Download Comptex Numbers

P . 4r r m
C o m o l e x N u m b e r s 39
ComptexNumbers
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 numberswere 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 addedto the system,forming the
set of real numbers.Later still, there was a need for anotherexpansionto the number
system.In this sectionwe study that expansion,the set of complex numbers.
Definitions
In Section P.3we saw that there are no even roots of negative numbers in the set of
real numbers. So, the real numbers are inadequateor 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 tyvosquareroots, three clbe
roots, four fourth roots, and so on. (Actually finding all of the roots of any real
number is done in trigonometry.)
The imaginary numbersare basedon the symbol V - l. Since there is no real
number whose squareis -1. a new number called i is defined such that i2 : -1.
Definition:
lrnaginaryNumberi
Theimaginary number i is definedby
i2 :
-1.
Wemay alsowrite t : \/ -1.
A complex number is formed as a real number plus a real multiple of i.
Definition:
ComplexNumbers
The set of complex numbers is the set of all numbersof the form a I bi,
wherea andb arerealnumbers.
In a * bi, a is called the real part and b is called the imaginary part. Two
complex numbers a * bi and c f di are equal if and only if their real parts are
equal (a : c) 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 * bi is the standard form of a complex numbeg 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 that part is omitted. For example,
0*3i:3i,
2*0i:2,
and
0+0i:0.
If b is a radical, then i is usually written before b. For examplg we write 2 + i\/t
rather than 2 + \,6i,which could be confused with2 + \/3i,If b is negative,a
subtraction symbol can be used to separate the real and imaginary parts as in
21i,
3 + (-2)i : 3 - 2i. A complex number with fractions, such as + may be
w r i t t e nu = .
40
ChapterP rrF
Prerequisites
fua.trV/a I
Standardform of a complexnumber
Determine whether each complex number is real or imaginary and write it in the
standardforma I bi.
a.3i
b.87
c.4-5i
d.0
e.
l*ni
2
Solution
a.
b.
c.
d.
Complex numbers
The complex number 3i is imaginary,and3i : 0 + 3i.
The complex number 87 is a real number,and 87 : 87 + 0i.
The complex number 4 - 5i is imaginary,and4 - 5i : 4 + (-5)t.
The complex number 0 is real, and 0 : 0 + 0i.
e. The complexnumberL+t
is imaginary,anaLld
:
) * T,.
4fV Tful. Determinewhetheri - 5 is real or imaginaryandwrite it in standard
r
torm.
, FigureP,23
The real numberscanbe classifiedasrationalor irrational.The complexnumberscan be classifiedas real or imaginary.The relationshipbetweenthesesetsof
numbersis shownin Fig.P.23.
Addition, Subtraction, and Multiptication
Now that we have defined complex numbers, we define the operationsof arithmetic
with them.
Definition: Addition,
Subtraction, and
Multiplication
If a * bi and c * di are complex numbers, we define their sum, difference,
and product as follows.
(a + bi) + (c + di) : (a.| c) * (b + Ai
(a + bi) * (c + di): (a - c) + (b * d)i
(a + bi)(c + di) : (ac - bd) + (bc + afii
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,replacingi 2 with - I whereverit occurs.
fuaw7/e [
0perationswith comptexnumbers
Performthe indicatedoperationswith the complexnumbers.
^. (-2+ 3, + (-4 - ei) b. (-1 - 5, - (3 - 2i) c. 2i(3+ i)
d. (3,)2 e. G3i)2 f. (s - zi)(s + 2i)
Sotution
^ . ( - 2 + 3 , )+ ( - 4 - 9 i ) : - 6 - 6 i
b. (-1 - s, - (3 - 2i)- -l - si - 3 i 2i : -4 - 3i
-2 + 6i
c.2i(3+ r: 6i + 2i2:6i + 2(-l):
d . G D z: 3 2 i 2 : 9 ( - l ) : - 9
P . 4r r r
C o m p l e x N u m b e r s41
e. (-3t)2: (-3)2i2: 9(-1) : -9
f. (5 - 2i)(s + 2i) : 25 - 4i2 - 25 - 4(-l)
: 2g
ffi Checktheseresultswith a calculatorthat handlescomplexnumbers,as in
Fig.P.24.
77V TiA.
Find theproduct (4 - 3i)(r + 2i).
r
r FigureP.24
We canfind whole-numberpowersof i by using the definition of multiplication.
