chapter 2 real and complex numbers...all numbers of the form p/q where p, q are integers and q is...
TRANSCRIPT
CHAPTER
2 REAL AND COMPLEX NUMBERS
Animation 2.1:Real And Complex numbersSource & Credit: eLearn.punjab
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Students Learning Outcomes
• Afterstudyingthisunit,thestudentswillbeableto:• Recallthesetofrealnumbersasaunionofsetsofrationaland
irrationalnumbers.• Depictrealnumbersonthenumberline.• Demonstrate a number with terminating and non-terminating
recurringdecimalsonthenumberline.• Givedecimalrepresentationofrationalandirrationalnumbers.• Knowthepropertiesofrealnumbers.• Explaintheconceptofradicalsandradicands.• Differentiatebetweenradical formandexponential formofan
expression.• Transformanexpressiongiveninradicalformtoanexponential
formandviceversa.• Recallbase,exponentandvalue.• Apply the laws of exponents to simplify expressions with real
exponents.• Definecomplexnumberzrepresentedbyanexpressionofthe
form z a ib= + ,whereaandbarerealnumbersand 1i = -• Recognizeaasrealpartandbasimaginarypartofz = a + ib.• Defineconjugateofacomplexnumber.• Knowtheconditionforequalityofcomplexnumbers.• Carry out basic operations (i.e., addition, subtraction,
multiplication and division) on complex numbers.
Introduction
Thenumbersare the foundationofmathematicsandweusedifferentkindsofnumbers inourdaily life.So it isnecessarytobefamiliarwithvariouskindsofnumbersInthisunitweshalldiscussrealnumbersandcomplexnumbersincludingtheirproperties.Thereisaone-onecorrespondencebetweenrealnumbersandthepointsontherealline.Thebasicoperationsofaddition,subtraction,multiplicationanddivisiononcomplexnumberswillalsobediscussedinthisunit.
2.1 Real Numbers
Werecallthefollowingsetsbeforegivingtheconceptofrealnumbers.
Natural NumbersThenumbers1,2,3,...whichweuseforcountingcertainobjectsare callednatural numbers or positive integers. The set of naturalnumbersisdenotedbyN.i.e.,N={1,2,3,....}
Whole NumbersIfweinclude0inthesetofnaturalnumbers,theresultingsetisthesetofwholenumbers,denotedbyW,i.e.,W={o,1,2,3,....}
IntegersThesetofintegersconsistofpositiveintegers,0andnegativeintegersandisdenotedbyZi.e.,Z={...,–3,–2,–1,0,1,2,3,...}
2.1.1 Set of Real Numbers
Firstwerecallaboutthesetofrationalandirrationalnumbers.
Rational Numbers Allnumbersof the formp/qwherep,qare integersandq isnotzeroarecalledrationalnumbers.ThesetofrationalnumbersisdenotedbyQ,
Irrational NumbersThenumberswhichcannotbeexpressedasquotientofintegersarecalledirrationalnumbers.ThesetofirrationalnumbersisdenotedbyQ’,
. ., | , 0 pi e Q p q Z qq
= ∈ ∧ ≠
| , , 0pQ x x p q Z qq
′ = ≠ ∈ ∧ ≠
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Forexample,thenumbers 2, 3, 5, p andeareallirrationalnumbers.Theunionofthesetofrationalnumbersandirrationalnumbers isknownasthesetofrealnumbers.ItisdenotedbyR,i.e.,R=QjQ/
HereQandQ’arebothsubsetofRandQkQ/=fNote:
(i) NfWfZfQ(ii) QandQ/aredisjointsets.(iii) foreachprimenumberp,
isanirrationalnumber.(iv) squarerootsofallpositive
non-squareintegersareirrational.
2.1.2 Depiction of Real Numbers on Number Line
Therealnumbersarerepresentedgeometricallybypointsonanumberlinelsuchthateachrealnumber‘a’correspondstooneandonlyonepointonnumberlinelandtoeachpointPonnumberlineltherecorrespondspreciselyonerealnumber.Thistypeofassociationorrelationshipiscalledaone-to-onecorrespondence.Weestablishsuchcorrespondenceasbelow.WefirstchooseanarbitrarypointO(theorigin)onahorizontallinelandassociatewithittherealnumber0.Byconvention,numberstotherightoftheoriginarepositiveandnumberstotheleftoftheoriginarenegative.Assignthenumber1tothepointAsothatthelinesegmentOArepresentsoneunitoflength.
