EXPONENTS AND RADICALS


 Amy Ford
 1 years ago
 Views:
Transcription
1 Expoets d Rdicls MODULE  EXPONENTS AND RADICALS We hve lert bout ultiplictio of two or ore rel ubers i the erlier lesso. You c very esily write the followig, d Thik of the situtio whe is to be ultiplied ties. How difficult is it to write?... ties? This difficulty c be overcoe by the itroductio of expoetil ottio. I this lesso, we shll expli the eig of this ottio, stte d prove the lws of expoets d ler to pply these. We shll lso ler to express rel ubers s product of powers of prie ubers. I the ext prt of this lesso, we shll give eig to the uber /q s qth root of. We shll itroduce you to rdicls, idex, rdicd etc. Agi, we shll ler the lws of rdicls d fid the siplest for of rdicl. We shll ler the eig of the ter rtiolisig fctor d rtiolise the deoitors of give rdicls. OBJECTIVES After studyig this lesso, you will be ble to write repeted ultiplictio i expoetil ottio d vicevers; idetify the bse d expoet of uber writte i expoetil ottio; express turl uber s product of powers of prie ubers uiquely; stte the lws of expoets; expli the eig of 0, p d q ; siplify expressios ivolvig expoets, usig lws of expoets; Mthetics Secodry Course
2 MODULE  Expoets d Rdicls idetify rdicls fro give set of irrtiol ubers; idetify idex d rdicd of surd; stte the lws of rdicls (or surds); express give surd i siplest for; clssify siilr d osiilr surds; reduce surds of differet orders to those of the se order; perfor the four fudetl opertios o surds; rrge the give surds i scedig/descedig order of gitude; fid rtiolisig fctor of give surd; rtiolise the deoitor of give surd of the for d b x x where x d y re turl ubers d d b re itegers; siplify expressios ivolvig surds. EXPECTED BACKGROUND KNOWLEDGE Prie ubers Four fudetl opertios o ubers Rtiol ubers Order reltio i ubers. y,. EXPONENTIAL NOTATION Cosider the followig products: (i) I (i), is ultiplied twice d hece is writte s. I, is ultiplied three ties d so is writte s. I, is ultiplied five ties, so is writte s. is red s rised to the power or secod power of. Here, is clled bse d is clled expoet (or idex) Siilrly, is red s rised to the power or third power of. Here, is clled the bse d is clled expoet. Siilrly, is red s rised to the power or Fifth power of. Agi is bse d is the expoet (or idex). 0 Mthetics Secodry Course
3 Expoets d Rdicls Fro the bove, we sy tht The ottio for writig the product of uber by itself severl ties is clled the Expoetil Nottio or Expoetil For. Thus,... 0 ties 0 d ( ) ( )... 0 ties ( ) 0 I 0, is the bse d expoet is 0. I ( ) 0, bse is d expoet is 0. Siilrly, expoetil ottio c be used to write precisely the product of rtioil uber by itself uber of ties. MODULE  Thus, d... ties...0 ties 0 I geerl, if is rtiol uber, ultiplied by itself ties, it is writte s. Here gi, is clled the bse d is clled the expoet Let us tke soe exples to illustrte the bove discussio: Exple.: Evlute ech of the followig: () i Solutio: (i) ( ) ( ) Exple.: Write the followig i expoetil for: (i) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) () Solutio: (i) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Mthetics Secodry Course
4 MODULE  Expoets d Rdicls Exple.: Express ech of the followig i expoetil ottio d write the bse d expoet i ech cse. (i) 0 Solutio: (i) 0 Altertively 0 () () Bse, expoet Here, bse d expoet Here, bse d expoet Here, bse d expoet Exple.: Siplify the followig: Solutio: Siilrly Exple.: Write the reciprocl of ech of the followig d express the i expoetil for: (i) Mthetics Secodry Course
5 Expoets d Rdicls Solutio: (i) MODULE  Reciprocl of Reciprocl of ( ) Reciprocl of Fro the bove exple, we c sy tht if q p is y ozero rtiol uber d is y positive iteger, the the reciprocl of p q q is. p CHECK YOUR PROGRESS.. Write the followig i expoetil for: (i) ( ) ( ) ( ) ( )... 0 ties... 0 ties. Write the bse d expoet i ech of the followig: Mthetics Secodry Course
6 MODULE  Expoets d Rdicls (i) ( ) (). Evlute ech of the followig (i). Siplify the followig: (i). Fid the reciprocl of ech of the followig: (i) ( ). PRIME FACTORISATION Recll tht y coposite uber c be expressed s product of prie ubers. Let us tke the coposite ubers, 0 d. (i) 0 We c see tht y turl uber, other th, c be expressed s product of powers of prie ubers i uique er, prt fro the order of occurrece of fctors. Let us cosider soe exples Exple.: Express 00 i expoetil for. Solutio: Mthetics Secodry Course
7 Expoets d Rdicls 00 Exple.: Express i expoetil for. Solutio: 0 0 MODULE  CHECK YOUR PROGRESS.. Express ech of the followig s product of powers of pries, i.e, i expoetil for: (i). Express ech of the followig i expoetil for: (i) (iv) 0 (v). LAWS OF EXPONENTS Cosider the followig (i) ( ) ( ) ( ) ( ) ( ) [( ) ( )] [( ) ( ) ( ) ( )] [ ( ) ( ) ( ) ( ) ( ) ( )] ( ) ( ) Mthetics Secodry Course
8 Expoets d Rdicls MODULE  Mthetics Secodry Course (iv) ( ) ( ) Fro the bove exples, we observe tht Lw : If is y ozero rtiol uber d d re two positive itegers, the Exple.: Evlute. Solutio: Here, d. Exple.: Fid the vlue of Solutio: As before, 0 0 Now study the followig: (i) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )( ) ( ) ( )
9 Expoets d Rdicls MODULE  Mthetics Secodry Course Fro the bove, we c see tht Lw : If is y ozero rtiol uber d d re positive itegers ( > ), the Exple.0: Fid the vlue of. Solutio: I Lw, < >, the ( ) Lw : Whe > Exple.: Fid the vlue of Solutio: Here, d. Let us cosider the followig: (i) ( )
