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58
C h a p t e r1 w ; * +
E q u a t i o nIsn, e q u a l i t i easn,d M o d e l i n g
24. Injuries in the Boston Marathon
Use the equation of
Exercise 23 to predict the percentageof runners injured in
age group 6 (60-plus). Actually, 2lo/o of the runners in the
60-plus age group were injured. Can you offer an explanation
for the difference between the predicted and the actual
figures?- 1.1%
Thinking Outside the Box V
Counting Coworkers Chris and Pat work at a Tokyo Telemarketing. One day Chris said to Pat, "l9 f 40 of my coworkersare female." Pat replied, "That's strange,12125of my coworkersare
female." If both are correct, then how many workers are there at
Tokyo Telemarketing and what are the genders of Chris and Pat?
201workers.Chrisfemale.Patmale
ffi-T-o-P-g*ir*
l . Use your calculatorto find the regression equation for the data
siven in the table: i' : 1.8.r+ 8 . 2
,
Use the regressionequation to estimatethe cost in 2010. 526.20
3. Usethe regressionequationto estimatethe cost in 1998.$a.60
Cost ($)
Our system of numbers developed as the need arose. Numbers were first used for
counting. As society advanced the rational numberswere formed to expressfractional
parts and ratios. Negative numbers were invented to expresslossesor debts.When it
was discoveredthat the exact size of some very real objects could not be expressed
with rational numbers, the irrational numbers were added to the system, forming the
set of real numbers. Later still, there was a need for another expansionto the number
system.In this sectionwe study that expansion,the set of complex numbers.
Definitions
There are no even roots of negativenumbers in the set of real numbers.So the real
numbers are inadequate or incomplete in this regard. The imaginary numbers were
invented to complete the set of real numbers. Using real and imaginary numbers,
every nonzero real number has two squareroots, three cube roots, -fow fourth roots,
and so on. (Finding all of the roots of any real number is
$gne in trigonometry.)
The imaginary numbers are basedon the symbol V - t. Since there is no real
number whose squareis - 1, a new number called i is defined such that i2 - - 1.
Defin ition:
lmaginaryNumberi
,
The imaginary number i is defined by
I
'
I w. may also write t :
i2 :
{ 4.
- 7.
i
1.5 sffiw
CompleN
x umbers
59
A complexnumberis formedasa real numberplus a real multipleof i.
Definition:
Complex Numbers
i The set of complexnumbersis the set of all numbersof the form a * bi, i
,
. wherea andb aierealnumbers.
In a -t bi, a is called the real part and b is called the imaginary part. Two
complex numbers a * bi and c * di are eqiral if and only if their real parts are
equal (o : ,) and their imaginary parts are equal (b : d).If b : 0, then a * bi is
a real number. If b * 0, then a * bi is an imaginary number.
The form a -l bi is the standard form of a complex number, but for convenience we use a few variations of that form. If either the real or imaginary part of a
complex number is 0, then thatpart is omitted. For example,
2t0i:2,
0+3i:3i,
and
0*0i:0.
If b is a radical, then i is usually written before b. For examplg we write 2 + iV3
rather than2 + V3i, which could be confusedwith 2 + V3i.If b is negative,a
subtraction symbol can be used to separate the real and imaginary parts as in
3 + (-2)i
3 -2i.Acomplexnumberwithfractions, suchur+ - tt,maybe
written ut*.
e*rr'fk
E Standardform of a comptexnumber
Determine whether each complex number is real or imaginary and write it in the
standardform a * bi.
r.3i
b.87 c.4-5i
d.0
| -f rri
e'2
$olution
Complex numbers
a.
b.
c.
d.
The complex number 3i is imaginary, and3i : 0 * 3i.
The complex number 87 is a real number, and 87 : 87 + 0i.
+ (-5) i.
Thecompl exnumber4- 5i i si magi nary and4 - 5i :4
:
The complex number 0 is real, and 0
0 * 0i.
e. Thecomplexnumbert+t
F i g u re 1 .4 4
is imaginary,
andt+t
:
L * Tr.
I
The real numbers can be classified as rational or irrational. The complex numbers can be classified as real or imaginary. The relationship between these sets of
numbersis shown in Fig. 1.44.
Addition, Subtraction,and Muttiptication
Now that we have defined complex numbers, we define the operations of arithmetic
with them.
Definition: Addition,
Subtraction, and
Multiplication
, lf a * bi and,c * di are complexnumbers,we definetheir sum,difference,
md productasfollows.
,
(a+bi)+(c +di):(a+c)+
(b+d)i
(a + bi) - (c + di): (a - c) + (b - d)i
(a + bi)(c + di) - (ac - bd) + (bc + ad)i
Chapter1 rxre
Equations,
Inequalities,
and Modeling
It is not necessaryto memorize these definitions, becausethe results can be
obtained by performing the operations as if the complex numbers were binomials
with i being a variable, replacing i2 with - I wherever it occurs.