S i n c e f l: i a n di 2 : - l , w e h a v e
i3 : il . i2 : i(-l) : -i
i4 : it . i3 : i(-i) : -i2 : l.
and
The first eightpowersof i arelistedhere.
.l
l- :
I
.)
l-:
.5.6..1..R
..1-:
I
l-:
-l .
-l
.?
l- :
-I
.4
:
l
L
-
-l
,- =
I
l
This list could be continuedin this pattern,but any other whole-numberpower of i
canbe obtainedfrom knowing the first four powers.We can simpliff a powerof i by
using the fact that i4 : I and(i1) = 1 for anyintegern.
simd*ying a powerof f
fua*Vla I
Simpliff.
-i83
a.
b. i-46
Solutioa
a. Divide 83 by 4 andwrite 83 : 4 . 20 * 3. So
i 8 3: ( i \ 2 0 . f - 1 1 0 i.3 : l . i 3 : _ i .
b. Since-46 : 4(-12) * 2, we have
i - 4 6- e + y - r z .i 2 - 1 ' t 2 . ; 2 : 1 ( _ t ) - _ 1 .
i 35.
TrV 77a1. Simplify
Division of ComptexNumbers
The complexnumbersa t bi anda - bi ne calledcomplexconjugatesof each
other.
fud*Vle
I
Gompleor
coniugates
Find the productof the given complexnumberand its conjugate.
a.3-i
b.4+2i
c.-i
Sohtion
- i2:10.
a . T h e c o n j u g a t e o- f 3i i s 3 * i , a n d ( 3- t ) ( 3 + t : 9
+ 2iis4 - 2i,and(4 + 2i)(4- 2i): 16 - 4i2:20.
b. Theconjugateof4
c. Theconjugate
of -i is i,and -i . i : -i2 : l.
Try
7fif5. Find theproductof 3 - 5i andits conjugate.
42
ChapterPrre
Prerequisites
In general we have the following theorem about complex conjugates.
Theorem:
ComplexConjugates
We use the theorem about complex conjugates to divide imaginary numbers, in a
processthat is similar to rationalizing a denominator.
fud*F/e
uiviaing imaginarynumbers
I
Write eachquotientin the form a -l bi.
8-t
I
A. z+t
D. _
)-+t
u.-
3-2i
t
Sotution
- i,theconjugate
a. Multiplythenumeratoranddenominatorby2
of 2 -l i:
- , ] _ t 6- t o i _i+2_ r s- r 0 ,
: -3 _ 2 i
1
?
?
!
?
(2+i)(2-i)
4-i2
s
=
:
2+i
(3 - 2i)(2 + r:8
Checkdivisionusingmultiplication:
.
v.-
s-2i,
t/(5-4i ) rFrEc
Sr'4l+4"/41i.
{3*21 }r[
-2-Si,
* FigureP.25
1
l(s+4,
s+4i
(s-4r(s+4r)
s-4i
ls
C
h e c kl :
"""""'
\+r
s+4i
2s+16
a \
)s
)o
+ -t l(5 - 4it: 1 +'-!i
4r')
4t
4t
41
-
-,.
5
4.
41 4t'
2 0 .- 1 6 . ,
i'
n'-
:4 + 1q: r
4t
41
You can alsocheckwith a calculatorthat handlescomplexnumbers,as in
ffi
Fig.P.25. tr
-3i + 2i2 -2 - 3i
(3 - 2t(-r)
3 - 2i
t
^-u
-t'
I
Check:(-2 - 3r(, : -3i2 - 2i : 3 - 2i.
4fV \fut. wite
#
in theforma * bi.
Rootsof Negative Numbers
-9. This
In Examples2(d) and 2(e), we saw that both (3t)2 : -9 and (-Zi)2:
means that in the complex number system there are two squareroots of -9,3i and
-3i. For any positivereal numberb,wehave (tli)':
-b and (-:t/t)t
: -b.
So there are two squareroots of - b, iV b and - i \/ b. We call i\/ b the principat
square root of -b and,make the following definition.
P.4ttr
Definition:SquareRootof
a NegativeNumber
ForThouqht
43
For anypositivereal numberb, f -b : i\/b.
In the real number system, f-z
number systemthey are defined ut f
u"d \F-g_ur"undefined, but in the complex
-z : i\/iand V-g
: i\,6.Even thoush
we now have meaning for a symbol such as \/-,
all operations with compiex
numbers must be performed after converting to the a + bi
form.If we perform oper_
ations with roots of negative numbers using properties of the real numbers, we
can
get contradictoryresults:
\/-. V:8 : V(-rX_s) : r/te : q
ifr.i\/8:iz.ft6:-4
Theproductrule fi
ercw/e
Incorrect.