10 Expoets d Rdicls MODULE  Mthetics Secodry Course 0 Fro the bove two cses, we c ifer the followig: Lw : If is y ozero rtiol uber d d re two positive itegers, the ( ) Let us cosider exple. Exple.: Fid the vlue of Solutio:.. Zero Expoet Recll tht, if >, if > Let us cosider the cse, whe 0 0 Thus, we hve other iportt lw of expoets,. Lw : If is y rtiol uber other th zero, the o. Exple.:Fid the vlue of (i) 0 0 Solutio: (i) Usig 0, we get 0
11 Expoets d Rdicls MODULE  Mthetics Secodry Course Agi usig 0, we get 0. CHECK YOUR PROGRESS.. Siplify d express the result i expoetil for: (i) () (). Siplify d express the result i expoetil for: (i) ( ) ( ). Siplify d express the result i expoetil for: (i) ( ) (iv) 0 (v) 0. Which of the followig stteets re true? (i) (iv) (v) 0 0 (vi) (vii) 0
12 MODULE  Expoets d Rdicls. NEGATIVE INTEGERS AS EXPONENTS i) We kow tht the reciprocl of is. We write it s d red it s rised to power. ii) The reciprocl of ( ) is power.. We write it s ( ) d red it s ( ) rised to the iii) The reciprocl of. We write it s d red it s rised to the power ( ). Fro the bove ll, we get If is y ozero rtiol uber d is y positive iteger, the the reciprocl of i. e. is writte s d is red s rised to the power ( ). Therefore, Let us cosider exple. Exple.: Rewrite ech of the followig with positive expoet: Solutio: (i) (i) ( ) Fro the bove exple, we get the followig result: If q p is y ozero rtiol uber d is y positive iteger, the p q q p q p. 0 Mthetics Secodry Course
13 Expoets d Rdicls MODULE  Mthetics Secodry Course. LAWS OF EXPONENTS FOR INTEGRAL EXPONENTS After givig eig to egtive itegers s expoets of ozero rtiol ubers, we c see tht lws of expoets hold good for egtive expoets lso. For exple. (iv) (i) Thus, fro the bove results, we fid tht lws to hold good for egtive expoets lso. For y ozero rtiol ubers d b d y itegers d,.. if > if >. ( ). ( b) b CHECK YOUR PROGRESS.. Express s rtiol uber of the for q p :
14 MODULE  Expoets d Rdicls. Express s power of rtiol uber with positive expoet: (i). Express s power of rtiol uber with egtive idex: (i) [ ] ( ). Siplify: (i). Which of the followig stteets re true? (i) ( ) b (b) (iv) b b (v) o. MEANING OF p/q You hve see tht for ll itegrl vlues of d, Wht is the ethod of defiig /q, if is positive rtiol uber d q is turl uber. Cosider the ultiplictio q q q q q q... q ties...q ties q q q Mthetics Secodry Course
15 Expoets d Rdicls MODULE  I other words, the qth power of q or i other words q is the qth root of d is writte s q. For exple, or is the fourth root of d is writte s, Let us ow defie rtiol powers of If is positive rel uber, p is iteger d q is turl uber, the p q q We c see tht p q p p q p q p q p p p p...q ties. q q q q q... q ties p p q q p p/q is the qth root of p Cosequetly, is the cube root of. Let us ow write the lws of expoets for rtiol expoets: (i) ( ) (iv) (b) b (v) b b Let us cosider soe exples to verify the bove lws: Exple.: Fid the vlue of (i) ( ) ( ) Mthetics Secodry Course /
16 Expoets d Rdicls MODULE  Mthetics Secodry Course Solutio: (i) ( ) ( ) ) ( ( ) ( ) ) ( CHECK YOUR PROGRESS.. Siplify ech of the followig: (i) ( ). Siplify ech of the followig: (i) ( ) ( ). SURDS We hve red i first lesso tht ubers of the type d, re ll irrtiol ubers. We shll ow study irrtiol ubers of prticulr type clled rdicls or surds.
17 Expoets d Rdicls MODULE  A surd is defied s positive irrtiol uber of the type x, where it is ot possible to fid exctly the th root of x, where x is positive rtiol uber. The uber x is surd if d oly if (i) it is irrtiol uber it is root of the positive rtiol uber.. Soe Teriology I the surd x, the sybol is clled rdicl sig. The idex is clled the order of the surd d x is clled the rdicd. Note: i) Whe order of the surd is ot etioed, it is tke s. For exple, order of ( ) is. ii) is ot surd s its vlue c be deteried s which is rtiol. iii), lthough irrtiol uber, is ot surd becuse it is the squre root of irrtiol uber.. PURE AND MIXED SURD i) A surd, with rtiol fctor is oly, other fctor beig rrtiol is clled pure surd. For exple, d 0 re pure surds. ii) A surd, hvig rtiol fctor other th logwith the irrtiol fctor, is clled ixed surd. For exple, d re ixed surds.. ORDER OF A SURD I the surd, is clled the coefficiet of the surd, is the order of the surd d is the rdicd. Let us cosider soe exples: Exple.: Stte which of the followig re surds? (i) (iv) Mthetics Secodry Course
18 MODULE  Expoets d Rdicls Solutio: (i), which is rtiol uber. is ot surd. is irrtiol uber. is surd., which is irrtiol is surd. (iv) is irrtiol. is surd, d (iv) re surds. Exple.: Fid idex d rdicd i ech of the followig: (i) (iv) Solutio: (i) idex is d rdicd is. idex is d rdicd is. idex is d rdicd is. (iv) idex is d rdicd is. Exple.: Idetify pure d ixed surds fro the followig: (i) Solutio: (i) is pure surd. is ixed surd. is ixed surd..0 LAWS OF RADICALS Give below re Lws of Rdicls: (without proof): (i) [ ] Mthetics Secodry Course
19 Expoets d Rdicls MODULE  b b b b where d b re positive rtiol ubers d is positive iteger. Let us tke soe exples to illustrte. Exple.: Which of the followig re surds d which re ot? Use lws of rdicls to scerti. (i) 0 0 (iv) Solutio: (i) which is rtiol uber. 0 is ot surd , which is irrtiol is surd. (iv) It is ot surd., which is irrtiol is surd. CHECK YOUR PROGRESS.. For ech of the followig, write idex d the rdicd: (i) Mthetics Secodry Course