Thenumber ‘a’ associatedwith a point P on l is called thecoordinateofP,andliscalledthecoordinatelineortherealnumberline.Foranyrealnumbera,thepointP’(– a)correspondingto–aliesatthesamedistancefromOasthepointP(a)correspondingtoabutintheoppositedirection.
2.1.3 Demonstration of a Number with Terminating and Non-Terminating decimals on the Number Line
Firstwe give the following concepts of rational and irrationalnumbers.
(a) Rational NumbersThedecimalrepresentationsofrationalnumbersareoftwotypes,terminatingandrecurring.
(i) Terminating Decimal FractionsThedecimalfractioninwhichtherearefinitenumberofdigitsinitsdecimalpartiscalledaterminatingdecimalfraction.Forexample
2 30.4 0.3755 8
and= =
(ii) Recurring and Non-terminating Decimal Fractions Thedecimal fraction (non-terminating) inwhich somedigitsarerepeatedagainandagaininthesameorderinitsdecimalpartiscalledarecurringdecimalfraction.Forexample
2 40.2222 0.363636...9 11
and= =
(b) Irrational Numbers Itmay be noted that the decimal representations for irrationalnumbersareneitherterminatingnorrepeatinginblocks.Thedecimalformofanirrationalnumberwouldcontinueforeverandneverbegintorepeatthesameblockofdigits.
p
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e.g., =1.414213562...,p= 3.141592654..., e= 2.718281829..., etc.
Obviously these decimal representations are neither terminatingnorrecurring.Weconsiderthefollowingexample.
Example
Expressthefollowingdecimalsintheform qp
wherep,qdZ
andqm0 (a) 0.3 0.333...= (b) 0.23 0.232323...=
Solution(a) Let 0.3x = whichcanberewrittenasx=0.3333... ……(i)Notethatwehaveonlyonedigit3repeatingindefinitely.So,wemultiplybothsidesof(i)by10,andobtain10x=(0.3333...)x10or 10x=3.3333... ……(ii)Subtracting(i)from(ii),wehave10x–x=(3.3333...)–(0.3333...)
or 9x=3⇒ 13
x =
Hence 10.33
=
(b) Let 0.23 0.23 23 23... x = =
Sincetwodigitblock23isrepeatingitselfindefinitely,sowemultiplybothsidesby100.Then100 23.23x =
100 23 0.23 23x x= + = +
100 23232399
99
x xx
x
⇒ - =⇒ =
⇒ =
Thus 230.2399
= isarationalnumber.
2.1.4 Representation of Rational and Irrational Numbers on Number Line
Inordertolocateanumberwithterminatingandnon-terminatingrecurringdecimalonthenumberline,thepointsassociatedwiththe
rationalnumbers mn
and mn
- wherem, narepositive integers,we
subdivideeachunitlengthintonequalparts.Thenthemthpointof
divisiontotherightoftheoriginrepresentsmnandthattotheleft
oftheoriginatthesamedistancerepresents mn
-
ExampleRepresentthefollowingnumbersonthenumberline.
2 15 7( ) ( ) ( ) 15 5 9
i ii iii- -
Solution
(i) Forrepresentingtherationalnumber 25
- onthenumberlinel,
dividetheunitlengthbetween0and–1intofiveequalpartsandtaketheendofthesecondpartfrom0toitsleftside.ThepointMinthe
followingfigurerepresentstherationalnumber 25
-
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15 1( ) 2 :7 7
ii = + itliesbetween2and3.
Dividethedistancebetween2and3intosevenequalparts.Thepoint
Prepresentsthenumber15 12 .7 7
=
(iii) For representing the rational number, 71 .9
- divide the unit
lengthbetween–1and–2intonineequalparts.Taketheendofthe7thpartfrom–1.ThepointMinthefollowingfigurerepresentsthe
rationalnumber, 71 .9
-
Irrationalnumberssuchas 2, 5 etc.canbelocatedonthelineℓbygeometricconstruction.Forexample,thepointcorrespondingto 2 maybeconstructedbyformingaright∆OABwithsides(containingtherightangle)eachoflength1asshowninthefigure.ByPythagorasTheorem,
2 2(1) (1) 2OB = + =
BydrawinganarcwithcentreatOandradiusOB= 2 ,wegetthepointPrepresenting 2 onthenumberline.