A-*r".f/t
A
Operationswith comptexnumbers
Performthe indicatedoperationswith the complexnumbers.
a. (-2 + 3r) + (-4 - ei) b. (-1 - si) - (3 - 2i) c. 2t(3 + t)
d. QD2 e. e3il2 f. 6 - 2i)(s + 2i)
Solution
a. (-2 + 3i) + (-4 - 9i) : -6 - 6i
b. (-1 - 5i) - (3 - 2i) : -l - 5i - 3 * 2i : -4 - 3i
c. 2i(3 * i) : 6i + 2i2: 6i + 2(-t1 : -2 + 6i
d . ( 3 i l 2: 3 2 i 2 : 9 ( - t ; : - 9
e. (4il2 : e3)2iz : 9(-1) : -9
f. (5 - 2i)(5 + 2i) : 25 - 4i2: 25 - 4(-r) : 29
Figure1.45
ru Check theseresultswith a calculatorthat handlescomplexnumbers,as in
Fig.1.45.
I
powersof i by usingthe definitionof multiplication.
Wecanfind whole-number
Sinceir : iandi2 - -l,wehave
i3:il
.i2:
t(-l) - -i
and
i4--it -i3:i(-i):
-i2:1.
The first eight powers of i are listed here.
i1:i
i2:-l
i3--i
ia:l
i5:i
i6:-l
i7--i
i8:1
This list could be continued in this pattern, but any other whole-number power of i
can be obtained from knowing the first four powers. We can simpli$r a power of i by
using the fact that ia : 1 and (io)o : 1 for any integer n.
ea",./t
E Simptifying a power of f
Simpliff.
a. i83 b. i-46
Sotution
a. Divide83 by 4 andwrite83 : 4 . 20 + 3. So
i83- (to)ro. i3 : 120. f : 1
b. Since-46 : 4(-12) * 2, we have
i - 4 6 - ( t + 1 - r z .i 2 :
1r2. i2:
| (_1):
Division of GomptexNumbers
The complex numbers a f bi and a - bi are called complex conjugates of each
other.
1.6 rrffi
e**rr./t
ComplexNumbers
61
Complex conjugates
A
Find the productof the givencomplexnumberandits conjugate.
n.3 . t b.4+2i
c. -i
Sotution
a. Theconjugate
of3 - i is 3 + i, and(3 -- r(3 +, - 9 - i2 : 10.
b. Theconjugate
of 4 + 2iis4 - 2i,and(4 + 2i)(4 - 2i): 76 - 4i2:20.
c. Theconjugate
of -i is i, and-i ' i : -i2 : l.
I
In generalwe havethe following theoremaboutcomplexconjugates.
Theorem:
Complex Conjugates
We use the theorem about complex conjugates to divide imaginary numbers, in a
processthat is similar to rationalizing a denominator.
tr Dividing imaginarynumbers
er*r"'fb
1.,
Write each quotient in the form a t bi.
I
!
I
I
8-r
I
'-i
a.-
I
2+i
u.-
5-4i
3-2i
i
1,.-
Solution
a. Multiply the numeratoranddenominatorby 2 (8 - t)(2 - i)
8- t
16 - r}i + i2
(2+i)Q-i):
2+t:
44
Checkdivisionusingmultiplication:(3 - 2i)(2 + i) : 8 - t.
1 _
1(5+4,
_ 5+4i _5+4i
5 - 4 i - ( 5 - 4 t ) ( s+ 4 t ) - 2 5 + 1 6
4r
ls
4 \
check:(A *
nt)6
_: 5 , 4 ,
n4t'
- 4i) -- 2 s * 2 0 . - 2 0 . - 1 6 . ,
;
;,
^t
;t'z
- 25 t6 t '
4r- 4rt - l
. . F i g u r e1 . 4 6
ru
t r
Youcanalsocheckwith a calculatorthathandlescomplexnumbers,asin
Fig. 1.46. tr
3 - 2i
(l - 2i)(-i)
i
i(-i)
C h e c k( - 2 - 3 i ) ( i ) :
-3i + 2i2
-2 - 3i
-i2
-3i2 -2i:3
I
-2i.
L
J a
I
52
Chapter1 wm,trrirs
Equations,
Inequalities,
and Modeling
Roots of Hegative Numbers
In Examples 2(d) and 2(e), we saw that both (3i)' : -9 and (4il2 : -9. This
means that in the complex number system there are two squareroots of -9, 3i and
-3i. For any positivereal numberb,wen2ve (i{i)':
-U atta(-i{i)'
: -b.