Correct.
. f b : t/ot i"used only for nonnegative
numbersa and,
b.
rootsof negative
numbers
@ Square
write eachexpressionin the form a t bi,where a andb arerealnumbers.
.l/la +
V-18
b.
".
Solution
-4 + \/4
The first step in each case is to replace the squareroots ofnegative numbers
by expressionswith i.
".
b.
Vr T + V- 18 : if8 + i\/n:2i\/i
+ 3i\/t
: 5i\/t
-q + \/-s0
-4 + i\/Eo -4 + 5i\/t
4
4
o,
:-r+i,fi
".
f4(fs
- f-)
: tit/i(t - i/t) : ei\/j - 3i2\/6
:3f 6+ sift
77V lbr. y,rir"?-!4
intheforma t bi.
For Thought
True or False?Explain.
1. The multiplicative inverseof i is -i. r
2. The conjugateof 1is -i. r
3. The set of complex numbersis a subsetof the set of real
numbers. p
4.(ft - ilr)(\/t + ifi):57
s.(2+5i)(2+5i):4+2sF
6.s-f4:5_
7.Gil2+9:0r
8.(-3r2+9:0r
f. ia :
lr
l0.ir8:1p
eip
ChapterP rril
M
Prerequisites
Exercises
Determine whether each complex number is real or imaginary and
write it in the standardform a * bi. (ExampleI)
t
1. 6i lmaginary.o
+ 6t 2. _3i + lft
,.
4. -72Rear,-72 + oi S. f7
Real,\4
6. -ift
Imaginary,V6 - 3i
7.
+ 0i
;Real,:,
I
r
t
Imaginary,
-l +l - t
JJ
Imaginary,
o - t/ii
+ 0i
8. 0 Reat,0+ 0t
- (3 + 2i)-2- 3i 12.(6 - 7i)_ (3 _ 4i)
13.(r - ^[D + e + zi\/r) o *,l,frtt
14.(5+ 3i\[, + e4 - si\/i) r * zixG
/'t- r\ - / - ,\r
r c . ( s + 1 ) _ 1 1 _ . 1 \ o *;r< 1 6(;
'
3
.
t')t
\
) \2
;') (' ;' )
17. -6i(3 - 2i) -t2 - tli
18. -3r(5 + 2i)6 _ tsi
19. (2 - 3i)(4 + 6i)26
20. (3 - i)(s - 2i) 13_ ll,
2r. (4 - 5r(6 + 2i) 34 - 22i
22. (3_{str)[z + si)
23. (s - 2i)(s + 2i) 2e
r
|
|
-2+i5
-3i3
r
{
3
52'
53' 5+a2e- ni
|
it- r'
;i
,/
,'/s4.--!i-!*6
5'
55.
-3+3i
3-3156.
t
-r-.410r,
-t
',.#+ -+' *.=+,i*lf,sr.
|
13
2S. (-6 - 2i)2tz + z+i
zs. (\/\ - zi)zr - ci\/i
30. (\6 + i\/t), z + oit/)
31. itl t
32. i24 t
33. ie8 - I
34. ite -i
3f. i-t -i
36. i-2 -r
40. i-27 i
i-4 I
41. i-38 -t
39. i-t3 -i
42. i-66 -l
Find the product ofthe given cornplex number and its conjugate.
(Example4)
43.3 - sisl
6r. \/-4 - t/-s -i
62.\/-16 + t/-zsgi
63. \/-4 - r/rc -4 + 2i
64.\/4.
as. $/-a)z -e
66. (V=)3 -sit/i
'/----'-
-r +1v3 7s.2-t1!-:-+,
2
7r. -3 + V! - 4(l)(s)
72.| - \4-rf:7(l)(l)
-l+i\/tt
- r
+ \/8)
,l
n tt.
/
44.4 + 3i2s as.
2it7l4 46.
ircte
)+
!27t4_
/
4s. -i\/,s
4s.3 - iftrrlro. 1 + iU
22
Evaluate the expressionll#3for
a, b, and c. (Example6)
7 5 .a :
l,b:2,c:5
t-it/i
- ,
t7
i;_t;!^,86,c:
z+x6
- \/=)
+ zit/z
each choice of
_4,c:
I
))
78.a:2,b:-4,c:5
z+ile
2
-t-t/ir-+*"
Evaluate the expression -=---;-Jbr
and c. (Example6)
81.a:-2,b:6,c:6
-
t/-o(t/i
_ l + 2 i 7 6 .a = 5 , b :
*2+l - t
7 7 ,a : 2 , b : 4 , c : 3
-z + it/i
'n'
-t
3
^
-2+\/-20
6s.
r/-t
-6 + V-J
--,
z t -*t 6 ,
6s.