EXERCISE 2.1
1. Identify which of the following are rational and irrationalnumbers.
1 153 7.25 296 2
(i) (ii) (iii) (iv) (v) (vi)p
2. Convertthefollowingfractionsintodecimalfractions.
17 19 57 205 5 2525 4 8 18 8 38
(i) (ii) (iii) (iv) (v) (vi)
3. Whichofthefollowingstatementsaretrueandwhicharefalse?
(i) 23 isanirrationalnumber.(ii)pisanirrationalnumber.
(iii) 19isaterminatingfraction.(iv) 3
4isaterminatingfraction.
(v) 45 isarecurringfraction.
4. Representthefollowingnumbersonthenumberline.
2 4 3 5 32 53 5 4 8 4
(i) (ii) (iii)1 (iv) (v)2 (vi)- -
5.Givearationalnumberbetween 34 and 5
9.
6.Expressthefollowingrecurringdecimalsastherationalnumber
pq
wherep, qareintegersandq≠0 67(i)0.5(ii)0.13(iii)0.
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2.2 Properties of Real Numbers
Ifa,barerealnumbers,theirsumiswrittenasa+bandtheirproductasaboraxbora.bor(a)(b).
(a) Properties of Real numbers with respect to Addition and Multiplication Properties of real numbers under addition are as follows:
(i) Closure Property
a+bdR, a,bdR
e.g., if-3and5dR,then-3+5=2dR
(ii) Commutative Property
a+b=b+a, a,bdR
e.g.,if2,3dR,then2+3=3+2or 5=5
(iii) Associative Property
(a+b)+c=a+(b+c), a,b,cdRe.g., if5,7,3dR,then(5+7)+3=5+(7+3)or 12+3=5+10or 15=15
(iv) Additive IdentityThereexistsauniquerealnumber0,calledadditiveidentity,suchthat
a+0=a=0+a, adR
(v) Additive Inverse
ForeveryadR,thereexistsauniquerealnumber–a,calledtheadditiveinverseofa,suchthata+(–a)=0=(–a)+ae.g.,additiveinverseof3is–3since3+(–3)=0=(–3)+(3)
Properties of real numbers under multiplication are as follows:
(i) Closure Property
abdR, a,bdRe.g., if-3,5dR,then (-3)(5)dRor -15dR
(ii) Commutative Property
(i) Associative Property
(ab)c=a(bc), a,b,cdR
e.g.,if2,3,5dR,then(2%3)%5=2%(3%5)or 6%5=2%15or 30=30
, ,1 3,3 2
1 3 3 13 2 2 3
1 12 2
e.g.,if
then
or
ab ba a b R
R
= ∀ ∈
∈
=
=
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(ii) Multiplicative IdentityThereexistsauniquerealnumber1,calledthemultiplicativeidentity,suchthat
a•1=a=1•a, adR
(iii) Multiplicative InverseForeverynon-zerorealnumber,thereexistsauniquereal
number calledmultiplicativeinverseofa,suchthat
So,5and aremultiplicativeinverseofeachother.
(vi) Multiplication is Distributive over Addition and Subtraction Foralla,b,cdR a(b+c)=ab+ac(Leftdistributivelaw) (a+b)c=ac+bc (Rightdistributivelaw) e.g., if2,3,5dR,then2(3+5)=2%3+2%5 or 2%8=6+10 or 16=16Andforalla,b,cdR a(b-c)=ab-ac (Leftdistributivelaw) (a- b)c=ac-bc(Rightdistributivelaw)e.g.,if2,5,3dR,then 2(5-3)=2%5-2%3 or 2%2=10-6 or 4=4
Note:
(b) Properties of Equality of Real NumbersPropertiesofequalityofrealnumbersareasfollows:
(i) Reflexive Property
a=a, adR
(ii) Symmetric Property
Ifa=b,thenb=a, a,bdR
(iii) Transitive Property Ifa =bandb=c,thena=c, a,b,cdR
(iv) Additive Property Ifa=b,thena+c=b+c, a,b,cdR
(v) Multiplicative Property
Ifa=b,thenac=bc, a,b, cdR
(vi) Cancellation Property for Addition
Ifa+c=b+c,thena=b, a,b,cdR
(vii) Cancellation Property for Multiplication
Ifac=bc,c≠0thena=b, a,b,cdR
(i) Thesymbol means“forall”,(ii) aisthemultiplicativeinverseofa–1,i.e.,a=(a–1)–1