So there are two squareroots of -b, iV b and -iY b. We call tY b the principal
square root of -b andmake the following definition.
i1.-':,:t
Definition: SquareRoot of
a Negative Number
For any positive real number b,
{-r = i{8.
In the realnumbersystem,f:t
and V-8 areundefined,but in the complex
: tfi and VJ
: t{8.Even though
numbersystemtheyaredefinedu, fi
we now havemeaningfor a symbol such as f-,
ail operationswith complex
numbersmustbepedormedafter convertingto thea -t biform. If we performoperationswith rootsof negativenumbersusingpropertiesof the real numbers,we can
getcontradictorY
results.
v-g
ift.
Theproductrule l;'
A-*r".f/t
if8
fb
: /1-2;1-9;
: {G
: i2 . 116 - -4
: liis
: 4 rncorrect.
correct.
usedonly fornonnegative
numbersa andb.
reusmhers
tr $quaretrsetsmfmegatflve
Write each expressionin the form a * bi, where a and b arereal numbers.
a.
V=
_4+f4
V=
c .f 4 ( l e - l = )
b.
Sc&ution
The first stepin eachcaseis to replacethe squarerootsof negativenumbersby expressionswith i.
a. V=
+ V-18 : r.r/8+ iVG:
2if2 + 3if,
:
-4+fq
-4+i,.7'n
sif,
- 4 + sift
- -1*lfi
c. f4({e
- l-)
:3ift(3 - ift) : sifl - 3ffa
: 3 . V 6 + sift
I
1.6 smw
63
Exercises
For Thought
True or False?Explain.
1. The multiplicative inverseof i is -i. r
6. 5 _ 14
2. The conjugateof i is -i. t
7 . ( Z i ) 2+ 9 : 0 r .
3. The set of complex numbersis a subsetof the set of real
numbers. F
4.
(ft - ift)(ft + ift):5r
f,.
(2 + si)(2+ 5r): 4 * 25p
f. ia: lt
10.i18:1F
Determine whether each complex number is real or imaginary and
write it in the standardform a * bi. (Examplet)
+ 6i 2. -3i + Y_6
1. 6i lmaginary.0
Imaginary,V6 - 3i
4. *72 Real,-72
lnear,|
+ oi
,.+
Imaginary,
l -i
-l +
6. -ift
+0i5.\n
Real,\4
+ 0;
Imaginary,
o-fii
8 . 0 R e a l , 0+ 0 t
1 0 .( - 3 + 2 i ) + ( 5 - 6 '
2-4i
11. (r - i) - (3 + 2i) -2 - 3i 12. (6 -"7t1 - Q - 4i)
9.(3-3i)+(4+si)1+2i
(t
s_-(t - 2.\ (^ - r)q
tu.(t
i,)- (3
\5 i,)- (; ,,)r*;,
*
l)
17. -6i(3 - 2i) -12 - rqi
2r. (4 - 5r)(6 + 2i) 34 - 22i
22. (3 + 7i)(2 + 5,
-29 + 29i
24. (4 + 3i)(4 - 3i) 2s
- r)(ft + i) +
2s.(ft
30. (fe
- 2i)'r - +ifi
* ift)'z + ai{i
3 3 . i e 8- t
31. iti t
32. i24 t
34. ile -i
35. i
37. i-3 t
38. i-4 r
39. i-t3 -i
40. i-27 i
41. i-38 -t
42. i-66 -t
-r -;
48.-i{t s
47.i t
36. i-2 -t
i,T
4s.3 - i{t n so.tr. ,Y
Writeeachquotientin theform a * bi. (Example
5)
-3i 3
I -z- -lt
I
2.-^
-s
51. 2-is
trA
Jr.
20.(3-r(5-2i)n*ni
2s.($
28. (-6 - 2i)2tz + z+i
18. -3r(5 + 2i)6 - rsi
19. (2 - 3i)(4 + 6i) 26
23. (s - 2i)(s + 2i) 2e
27. (3 + 4i)2-t + z+i
Find the product of the given complex number and its conjugate.
(Example4)
I
I
43.3 - 9ie0 44.4 + 3i2s 45.; + Ztnl+ 46.: - ttols
z
r
ztlq-
13.(r - ifil + (3+ zifb o * ,i[, "
14.(5+ zifi) + (-4 - srVt t - zi{i
(-
- ili) s
+ iVi)(fi
26. (f,
J J
Performtheindicatedoperationsand writeyour answersin the
2 and3)
form a * bi,wherea andb arereal numbers.(Examples
ls.
8. (-3i)'* 9 : 0r
Exercises
ffi
7.
: 5 _ eip
3i76
-z
t
---t
+ I J
57.
l-t
r
3+2it3
60.
