chapter 2 real and complex numbers...all numbers of the form p/q where p, q are integers and q is...

CHAPTER

2 REAL AND COMPLEX NUMBERS

Animation 2.1:Real And Complex numbersSource & Credit: eLearn.punjab

version: 1.1

http://elearn.punjab.gov.pk/

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2. Real and Complex Numbers eLearn.Punjab

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Version: 1.1Version: 1.1

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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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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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

chapter 2 real and complex numbers...all numbers of the form p/q where p, q are integers and q is...

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