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(c) Properties of Inequalities of Real NumbersPropertiesofinequalitiesofrealnumbersareasfollows:(i) Trichotomy Property
a,bdRa<bora=bora>b
(ii) Transitive Property
a,b,cdR
(a) a<bandb<c⇒a<c (b) a>bandb>c⇒a>c
(iii) Additive Property
[ a,b,cdR
a<b⇒a+c<b+ca<b⇒c+a<c+b
and (b)a>b⇒a+c>b+ca>b⇒c+a>c+b
(iv) Multiplicative Property
(a) a,b,cdRandc>0
(i)a>b⇒ac>bca>b⇒ca>cb
(ii)a<b⇒ac<bca<b⇒ca<cb
(b) a,b,cdRandc<0
(i)a>b⇒ac<bca>b⇒ca<cb
(ii)a<b⇒ac>bca<b⇒ca>cb
(v) Multiplicative Inverse Propertya,bdRanda≠0,b≠0
EXERCISE 2.2
1. Identifythepropertyusedinthefollowing(i) a+b=b+a (ii) (ab)c=a(bc)(iii) 7%1=7 (iv) x>yorx=yorx<y(v) ab=ba (vi) a+c=b+c⇒a=b
(vii) 5+(-5)=0 (viii)
(ix) a>b⇒ac>bc(c>0)2. Fill in the following blanks by stating the properties of realnumbersused. 3x+3(y-x) =3x+3y-3x, …….. =3x–3x+3y, ……..=0+3y, ……..=3y ……..3. Givethenameofpropertyusedinthefollowing.
2.3 Radicals and Radicands
2.3.1 Concept of Radicals and Radicands
Ifnisapositiveintegergreaterthan1andaisarealnumber,thenanyrealnumberxsuchthatxn=aiscalledthenthrootofa,andinsymbolsiswrittenas
1/, ( ) , nnx a or x a= =
In the radical ,n a the symbol is called the radical sign, n is
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calledtheindexoftheradicalandtherealnumberaundertheradicalsigniscalledtheradicandorbase.
Note:
2.3.2 Difference between Radical form and Exponential form Inradicalform,radicalsignisused
e.g., isaradicalform. areexamplesofradicalform.
Inexponentialform,exponentialisusedinplaceofradicals, e.g., x=(a)1/nisexponentialform. x3/2,z2/7areexamplesofexponentialform.
Properties of Radicals Leta,bdRandm,nbepositiveintegers.Then,
( )
(i) (ii)
(iii) (iv) (v)
nn n n n
n
m m nn nn m nm n
a aab a bb b
a a a a a a
= =
= = =
2.3.3 Transformation of an Expression given in Radical form to Exponential form and vice versa Themethod of transforming expression in radical form toexponentialformandviceversaisexplainedinthefollowingexamples.
Example 1Write each radical expression in exponential notation and eachexponentialexpressioninradicalnotation.Donotsimplify.
Solution
Example 2
Solution
EXERCISE 2.3
1. Writeeachradicalexpressioninexponentialnotationandeachexponentialexpressioninradicalnotation.Donotsimplify.
2. Tellwhetherthefollowingstatementsaretrueorfalse?
3. Simplifythefollowingradicalexpressions.
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2.4 Laws of Exponents / Indices
2.4.1 Base and Exponent
Intheexponentialnotationan(readasatothenthpower)wecall‘a’asthebaseand‘n’astheexponentorthepowertowhichthebaseisraised. Fromthisdefinition, recall that,wehave the following lawsofexponents.Ifa, bdRandm,narepositiveintegers,then
I am•an=am+n II (am)n=amn
III (ab)n=anbn IV
V =am/an,am-n,a≠0 VI a0=1,wherea≠0
VII1 , 0wheren
na aa
- = ≠
2.4.2 Applications of Laws of Exponents
Themethodofapplyingthelawsofindicestosimplifyalgebraicexpressionsisexplainedinthefollowingexamples.
Example 1Userulesofexponentstosimplifyeachexpressionandwritetheanswerintermsofpositiveexponents.