Vrso - ro
67.\/-.
lt4
27. (3 + 4i)2-t + z+i
37. i-3 i
Ilrite each expression in theform a * bi, where a and b are real
numbers. (Example6)
73. V-8(V-2
- it/i) s
'38.
7
60.4+2i
+1.
t75-3t17
24. (4 + 3i)(4 _ 3i) 2s
- i)(t/l + i) +
26.(\/, + i\/r)(\/,
47.i1
I
, -,1+
2-i
e. (3 - 3i) + (4 + 5i)7+ 2i r0. (-32+_2i)+ (5 _ 6t
2s.($
st'
3ao*-*r
Perform the indicated operations and write your answers in the
-t bi, where a and b are real
numbers. (Examples
forn a
2 and 3)
11.(l -,
Write each quotient in theform a t bi. (Exampte5)
'o'
,=
each choice of a, b.
Zi ],,8
-12,c:84
8 2 .a : 3 , b : 6 , c : 8
-t - it/G
3
2
Pedorm the indicated operations. Write the answers in the fonn
a * bi where a and b are real numbers.
83. (3 - sr)(3+ 51 34
84. (2 - 4i)(2 + 4i) 20
tDue to spaceconstrictions,
answers to these exercisesmay be found in the complete Answers beginning on page A-l in the
back ofthe book.
P.5rrr
8s. (3 - 5, + (3 + 5t6
?-5t
a
s 7 . - ;- -r - - ]
J
tt
)t
8 9 .( 6 - 2 i ) - ( 7
86.(2-4i)+(2+4i)4
r<
8s.2#
- li
tl
-3t-1
+,
91. is(i2 - 3i)3 - i
34
_T
e 0 -. 3( 5
-6r-(8-er)
+31
9 2 . 3 i 7 ( i- 5 ; 3 )r s
For Writing/Discussion
Polynomials 45
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
+l' +i' Fourthrootsof 1 are +l and
anypositiverealnumber.
+ i. Fourthrootsof x for x ) 0 are + {x and,+ i{x.
+ il00!,wheren!(read"nfac99. Evaluatei0!+ il! + i2! +...
torial") is the product ofthe integers from I through n if n > 1
a n d 0 !: l . g s + 2 i
93. Explain in detail how to find in for anypositive integern.
7hwrry Ocr#nafha Ea'r lf e ftI
94. Fin! a nurqbera * bi suchthata2 + b2 is irrational.
Reversingthe Digits Find a four-digit integerx suchthat 4x is another four-digit integerwhosedigits arein the reverseorder ofthe
digitsofx. 2178
tlz + itlz
95. Letw = a * bi and,w: a - bi,wherea andb arerealnumbers.Showthat w * w is real andthat w - w is imaginary.
(containingno mathematical
Write sentences
symbols)stating
theseresults.
SummingReciprocals Thereis only oneway to write I asa sum
ofthe reciprocalsofthree differentpositiveintegers:
1*l*1=',
236-
96. Is it truethattheproductof a complexnumberandits conjugateis a real number?Explain. True
97. Provethatthereciproc.al
ofa t bi,wherea utdbare notboth
.ab
Z€iO.
__i_
_
. arS. + b . d . + b .
___;___
- r.
-_
Find all possiblewaysto write I asa sumof thereciprocalsof four
differentpositiveintegers.
l
I
I
2
3
l0
t5'2 3
l
I
I
t1
I
_r_r-r-_r-r-r--r-r-r2' 3' 7 42'2' 4'
I
9
_ f
I
_ I - I - _ I - l - l - - I - f
ll
I
6'
ll
1
r8'2 3
1l
I
t2'2' 4'
I
I
8
24'
I
20
_ I -
I
s'
fMPop Quiz
l . Findthesumof 3 + 2i and4 - i.'7 + i
, Findtheproductof 4 - 3i and2 + i. 11 - 2i.
3. Findtheproductof2 - 3i andits conjugate.13
4.
wite
VlTrin
theforma t bi.'|| .
f
Find i27. -i
6. What are the two squareroots of -16? +4i
,