i 22. 12 2 16i 23. 1 2 i Ï}

3

24. Ï}

5 1 2i Ï}

2 25. 2 2 Ï}

3 }4 i 26. Ï

}

7 1 2Ï

}

5 }9 i

Determine whether a and b are greater than zero, less than zero, or

equal to zero for the given complex number a 1 bi.

27. a 1 bi lies in the fi rst quadrant of the complex plane.

28. a 1 bi lies in the second quadrant of the complex plane.

29. a 1 bi lies in the third quadrant of the complex plane.

30. a 1 bi lies in the fourth quadrant of the complex plane.

31. a 1 bi lies on the negative real axis of the complex plane.

32. a 1 bi lies on the positive imaginary axis of the complex plane.



1. Ï}

525 2. Ï}

567 3. Ï}

192

4. Î} 49}81 5. Î} 128}

25 6. Î} 53}

9


7. Ï}

299 8. Ï}

2196 9. Ï}

280

10. 2 1 Ï}

227 11. 6 2 Ï}

2162 12. 23 1 Ï}

244


13. 6i 1 (22 2 7i) 14. (1 2 4i) 2 (1 1 3i)

15. (2 1 5i) 1 (5 2 2i) 16. (27 2 12i) 1 (4 1 5i)

17. (1 2 i) 2 (6 1 i) 18. (9 2 8i) 2 (4 2 13i)

19. (22 1 3i) 1 (2 2 3i) 20. (4 2 i) 2 (26 1 7i)

21. 6i 1 (27 1 i) 2 2 22. 7 2 (210 1 i) 1 4i

23. (1 1 3i)(22 1 i) 24. (22 2 5i)(2 2 2i)

25. (3 2 2i)7i 26. (7 1 i)5i

27. (22 1 2i)2 28. (1 2 i)(2 2 6i)

29. 2(2 1 3i)(1 2 4i) 30. (3 1 5i)2

31. 4i(3 2 i)(22 1 8i) 32. 23i(2 2 i)(4 2 5i)

33. 3i}4 2 i

34. 6 1 2i}4 1 8i

35. 1 1 2i}2 2 4i

3 2 1 4i}1 2 2i

36. 23 1 i}5 1 2i

1 (2 1 2i)

Identify the complex numbers plotted in the complex plane. Then fi nd the

absolute value of each complex number.

37. imaginary

real

C

B

A

1

i

38. imaginary

real

B

C

A

1

i


algebra 2 book - · pdf filea. 2x1 2yi 5 6 1 4i b. 3x1 5yi 5 9 2 15i solution because the...

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