22 3 7 3 0
3 4 5
49
(i) (ii)x x y a bx y a
-- -
- -
Solution
Example 2Simplifythefollowingbyusinglawsofindices:
4/3
1
8 4(3)125 3 3
(i) (ii) n
n n
-
+ -
SolutionUsingLawsofIndices,
EXERCISE 2.4
1. Uselawsofexponentstosimplify:
2 3 7 5 7
3 4 3 4
7 4 3
3 5 2
2 23 0 3 5
5
( )
4 4 19 9
(i)
(ii)
m n m n
mm n
n
x x y x y a a ax y x y
y y a ax x a
a b aa
- - -+
- -
--
- +
- -+
-
= =
= = =
×=
0
2 28
8
16
, 1
4 99 4
8116
mm n
n
n n
n n
n
a a ba
a a ba b a
a aa b b
-
- + -
= =
= = =
= =
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4/32 1 4 5 4 1
4 3 0 2 3
(81) .3 (3) (243)(9 )(3 )
(iii) (iv)n n
nx y zx y z
-- - - -
-
-
2. Showthat
1 a b b c c aa b c
b c ax x xx x x
+ + +
× × =
3. Simplify
2
1/3 1/3 1/2 2/3 1/2
1/2 1/3 1/4 1/2
3 2 3
2 (27) (60) (216) (25)(180) (4) (9) (.04)
( ) , 032 2 3
(i) (ii)
(iii)5 (5 ) (iv) x x x
- -
× × ×× ×
÷ ÷ ≠
2.5 Complex Numbers Werecallthatthesquareofarealnumberisnon-negative.Sothesolutionoftheequationx2+1=0orx2=–1doesnotexistinR.Toovercome this inadequacyof realnumbers,weneedanumberwhose square is –1. Thus the mathematicians were tempted tointroducealargersetofnumberscalledthesetofcomplexnumberswhichcontainsRandeverynumberwhosesquareisnegative.Theyinventedanewnumber–1,calledthe imaginaryunit,anddenoteditbytheletter i (iota)havingthepropertythat 2 1i = - .Obviously i isnota realnumber. It isanewmathematicalentity thatenablesustoenlargethenumbersystemtocontainsolutionofeveryalgebraicequation of the form x2 = –a,wherea > 0. By taking newnumber
1i = - ,thesolutionsetofx2+1=0is
Note:
Integral Powers of i Byusing 1i = - ,wecaneasilycalculatetheintegralpowersofi .
e.g., 2 3 2 4 2 2 8 2 4 41, , ( 1)( 1) 1, ( ) ( 1) 1, i i i i i i i i i i= - = × = - = × = - - = = = - =10 2 5 5( ) ( 1) 1, etc.i i= = - = - Apure imaginarynumber isthesquarerootofanegativerealnumber.
2.5.1 Definition of a Complex Number
Anumberoftheformz=a+biwhereaandbarerealnumbersand 1i = - ,iscalledacomplexnumberandisrepresentedbyzi.e.,z=a+ib
2.5.2 Set of Complex Numbers
ThesetofallcomplexnumbersisdenotedbyC,and
{ | , , 1}where andC z z a bi a b R i= = + ∈ = -
Thenumbersaandb,calledtherealandimaginarypartsofz,aredenotedasa=R(z)andb=lm(z).Observe that:(i) EveryadRmaybeidentifiedwithcomplexnumbersoftheforma+Oitakingb=0.Therefore,everyrealnumberisalsoacomplexnumber.ThusRfC.Notethateverycomplexnumberisnotarealnumber.(ii) Ifa=0,thena+bireducestoapurelyimaginarynumberbi.ThesetofpurelyimaginarynumbersisalsocontainedinC.
TheSwissmathematicianLeonardEuler(1707–1783)wasthefirsttousethesymboliforthenumber 1-
Numberslike 1, 5- - etc.arecalledpureimaginarynumbers.
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2x=4andy2=9,i.e.,x=2andy=±3
PropertiesofrealnumbersRarealsovalidforthesetofcomplexnumbers.
(i) 1 1z z= (Reflexivelaw)(ii) Ifz1=z2,thenz2=z1 (Symmetriclaw)(iii) Ifz1=z2andz2=z3,thenz1=z3 (Transitivelaw)