20 MODULE  Expoets d Rdicls. Stte which of the followig re surds: (i) (iv) (v). Idetify pure d ixed surds out of the followig: (i) (iv). LAWS OF SURDS Recll tht the surds c be expressed s ubers with frctiol expoets. Therefore, lws of idices studied i this lesso before, re pplicble to the lso. Let us recll the here: (i) x. y xy or x.y ( xy) x y x y or x y x y x x x or x x x x x or x (iv) ( ) (v) p p p p ( ) ( ) x x x p p or x x x Here, x d y re positive rtiol ubers d, d p re positive itegers. Let us illustrte these lws by exples: (i) ( ) x () () Mthetics Secodry Course
21 Expoets d Rdicls (iv) ( ) MODULE  Thus, we see tht the bove lws of surds re verified. A iportt poit: The order of surd c be chged by ultiplyig the idex of the surd d idex of the rdicd by the se positive uber. For exple d. SIMILAR (OR LIKE) SURDS Two surds re sid to be siilr, if they c be reduced to the se irrtiol fctor, without cosidertio for coefficiet. For exple, d re siilr surds. Agi cosider d. Now d re expressed s d. Thus, they re siilr surds.. SIMPLEST (LOWEST) FORM OF A SURD A surd is sid to be i its siplest for, if it hs ) sllest possible idex of the sig b) o frctio uder rdicl sig c) o fctor of the for, where is positive iteger, uder the rdicl sig of idex. For exple, Let us tke soe exples. Exple.0: Express ech of the followig s pure surd i the siplest for: (i) Mthetics Secodry Course
22 MODULE  Expoets d Rdicls Solutio: (i), which is pure surd., which is pure surd., which is pure surd. Exple.: Express s ixed surd i the siplest for: (i) 0 0 Solutio: (i), which is ixed surd. 0, which is ixed surd. 0, which is ixed surd. CHECK YOUR PROGRESS.. Stte which of the followig re pirs of siilr surds: (i),, 0,. Express s pure surd: (i). Express s ixed surd i the siplest for: (i) 0. FOUR FUNDAMENTAL OPERATIONS ON SURDS.. Additio d Subtrctio of Surds As i rtiol ubers, surds re dded d subtrcted i the se wy. 0 Mthetics Secodry Course
23 Expoets d Rdicls MODULE  For exple, ( ) d [ ] For ddig d subtrctig surds, we first chge the to siilr surds d the perfor the opertios. For exple i) 0 ( ) ii) ( ) Exple.: Siplify ech of the followig: (i) Solutio: (i) 0 Exple.: Show tht 0 0 Solutio: 0 Mthetics Secodry Course
24 MODULE  Expoets d Rdicls [ ] 0 0 RHS Exple.: Siplify: 000 Solutio: Required expressio 0 ( 0 ) CHECK YOUR PROGRESS. Siplify ech of the followig: Mthetics Secodry Course
25 Expoets d Rdicls MODULE .. Multiplictio d Divisio i Surds Two surds c be ultiplied or divided if they re of the se order. We hve red tht the order of surd c be chged by ultiplyig or dividig the idex of the surd d idex of the rdicd by the se positive uber. Before ultiplyig or dividig, we chge the to the surds of the se order. Let us tke soe exples: [ d re of se order] Let us ultiply d 0 d Let us cosider exple: Exple.:(i) Multiply d 0. Divide by. Solutio: (i) Exple.: Siplify d express the result i siplest for: 0 Mthetics Secodry Course
26 MODULE  Expoets d Rdicls Solutio: 0 0 Give expressio 0 0. COMPARISON OF SURDS To copre two surds, we first chge the to surds of the se order d the copre their rdicds log with their coefficiets. Let us tke soe exples: Exple.: Which is greter or? Solutio: > > > Exple.: Arrge i scedig order:, d. Solutio: LCM of,, d is. Now < < < < Mthetics Secodry Course
27 Expoets d Rdicls MODULE  CHECK YOUR PROGRESS.. Multipliy d.. Multipliy d.. Divide by.. Divide by 0.. Which is greter or?. Which i sller: 0 or?. Arrge i scedig order:,,. Arrge i descedig order:,,. RATIONALISATION OF SURDS Cosider the products: (i) I ech of the bove three ultiplictios, we see tht o ultiplyig two surds, we get the result s rtiol uber. I such cses, ech surd is clled the rtiolisig fctor of the other surd. (i) is rtiolisig fctor of d vicevers. is rtiolisig fctor of d vicevers. is rtiolisig fctor of d vicevers. Mthetics Secodry Course
28 MODULE  Expoets d Rdicls I other words, the process of covertig surds to rtiol ubers is clled rtiolistio d two ubers which o ultiplictio give the rtiol uber is clled the rtiolistio fctor of the other. For exple, the rtiolisig fctor of x is x, of is. Note: (i) The qutities x y d x y re clled cojugte surds. Their su d product re lwys rtiol. Rtiolistio is usully doe of the deoitor of expressio ivolvig irrtiol surds. Let us cosider soe exples. Exple.: Fid the rtiolisig fctors of d. Solutio: Rtiolisig fctor is.. Rtiolisig fctor is. Exple.0: Rtiolise the deoitor of Solutio: ( )( ) ( )( ). ( ) 0 0 Exple.: Rtiolise the deoitor of Solutio: ( )( ) ( )( ). Mthetics Secodry Course
29 Expoets d Rdicls MODULE  Mthetics Secodry Course Exple.: Rtiolise the deoitor of. Solutio: ( ) ( ) [ ]( ) [ ] ( ) Exple.: If, b fid the vlues of d b. Solutio: b b, CHECK YOUR PROGRESS.0. Fid the rtiolisig fctor of ech of the followig: (i) xy y x. Siplify by rtiolisig the deoitor of ech of the followig: (i) (iv)
30 MODULE  Expoets d Rdicls. Siplify:. Rtiolise the deoitor of. If. Fid.. If x y, fid x d y. LET US SUM UP... ties is the expoetil for, where is the bse d is the expoet. Lws of expoet re: (i) (b) b (iv) b b (vi) o (vii) (v) ( ) p q q p A irrtiol uber x is clled surd, if x is rtiol uber d th root of x is ot rtiol uber. I x, is clled idex d x is clled rdicd. A surd with rtiol coefficiet (other th ) is clled ixed surd. The order of the surd is the uber tht idictes the root. The order of x is Lws of rdicls ( > 0, b > 0) b b b (i) [ ] b Mthetics Secodry Course