4+2i't
5-3it7
52.
s
":
)
5+2i29
29
-3+3t
55.-rt:r50.-+-2t
i
l-i2
3
2
-2-4i
-i
s
4 + 2 i 2 - r - , j 5 t962-i
I
'"'
'
13
2 - 3i 13 t3'-'' 3 + sti11
+ "i
r7
lDue to spaceconstrictions,answersto theseexercisesmay be found in the complete Answers beginning on page A-l in the back of the book.
t3
it
Chapter1 wffi*
64
Equations,
Inequalities,
and Modeling
Write each expressionin theform a * bi, where a and b are real
numbers. (Example 6)
il. f4-
es. f *a, - fG
-4 + 2i
-u
6s.(f-),
. {4
64. {1
ut.
-6 + f-
iU.
71. - 3 +
72.t-ffi
l?-4loto
73. f
9-V-ls
- 6. 2 2
-8(f -t * \6)
\/5
7 9 .a : I , b : 6 , c :
-3 - 2ift
8l.a--2,b-6,c:6
-o-fir-q*
17
z+fn
\,5
^,
for each choice of
for each choice ofa, b,
8 0 .a : l , b - - 1 2 , c : 8 4
6 - 4ift
8 2 .a : 3 , b :
6,c:8
*3 - i{G
23
Perform the indicated operations. Write the answers in theform
a -l bi where a and b are real numbers.
83. (3 - st(3 + si)34
84. (2 - 4i)(2 + 4i) 20
8s. (3 - si) + (3 + si)6
86.(2-4i)+(2+4i)4
ffi
89.(6-2i)-(7-3i)-r+i
eo.(5-6i)-(8-e,
9l.isu2-3r) 3-t
92. 3i7Q - 5i3)18
-3+3i
93. Explain in detail how to find i' for any positive integer n.
94. Findanumber a -t bisuchthat a2 + b2 isirrational.
- :3
-
75.a : I,b : 2,c - 5 -l + 2i 76. a : 5,b - -4,c : I
-2+l - i
55
4,c:3
78.a:2,b - -4,c: 5
7 7 .a : 2 , b :
-2 + if,
Z + if,G
2
2
Evaluate the expression
and c. (Example6)
{i + ixfi
95. Letw: a -f biandw: a - bi,whereaandbarerealnumbers. Show thatw * w is real and thatw - w is imaginary.
Write sentences(containing no mathematical symbols) stating
theseresults.
96. Is it true that the product of a complex number and its conjugate is a real number? Explain. True
97. Prove that the reciprocal of a -t bi,where a and b are not both
-ab
zero,ts ).
na- -t D-
,.
ui.
a- -r D-
98. Cooperative Learning Work in a small group to find the two
squareroots of 1 and two squareroots of - 1 in the complex
number system. How many fourth roots of I are there in the
complex number system and what are they? Explain how to
find all of the fourth roots in the complex number system for
* *
+
any positive real number. l, i, fourth roots of I are 1 and
+i. Fourthrootsof x for x ) 0 are+ tfx and.!i\in.
9 9 . E v a l u a t e i 0+
! i l t + i 2 t + . . . * i t o o ' , w h e rne! ( r e a d " n f a c torial") is the product of the integers from 1 through n if n > |
a n d 0 !: I . 9 5 + 2 i
Thinking Outside the Box VI
SummingReciprocals Thereis only oneway to write 1 asa sum
of the reciprocalsof threedifferentpositiveintegers:
111
t+ i
*
u
: t
Find all possible ways to write 1 as a sum of the reciprocals of four
different positive integers.
I
I
-I + -I + - I + - l. 1- + -1 + -I + - 1. -l + I
g
2
3
l0
t5'2 3
t8'2 t-8-24'
-I + -I + -'I, 7+ - 1. -1 + -I + -I + - 1. -l + -I + -I + - I
2
42'2 4
4
20
3
6
t2'2
5
PopQuiz
1 . F i n d t h e s u m o f3 * 2 i a n d 4 - i . l + i
2. Find the product of 4 - 3i and2 + i. ll - 2i.
3. Find the product of 2 * 3i and its conjugate. 13
4
s
-i
74.f-(/t-.l/4)
tfi + 2i$
-b+\6r-4a;
2-4i
7
88._____i
s
2+4i
ls
17
For Writing/Discussion
r-i$
-4+8i
Evaluate the expression
a, b, and c. (Example6)
-2 -r
,
-2 + f 1o
- l + r vr5
69.
2
-z + it/i
-3
-sifi
66.({-l
V - s o- r o
67. f - .
+ f -zs gi
62. f4
f--i
3-5i
8
87.--'-'-i
17
3+5i
4. Write
5101s
in the forma + bi.: + : i
13 13
2-3i
5. Find i"'. -i
6. What are the two squareroots of -16? *+t