EXERCISE 2.5
1. Evaluate(i) i7 (ii) i50 (iii) i12
(iv) (-)8 (v) (-i)5 (vi) i27
2. Writetheconjugateofthefollowingnumbers.(i) 2+3i (ii) 3-5i (iii) -i(iv) -3+4i (v) -4-i (vi) i-3
3. Writetherealandimaginarypartofthefollowingnumbers.(i) 1+i (ii) -1+2i (iii) -3i+2(iv) -2-2i (v) -3i (vi) 2+0i
4. Findthevalueofxandyifx+iy+1=4-3i.
2.6 Basic Operations on Complex Numbers
(i) AdditionLetz1=a+ibandz2=c+idbetwocomplexnumbersanda,b,c,ddR.Thesumoftwocomplexnumbersisgivenby z1+z2=(a+bi)+(c+di)=(a+c)+(b+d)i i.e., the sum of two complex numbers is the sum of thecorrespondingrealandtheimaginaryparts.e.g.,(3-8i)+(5+2i)=(3+5)+(-8+2)i=8-6i
(i) MultiplicationLetz1=a+ibandz2=c+idbetwocomplexnumbers.Theproductsarefoundas
(iii) Ifa=b=0,thenz=0+i0iscalledthecomplexnumber0.Thesetofcomplexnumbersisshowninthefollowingdiagram
2.5.3 Conjugate of a Complex Number
Ifwechange i to–i inz=a+bi,weobtainanothercomplexnumbera–bicalledthecomplexconjugateofzandisdenotedbyz(readzbar).
Thenumbersa+bianda—biarecalledconjugatesofeachother.
Note that:
2.5.4 Equality of Complex Numbers and its Properties
Foralla,b,c,ddR,a+bi=c+diifandonlyifa=candb=d.e.g., 2x+y2i=4+9iifandonlyif
(i) z = z(ii)Theconjugateofarealnumberz = a + oicoincideswiththenumberitself,sincez = a + 0i = a - 0i..(iii)conjugateofarealnumberisthesamerealnumber.
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(i) IfkdR,kz1=k(a+bi)=ka+kbi.(Multiplicationofacomplexnumberwithascalar)(ii) z1z2=(a+bi)(c+di)=(ac-bd)+(ad+bc)i(Multiplicationoftwocomplexnumbers) Themultiplication of any two complex numbers (a + bi) and(c+di)isexplainedas z1z2=(a+bi)(c+di)=a(c+di)+bi(c+di) =ac+adi+bci+bdi2
=ac+adi+bci+bd(-1) (sincei2=-1) =(ac-bd)+(ad+bc)i(combiningliketerms)e.g.,(2-3i)(4+5i)=8+10i-12i-15i2=23-2i. (sincei2=-1)
(iii) SubtractionLetz1=a+ibandz2=c+idbetwocomplexnumbers.Thedifferencebetweentwocomplexnumbersisgivenby z1-z2=(a+bi)-(c+di)=(a-c)+(b-d)i e.g.,(-2+3i)-(2+i)=(-2-2)+(3-1)i=-4+2ii.e.,thedifferenceoftwocomplexnumbersisthedifferenceofthecorrespondingrealandimaginaryparts.
(iv) DivisionLetz1=a+ibandz2=c+idbetwocomplexnumberssuchthatz2≠0.Thedivisionofa+bibyc+diisgivenby
(Multiplying the numeratoranddenominatorby c- di, thecomplexconjugateofc+di).
Operationsareexplainedwiththehelpoffollowingexamples.
Example 1Separatetherealandimaginarypartsof
Solution
Example 2
Solution
Example 3
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Solution
2 2
2 2 2
4 5 1 4 5(4 5 ).4 5 4 5 4 5
(4 5)(4 5 ) 16 40 25
(4) (5 ) 16 25
(multiplyinganddividingbythe
conjugateof )
(simplifying
i iii i i
ii i ii i
+ += + ×
- - +-
+ + += =
- -
216 40 25 , 116 25
9 40 9 4041 41 41
)
(since )i i
i i
+ += = -
-- +
= = - +
Example 4 Solve(3-4i)(x+yi)=1+0.iforrealnumbersxandy,where
1i = - .
Solution Wehave (3-4i)(x+yi) =1+0.i or 3x+3iy-4ix-4i2y =1+0.i or 3x+4y+(3y-4x)i =1+0.i Equatingtherealandimaginaryparts,weobtain 3x+4y=1and3y-4x=0olvingthesetwoequationssimultaneously,wehave
EXERCISE 2.6
1. Identifythefollowingstatementsastrueorfalse.
(i) (ii) i73=-i (iii) i10=-1
(iv) Complexconjugateof(-6i+i2)is(-1+6i)(v) Differenceofacomplexnumberz=a+bianditsconjugate
isarealnumber.(vi) If(a–1)–(b+3)i=5+8i,thena =6andb=-11.(vii) Productofacomplexnumberanditsconjugateisalways
anon-negativerealnumber.2. Expresseachcomplexnumberinthestandardforma +bi,
whereaandbarerealnumbers.