31 Expoets d Rdicls MODULE  Opertios o surds x y ( xy) ; x x x ; x y x y ( x ) x ; x x or ( x ) x x ( x ) Surds re siilr if they hve the se irrtiol fctor. Siilr surds c be dded d subtrcted. Orders of surds c be chged by ultiplyig idex of the surds d idex of the rdicd by the se positive uber. Surds of the se order re ultiplied d divided. To copre surds, we chge surds to surds of the se order. The they re copred by their rdicds logwith coefficiets. If the product of two surds is rtiol, ech is clled the rtiolisig fctor of the other. x y is clled rtiolisig fctor of x y d vicevers. TERMINAL EXERCISE. Express the followig i expoetil for: (i). Siplify the followig: (i). Siplify d express the result i expoetil for: (i) ( ) ( ) ( ) 0 Mthetics Secodry Course
32 MODULE  Expoets d Rdicls 0. Siplify ech of the followig: (i) o o o ( o o ) ( o o ). Siplify the followig: 0 (i) ( ) ( ) ( ) ( ). Fid x so tht. Fid x so tht x x. Express s product of pries d write the swers of ech of the followig i expoetil for: (i) The str sirus is bout. 0 k fro the erth. Assuig tht the light trvels t.0 0 k per secod, fid how log light fro sirus tkes to rech erth. 0. Stte which of the followig re surds: (i). Express s pure surd: (i) (iv). Express s ixed surd i siplest for: (i) 0 0. Which of the followig re pirs of siilr surds? (i),,, 0 0 Mthetics Secodry Course
33 Expoets d Rdicls.Siplify ech of the followig: (i) MODULE  0. Which is greter? (i) or or. Arrge i descedig order: (i),,,,. Arrge i scedig order:,, 0. Siplify by rtiolisig the deoitor: (i). Siplify ech of the followig by rtiolisig the deoitor: (i) 0. If b, fid the vlues of d b, where d b re rtiol ubers.. If x, fid the vlue of x. x. ANSWERS TO CHECK YOUR PROGRESS. (i) ( ) 0 0 Mthetics Secodry Course
34 MODULE  Expoets d Rdicls. Bse Expoet (i). (i) 0. (i). (i).. (i). (i) (iv) (v) ( ).. (i) (). (i) ( ). (i) (iv) (v). True: (i),, (vii) Flse:, (iv), (v), (vi) Mthetics Secodry Course
35 Expoets d Rdicls. MODULE .. (i). (i). (i). True:,, (iv) (i). (i).. (i),,,.,(iv). Pure: (i), (iv) Mixed:,.. (i),. (i). (i)..... Mthetics Secodry Course
36 MODULE  Expoets d Rdicls ,,.,,.0. (i) x y. (i).. [ ]. (iv). 0 ANSWERS TO TERMINAL EXERCISE. (i). (i) 0. (i). (i) zero zero. (i) () Mthetics Secodry Course
37 Expoets d Rdicls. x. x.. 0 secods 0.,, (iv) MODULE . (i) (i) 0. (i),. (i) zero. (i). (i),,,,., 0,. (i) ( ) ( ). (i) , b. Mthetics Secodry Course
Repeated multiplication is represented using exponential notation, for example:
Appedix A: The Lws of Expoets Expoets re shorthd ottio used to represet my fctors multiplied together All of the rules for mipultig expoets my be deduced from the lws of multiplictio d divisio tht you
More informationChapter 3 Section 3 Lesson Additional Rules for Exponents
Chpter Sectio Lesso Additiol Rules for Epoets Itroductio I this lesso we ll eie soe dditiol rules tht gover the behvior of epoets The rules should be eorized; they will be used ofte i the reiig chpters
More informationMATH 90 CHAPTER 5 Name:.
MATH 90 CHAPTER 5 Nme:. 5.1 Multiplictio of Expoets Need To Kow Recll expoets The ide of expoet properties Apply expoet properties Expoets Expoets me repeted multiplictio. 3 4 3 4 4 ( ) Expoet Properties
More informationm n Use technology to discover the rules for forms such as a a, various integer values of m and n and a fixed integer value a.
TIth.co Alger Expoet Rules ID: 988 Tie required 25 iutes Activity Overview This ctivity llows studets to work idepedetly to discover rules for workig with expoets, such s Multiplictio d Divisio of Like
More informationA function f whose domain is the set of positive integers is called a sequence. The values
EQUENCE: A fuctio f whose domi is the set of positive itegers is clled sequece The vlues f ( ), f (), f (),, f (), re clled the terms of the sequece; f() is the first term, f() is the secod term, f() is
More informationMATHEMATICS FOR ENGINEERING BASIC ALGEBRA
MATHEMATICS FOR ENGINEERING BASIC ALGEBRA TUTORIAL  INDICES, LOGARITHMS AND FUNCTION This is the oe of series of bsic tutorils i mthemtics imed t begiers or yoe wtig to refresh themselves o fudmetls.
More informationArithmetic Sequences
Arithmetic equeces A simple wy to geerte sequece is to strt with umber, d dd to it fixed costt d, over d over gi. This type of sequece is clled rithmetic sequece. Defiitio: A rithmetic sequece is sequece
More informationWe will begin this chapter with a quick refresher of what an exponent is.
.1 Exoets We will egi this chter with quick refresher of wht exoet is. Recll: So, exoet is how we rereset reeted ultilictio. We wt to tke closer look t the exoet. We will egi with wht the roerties re for
More informationUNIT FIVE DETERMINANTS
UNIT FIVE DETERMINANTS. INTRODUTION I uit oe the determit of mtrix ws itroduced d used i the evlutio of cross product. I this chpter we exted the defiitio of determit to y size squre mtrix. The determit
More informationSquare & Square Roots
Squre & Squre Roots Squre : If nuber is ultiplied by itself then the product is the squre of the nuber. Thus the squre of is x = eg. x x Squre root: The squre root of nuber is one of two equl fctors which
More informationSummation Notation The sum of the first n terms of a sequence is represented by the summation notation i the index of summation
Lesso 0.: Sequeces d Summtio Nottio Def. of Sequece A ifiite sequece is fuctio whose domi is the set of positive rel itegers (turl umers). The fuctio vlues or terms of the sequece re represeted y, 2, 3,...,....
More informationExample 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers.
2 Rtionl Numbers Integers such s 5 were importnt when solving the eqution x+5 = 0. In similr wy, frctions re importnt for solving equtions like 2x = 1. Wht bout equtions like 2x + 1 = 0? Equtions of this
More information5.6 POSITIVE INTEGRAL EXPONENTS
54 (5 ) Chpter 5 Polynoils nd Eponents 5.6 POSITIVE INTEGRAL EXPONENTS In this section The product rule for positive integrl eponents ws presented in Section 5., nd the quotient rule ws presented in Section
More informationChapter 04.05 System of Equations
hpter 04.05 System of Equtios After redig th chpter, you should be ble to:. setup simulteous lier equtios i mtrix form d vicevers,. uderstd the cocept of the iverse of mtrix, 3. kow the differece betwee
More informationSPECIAL PRODUCTS AND FACTORIZATION
MODULE  Specil Products nd Fctoriztion 4 SPECIAL PRODUCTS AND FACTORIZATION In n erlier lesson you hve lernt multipliction of lgebric epressions, prticulrly polynomils. In the study of lgebr, we come
More informationMATH 150 HOMEWORK 4 SOLUTIONS
MATH 150 HOMEWORK 4 SOLUTIONS Section 1.8 Show tht the product of two of the numbers 65 1000 8 2001 + 3 177, 79 1212 9 2399 + 2 2001, nd 24 4493 5 8192 + 7 1777 is nonnegtive. Is your proof constructive
More informationn Using the formula we get a confidence interval of 80±1.64
9.52 The professor of sttistics oticed tht the rks i his course re orlly distributed. He hs lso oticed tht his orig clss verge is 73% with stdrd devitio of 12% o their fil exs. His fteroo clsses verge
More informationExample 2 Find the square root of 0. The only square root of 0 is 0 (since 0 is not positive or negative, so those choices don t exist here).
BEGINNING ALGEBRA Roots ad Radicals (revised summer, 00 Olso) Packet to Supplemet the Curret Textbook  Part Review of Square Roots & Irratioals (This portio ca be ay time before Part ad should mostly
More informationPolynomial Functions. Polynomial functions in one variable can be written in expanded form as ( )
Polynomil Functions Polynomil functions in one vrible cn be written in expnded form s n n 1 n 2 2 f x = x + x + x + + x + x+ n n 1 n 2 2 1 0 Exmples of polynomils in expnded form re nd 3 8 7 4 = 5 4 +
More informationHermitian Operators. Eigenvectors of a Hermitian operator. Definition: an operator is said to be Hermitian if it satisfies: A =A
Heriti Opertors Defiitio: opertor is sid to be Heriti if it stisfies: A A Altertively clled self doit I QM we will see tht ll observble properties st be represeted by Heriti opertors Theore: ll eigevles
More informationSection 6.1 Radicals and Rational Exponents
Sectio 6.1 Radicals ad Ratioal Expoets Defiitio of Square Root The umber b is a square root of a if b The priciple square root of a positive umber is its positive square root ad we deote this root by usig
More informationPROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY
MAT 0630 INTERNET RESOURCES, REVIEW OF CONCEPTS AND COMMON MISTAKES PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY Contents 1. ACT Compss Prctice Tests 1 2. Common Mistkes 2 3. Distributive
More informationMath 135 Circles and Completing the Square Examples
Mth 135 Circles nd Completing the Squre Exmples A perfect squre is number such tht = b 2 for some rel number b. Some exmples of perfect squres re 4 = 2 2, 16 = 4 2, 169 = 13 2. We wish to hve method for
More informationSoving Recurrence Relations
Sovig Recurrece Relatios Part 1. Homogeeous liear 2d degree relatios with costat coefficiets. Cosider the recurrece relatio ( ) T () + at ( 1) + bt ( 2) = 0 This is called a homogeeous liear 2d degree
More informationBINOMIAL EXPANSIONS 12.5. In this section. Some Examples. Obtaining the Coefficients
652 (1226) Chapter 12 Sequeces ad Series 12.5 BINOMIAL EXPANSIONS I this sectio Some Examples Otaiig the Coefficiets The Biomial Theorem I Chapter 5 you leared how to square a iomial. I this sectio you
More informationDEPARTMENT OF ACTUARIAL STUDIES RESEARCH PAPER SERIES
DEPARTMENT OF ACTUARIAL STUDIES RESEARCH PAPER SERIES The ultibioil odel d pplictios by Ti Kyg Reserch Pper No. 005/03 July 005 Divisio of Ecooic d Ficil Studies Mcqurie Uiversity Sydey NSW 09 Austrli
More informationTrigonometric Form of a Complex Number. The Complex Plane. axis. ( 2, 1) or 2 i FIGURE 6.44. The absolute value of the complex number z a bi is
0_0605.qxd /5/05 0:45 AM Page 470 470 Chapter 6 Additioal Topics i Trigoometry 6.5 Trigoometric Form of a Complex Number What you should lear Plot complex umbers i the complex plae ad fid absolute values
More informationDEFINITION OF INVERSE MATRIX
Lecture. Iverse matrix. To be read to the music of Back To You by Brya dams DEFINITION OF INVERSE TRIX Defiitio. Let is a square matrix. Some matrix B if it exists) is said to be iverse to if B B I where
More informationOperations with Polynomials
38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: Write polynomils in stndrd form nd identify the leding coefficients nd degrees of polynomils Add nd subtrct polynomils Multiply
More informationA. Description: A simple queueing system is shown in Fig. 161. Customers arrive randomly at an average rate of
Queueig Theory INTRODUCTION Queueig theory dels with the study of queues (witig lies). Queues boud i rcticl situtios. The erliest use of queueig theory ws i the desig of telehoe system. Alictios of queueig
More information4.11 Inner Product Spaces
314 CHAPTER 4 Vector Spces 9. A mtrix of the form 0 0 b c 0 d 0 0 e 0 f g 0 h 0 cnnot be invertible. 10. A mtrix of the form bc d e f ghi such tht e bd = 0 cnnot be invertible. 4.11 Inner Product Spces
More informationLecture 5. Inner Product
Lecture 5 Inner Product Let us strt with the following problem. Given point P R nd line L R, how cn we find the point on the line closest to P? Answer: Drw line segment from P meeting the line in right