(i) (2+3i)+(7-2i) (ii) 2(5+4i)-3(7+4i) (iii) -(-3+5i)-(4+9i)(iv) 2i2+6i3+3i16-6i19+4i25
3. Simplifyandwriteyouranswerintheforma+bi.
4. Simplifyandwriteyouranswerintheforma+bi.
5. Calculate(a)z(b)z + z(c)z - z(d)z z,foreachofthefollowing
(i) z=-i (ii) z=2+i
(iii) (iv)
6. Ifz=2+3iandw=5-4i,showthat
7. Solvethefollowingequationsforrealxandy.(i) (2-3i)(x+yi)=4+i(ii) (3-2i)(x+yi)=2(x-2yi)+2i-1(iii) (3+4i)2-2(x-yi)=x+yi
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2. Real and Complex Numbers eLearn.Punjab
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2. Real and Complex Numbers eLearn.Punjab
Version: 1.1Version: 1.1
REVIEW EXERCISE 2
1. Multiple Choice Questions. Choose the correct answer.
2. True or false? Identify. (i) Divisionisnotanassociativeoperation. ......... (ii) Everywholenumberisanaturalnumber. ......... (iii) Multiplicativeinverseof0.02is50. ......... (iv) p isarationalnumber. ......... (v) Everyintegerisarationalnumber. ......... (vi) Subtractionisacommutativeoperation. ......... (vii) Everyrealnumberisarationalnumber. ......... (viii) Decimal representation of a rational number is either terminatingorrecurring. .........3. Simplifythefollowing:
12 8 10 84
3 4 5 6 4
2 1 5 4 4
81 25
32625
1/5 2/5
(i) (ii)
(iii) (iv)
n my x x y
x y z x y zx y z x y z
- -
- -
- - - -
4. Simplify2/3 1/2
3/2
(216) (25)(0.04)
-
×
5. Simplify
. 5( . ) , 0 p q q rp q
p r p rq r
a a a a aa a
+ +
- ÷ ≠
6. Simplify2 2 2l m n
l m m n n la a aa a a+ + +
7. Simplify 3 3 3l m n
m n la a aa a a
× ×
SUMMARY
*SetofrealnumbersisexpressedasR=QUQ/where
{ }| , 0 , | . isnotrationalpQ p q Z q Q x x
q
= ∈ ∧ ≠ =
* Propertiesofrealnumbersw.r.t.additionandmultiplication:Closure:a + b d R, ab d R, a, b d RAssociative:
(a + b) + c = a + (b + c), (ab)c = a(bc), a , b , c d R
Commutative:
a + b = b + a, ab = ba, a, b d R
AdditiveIdentity:
a + 0 = a = 0 + a, a dR
MultiplicativeIdentity:
a . 1 = a = 1 . a, a d R
AdditiveInverse:
a + (-a) = 0 = (-a) + a, a d R
MultiplicativeInverse:
Multiplicationisdistributiveoveradditionandsubtraction: a(b + c) = ab + ac, [ a, b, c d R (b + c)a = ba + ca [ a, b, c d R a(b - c) = ab - ac [ a, b, c d R (a - b)c = ac - bc [ a, b, c d R
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2. Real and Complex Numbers eLearn.Punjab
Version: 1.1
* PropertiesofequalityinRReflexive:a = a,[a d R Symmetric:a = b⇒b=a,[a, b d RTransitive:a = b, b = c⇒ a = c, [a, b,c d R Additiveproperty:Ifa=b,thena + c = b + c,[a, b, c d RMultiplicativeproperty:Ifa = b,thenac = bc,[a, b ,c d RCancellationproperty:Ifac = be, cm0,thena = b,[a, b,c d R*Intheradical isradicalsign,xisradicandorbaseandnisindexofradical.*Indicesandlawsofindices:[a, b,c d Randm, n d z,(am)n=amn,(ab)n=anbn
, 0n n
na a bb b
= ≠
aman=am+n
0
, 0
1 , 0
1
mm n
n
nn
a a aa
a aa
a
-
-
= ≠
= ≠
=
*Complexnumberz=a+biisdefinedusingimaginaryunit 1i = - . where a, b d Randa=Re(z),b=Im(z)*Conjugateofz =a+biisdefinedasz=a - bi