More informationTHE ARITHMETIC OF INTEGERS.  multiplication, exponentiation, division, addition, and subtraction
THE ARITHMETIC OF INTEGERS  multiplicatio, expoetiatio, divisio, additio, ad subtractio What to do ad what ot to do. THE INTEGERS Recall that a iteger is oe of the whole umbers, which may be either positive,
More informationFactoring Polynomials
Fctoring Polynomils Some definitions (not necessrily ll for secondry school mthemtics): A polynomil is the sum of one or more terms, in which ech term consists of product of constnt nd one or more vribles
More informationModule 4: Mathematical Induction
Module 4: Mathematical Iductio Theme 1: Priciple of Mathematical Iductio Mathematical iductio is used to prove statemets about atural umbers. As studets may remember, we ca write such a statemet as a predicate
More information1. MATHEMATICAL INDUCTION
1. MATHEMATICAL INDUCTION EXAMPLE 1: Prove that for ay iteger 1. Proof: 1 + 2 + 3 +... + ( + 1 2 (1.1 STEP 1: For 1 (1.1 is true, sice 1 1(1 + 1. 2 STEP 2: Suppose (1.1 is true for some k 1, that is 1
More informationThe second difference is the sequence of differences of the first difference sequence, 2
Differece Equatios I differetial equatios, you look for a fuctio that satisfies ad equatio ivolvig derivatives. I differece equatios, istead of a fuctio of a cotiuous variable (such as time), we look for
More informationSection A4 Rational Expressions: Basic Operations
A Appendi A A BASIC ALGEBRA REVIEW 7. Construction. A rectngulr opentopped bo is to be constructed out of 9 by 6inch sheets of thin crdbord by cutting inch squres out of ech corner nd bending the
More informationApplication: Volume. 6.1 Overture. Cylinders
Applictio: Volume 61 Overture I this chpter we preset other pplictio of the defiite itegrl, this time to fid volumes of certi solids As importt s this prticulr pplictio is, more importt is to recogize
More informationCHAPTER10 WAVEFUNCTIONS, OBSERVABLES and OPERATORS
Lecture Notes PH 4/5 ECE 598 A. L Ros INTRODUCTION TO QUANTUM MECHANICS CHAPTER0 WAVEFUNCTIONS, OBSERVABLES d OPERATORS 0. Represettios i the sptil d mometum spces 0..A Represettio of the wvefuctio i
More informationAlgebra Review. How well do you remember your algebra?
Algebr Review How well do you remember your lgebr? 1 The Order of Opertions Wht do we men when we write + 4? If we multiply we get 6 nd dding 4 gives 10. But, if we dd + 4 = 7 first, then multiply by then
More informationMATH 181Exponents and Radicals ( 8 )
Mth 8 S. Numkr MATH 8Epots d Rdicls ( 8 ) Itgrl Epots & Frctiol Epots Epotil Fuctios Epotil Fuctios d Grphs I. Epotil Fuctios Th fuctio f ( ), whr is rl umr, 0, d, is clld th potil fuctio, s. Rquirig
More informationThe Field Q of Rational Numbers
Chapter 3 The Field Q of Ratioal Numbers I this chapter we are goig to costruct the ratioal umber from the itegers. Historically, the positive ratioal umbers came first: the Babyloias, Egyptias ad Grees
More informationSection 8.3 : De Moivre s Theorem and Applications
The Sectio 8 : De Moivre s Theorem ad Applicatios Let z 1 ad z be complex umbers, where z 1 = r 1, z = r, arg(z 1 ) = θ 1, arg(z ) = θ z 1 = r 1 (cos θ 1 + i si θ 1 ) z = r (cos θ + i si θ ) ad z 1 z =
More informationInfinite Sequences and Series
CHAPTER 4 Ifiite Sequeces ad Series 4.1. Sequeces A sequece is a ifiite ordered list of umbers, for example the sequece of odd positive itegers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29...
More informationUnit 6: Exponents and Radicals
Eponents nd Rdicls : The Rel Numer Sstem Unit : Eponents nd Rdicls Pure Mth 0 Notes Nturl Numers (N):  counting numers. {,,,,, } Whole Numers (W):  counting numers with 0. {0,,,,,, } Integers (I): 
More informationSOME IMPORTANT MATHEMATICAL FORMULAE
SOME IMPORTANT MATHEMATICAL FORMULAE Circle : Are = π r ; Circuferece = π r Squre : Are = ; Perieter = 4 Rectgle: Are = y ; Perieter = (+y) Trigle : Are = (bse)(height) ; Perieter = +b+c Are of equilterl
More informationSECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES
SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,
More informationSINCLAIR COMMUNITY COLLEGE DAYTON, OHIO DEPARTMENT SYLLABUS FOR COURSE IN MAT 1355  INTERMEDIATE ALGEBRA I (3 CREDIT HOURS)
SINCLAIR COMMUNITY COLLEGE DAYTON OHIO DEPARTMENT SYLLABUS FOR COURSE IN MAT 1355  INTERMEDIATE ALGEBRA I (3 CREDIT HOURS) 1. COURSE DESCRIPTION: Ftorig; opertios with polyoils d rtiol expressios; solvig
More informationThe Euler Totient, the Möbius and the Divisor Functions
The Euler Totiet, the Möbius ad the Divisor Fuctios Rosica Dieva July 29, 2005 Mout Holyoke College South Hadley, MA 01075 1 Ackowledgemets This work was supported by the Mout Holyoke College fellowship
More information.04. This means $1000 is multiplied by 1.02 five times, once for each of the remaining sixmonth
Questio 1: What is a ordiary auity? Let s look at a ordiary auity that is certai ad simple. By this, we mea a auity over a fixed term whose paymet period matches the iterest coversio period. Additioally,
More informationA Resource for Freestanding Mathematics Qualifications
A pie chrt shows how somethig is divided ito prts  it is good wy of showig the proportio (or frctio) of the dt tht is i ech ctegory. To drw pie chrt:. Fid the totl umer of items.. Fid how my degrees represet
More informationCOMPLEX FRACTIONS. section. Simplifying Complex Fractions
58 (66) Chpter 6 Rtionl Epressions undles tht they cn ttch while working together for 0 hours. 00 600 6 FIGURE FOR EXERCISE 9 95. Selling. George sells one gzine suscription every 0 inutes, wheres Theres
More informationGraphs on Logarithmic and Semilogarithmic Paper
0CH_PHClter_TMSETE_ 3//00 :3 PM Pge Grphs on Logrithmic nd Semilogrithmic Pper OBJECTIVES When ou hve completed this chpter, ou should be ble to: Mke grphs on logrithmic nd semilogrithmic pper. Grph empiricl
More informationIntegration by Substitution
Integrtion by Substitution Dr. Philippe B. Lvl Kennesw Stte University August, 8 Abstrct This hndout contins mteril on very importnt integrtion method clled integrtion by substitution. Substitution is
More informationFactors of sums of powers of binomial coefficients
ACTA ARITHMETICA LXXXVI.1 (1998) Factors of sums of powers of biomial coefficiets by Neil J. Cali (Clemso, S.C.) Dedicated to the memory of Paul Erdős 1. Itroductio. It is well ow that if ( ) a f,a = the
More informationBinary Representation of Numbers Autar Kaw
Binry Representtion of Numbers Autr Kw After reding this chpter, you should be ble to: 1. convert bse rel number to its binry representtion,. convert binry number to n equivlent bse number. In everydy
More informationLesson 17 Pearson s Correlation Coefficient
Outlie Measures of Relatioships Pearso s Correlatio Coefficiet (r) types of data scatter plots measure of directio measure of stregth Computatio covariatio of X ad Y uique variatio i X ad Y measurig
More informationLecture 3 Gaussian Probability Distribution
Lecture 3 Gussin Probbility Distribution Introduction l Gussin probbility distribution is perhps the most used distribution in ll of science. u lso clled bell shped curve or norml distribution l Unlike
More informationMODULE 3. 0, y = 0 for all y
Topics: Inner products MOULE 3 The inner product of two vectors: The inner product of two vectors x, y V, denoted by x, y is (in generl) complex vlued function which hs the following four properties: i)
More informationTHE REGRESSION MODEL IN MATRIX FORM. For simple linear regression, meaning one predictor, the model is. for i = 1, 2, 3,, n
We will cosider the liear regressio model i matrix form. For simple liear regressio, meaig oe predictor, the model is i = + x i + ε i for i =,,,, This model icludes the assumptio that the ε i s are a sample
More informationINVESTIGATION OF PARAMETERS OF ACCUMULATOR TRANSMISSION OF SELF MOVING MACHINE
ENGINEEING FO UL DEVELOENT Jelgv, 28.29.05.2009. INVESTIGTION OF ETES OF CCUULTO TNSISSION OF SELF OVING CHINE leksdrs Kirk Lithui Uiversity of griculture, Kus leksdrs.kirk@lzuu.lt.lt bstrct. Uder the
More informationPREMIUMS CALCULATION FOR LIFE INSURANCE
ls of the Uiversity of etroşi, Ecoomics, 2(3), 202, 97204 97 REIUS CLCULTIO FOR LIFE ISURCE RE, RI GÎRBCI * BSTRCT: The pper presets the techiques d the formuls used o itertiol prctice for estblishig
More informationMATHEMATICAL INDUCTION
MATHEMATICAL INDUCTION. Itroductio Mthemtics distiguishes itself from the other scieces i tht it is built upo set of xioms d defiitios, o which ll subsequet theorems rely. All theorems c be derived, or
More informationFUNCTIONS AND EQUATIONS. xεs. The simplest way to represent a set is by listing its members. We use the notation
FUNCTIONS AND EQUATIONS. SETS AND SUBSETS.. Definition of set. A set is ny collection of objects which re clled its elements. If x is n element of the set S, we sy tht x belongs to S nd write If y does
More informationx(x 1)(x 2)... (x k + 1) = [x] k n+m 1
1 Coutig mappigs For every real x ad positive iteger k, let [x] k deote the fallig factorial ad x(x 1)(x 2)... (x k + 1) ( ) x = [x] k k k!, ( ) k = 1. 0 I the sequel, X = {x 1,..., x m }, Y = {y 1,...,
More informationRIGHT TRIANGLES AND THE PYTHAGOREAN TRIPLETS
RIGHT TRIANGLES AND THE PYTHAGOREAN TRIPLETS Known for over 500 yers is the fct tht the sum of the squres of the legs of right tringle equls the squre of the hypotenuse. Tht is +b c. A simple proof is
More informationName: Period GL SSS~ Dates, assignments, and quizzes subject to change without advance notice. Monday Tuesday Block Day Friday
Ne: Period GL UNIT 5: SIMILRITY I c defie, idetify d illustrte te followig ters: Siilr Cross products Scle Fctor Siilr Polygos Siilrity Rtio Idirect esureet Rtio Siilrity Stteet ~ Proportio Geoetric Me
More informationLecture 4: Cheeger s Inequality
Spectral Graph Theory ad Applicatios WS 0/0 Lecture 4: Cheeger s Iequality Lecturer: Thomas Sauerwald & He Su Statemet of Cheeger s Iequality I this lecture we assume for simplicity that G is a dregular
More informationLecture 15  Curve Fitting Techniques
Lecture 15  Curve Fitting Techniques Topics curve fitting motivtion liner regression Curve fitting  motivtion For root finding, we used given function to identify where it crossed zero where does fx
More information3. Greatest Common Divisor  Least Common Multiple
3 Greatest Commo Divisor  Least Commo Multiple Defiitio 31: The greatest commo divisor of two atural umbers a ad b is the largest atural umber c which divides both a ad b We deote the greatest commo gcd
More informationThe Binomial Multi Section Transformer
4/15/21 The Bioial Multisectio Matchig Trasforer.doc 1/17 The Bioial Multi Sectio Trasforer Recall that a ultisectio atchig etwork ca be described usig the theory of sall reflectios as: where: Γ ( ω
More information6.2 Volumes of Revolution: The Disk Method
mth ppliction: volumes of revolution, prt ii Volumes of Revolution: The Disk Method One of the simplest pplictions of integrtion (Theorem ) nd the ccumultion process is to determine soclled volumes of
More informationFinite Automata. Informatics 2A: Lecture 3. John Longley. 25 September School of Informatics University of Edinburgh
Lnguges nd Automt Finite Automt Informtics 2A: Lecture 3 John Longley School of Informtics University of Edinburgh jrl@inf.ed.c.uk 25 September 2015 1 / 30 Lnguges nd Automt 1 Lnguges nd Automt Wht is
More informationRepeating Decimals are decimal numbers that have number(s) after the decimal point that repeat in a pattern.
5.5 Fractios ad Decimals Steps for Chagig a Fractio to a Decimal. Simplify the fractio, if possible. 2. Divide the umerator by the deomiator. d d Repeatig Decimals Repeatig Decimals are decimal umbers
More informationCS103X: Discrete Structures Homework 4 Solutions
CS103X: Discrete Structures Homewor 4 Solutios Due February 22, 2008 Exercise 1 10 poits. Silico Valley questios: a How may possible sixfigure salaries i whole dollar amouts are there that cotai at least
More informationLinear Equations in Two Variables
Liner Equtions in Two Vribles In this chpter, we ll use the geometry of lines to help us solve equtions. Liner equtions in two vribles If, b, ndr re rel numbers (nd if nd b re not both equl to 0) then
More informationCurve Sketching. 96 Chapter 5 Curve Sketching
96 Chpter 5 Curve Sketching 5 Curve Sketching A B A B A Figure 51 Some locl mximum points (A) nd minimum points (B) If (x, f(x)) is point where f(x) reches locl mximum or minimum, nd if the derivtive of
More information3 The Utility Maximization Problem
3 The Utility Mxiiztion Proble We hve now discussed how to describe preferences in ters of utility functions nd how to forulte siple budget sets. The rtionl choice ssuption, tht consuers pick the best
More informationMATHEMATICS SYLLABUS SECONDARY 7th YEAR
Europe Schools Office of the SecretryGeerl Pedgogicl developmet Uit Ref.: 201101D41e2 Orig.: DE MATHEMATICS SYLLABUS SECONDARY 7th YEAR Stdrd level 5 period/week course Approved y the Joit Techig
More informationAsymptotic Growth of Functions
CMPS Itroductio to Aalysis of Algorithms Fall 3 Asymptotic Growth of Fuctios We itroduce several types of asymptotic otatio which are used to compare the performace ad efficiecy of algorithms As we ll
More informationAppendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered:
Appendi D: Completing the Squre nd the Qudrtic Formul Fctoring qudrtic epressions such s: + 6 + 8 ws one of the topics introduced in Appendi C. Fctoring qudrtic epressions is useful skill tht cn help you
More informationPHY 140A: Solid State Physics. Solution to Homework #2
PHY 140A: Solid Stte Physics Solution to Homework # TA: Xun Ji 1 October 14, 006 1 Emil: jixun@physics.ucl.edu Problem #1 Prove tht the reciprocl lttice for the reciprocl lttice is the originl lttice.
More informationMathematics. Vectors. hsn.uk.net. Higher. Contents. Vectors 128 HSN23100
hsn.uk.net Higher Mthemtics UNIT 3 OUTCOME 1 Vectors Contents Vectors 18 1 Vectors nd Sclrs 18 Components 18 3 Mgnitude 130 4 Equl Vectors 131 5 Addition nd Subtrction of Vectors 13 6 Multipliction by
More informationExample A rectangular box without lid is to be made from a square cardboard of sides 18 cm by cutting equal squares from each corner and then folding
1 Exmple A rectngulr box without lid is to be mde from squre crdbord of sides 18 cm by cutting equl squres from ech corner nd then folding up the sides. 1 Exmple A rectngulr box without lid is to be mde
More informationReasoning to Solve Equations and Inequalities
Lesson4 Resoning to Solve Equtions nd Inequlities In erlier work in this unit, you modeled situtions with severl vriles nd equtions. For exmple, suppose you were given usiness plns for concert showing
More information{{1}, {2, 4}, {3}} {{1, 3, 4}, {2}} {{1}, {2}, {3, 4}} 5.4 Stirling Numbers
. Stirlig Numbers Whe coutig various types of fuctios from., we quicly discovered that eumeratig the umber of oto fuctios was a difficult problem. For a domai of five elemets ad a rage of four elemets,
More informationHomework 3 Solutions
CS 341: Foundtions of Computer Science II Prof. Mrvin Nkym Homework 3 Solutions 1. Give NFAs with the specified numer of sttes recognizing ech of the following lnguges. In ll cses, the lphet is Σ = {,1}.
More informationTHE RATIONAL NUMBERS CHAPTER
CHAPTER THE RATIONAL NUMBERS When divided by b is not n integer, the quotient is frction.the Bbylonins, who used number system bsed on 60, epressed the quotients: 0 8 s 0 60 insted of 8 s 7 60,600 0 insted
More informationOur aim is to show that under reasonable assumptions a given 2πperiodic function f can be represented as convergent series
8 Fourier Series Our aim is to show that uder reasoable assumptios a give periodic fuctio f ca be represeted as coverget series f(x) = a + (a cos x + b si x). (8.) By defiitio, the covergece of the series
More informationRegular Sets and Expressions
Regulr Sets nd Expressions Finite utomt re importnt in science, mthemtics, nd engineering. Engineers like them ecuse they re super models for circuits (And, since the dvent of VLSI systems sometimes finite
More informationChapter 13 Volumetric analysis (acid base titrations)
Chpter 1 Volumetric lysis (cid se titrtios) Ope the tp d ru out some of the liquid util the tp coectio is full of cid d o ir remis (ir ules would led to iccurte result s they will proly dislodge durig
More informationHelicopter Theme and Variations
Helicopter Theme nd Vritions Or, Some Experimentl Designs Employing Pper Helicopters Some possible explntory vribles re: Who drops the helicopter The length of the rotor bldes The height from which the
More information8.1 Arithmetic Sequences
MCR3U Uit 8: Sequeces & Series Page 1 of 1 8.1 Arithmetic Sequeces Defiitio: A sequece is a comma separated list of ordered terms that follow a patter. Examples: 1, 2, 3, 4, 5 : a sequece of the first
More informationWHEN IS THE (CO)SINE OF A RATIONAL ANGLE EQUAL TO A RATIONAL NUMBER?
WHEN IS THE (CO)SINE OF A RATIONAL ANGLE EQUAL TO A RATIONAL NUMBER? JÖRG JAHNEL 1. My Motivatio Some Sort of a Itroductio Last term I tought Topological Groups at the Göttige Georg August Uiversity. This
More information5.3. Generalized Permutations and Combinations
53 GENERALIZED PERMUTATIONS AND COMBINATIONS 73 53 Geeralized Permutatios ad Combiatios 53 Permutatios with Repeated Elemets Assume that we have a alphabet with letters ad we wat to write all possible
More informationHere are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed.
This documet was writte ad copyrighted by Paul Dawkis. Use of this documet ad its olie versio is govered by the Terms ad Coditios of Use located at http://tutorial.math.lamar.edu/terms.asp. The olie versio
More informationSEQUENCES AND SERIES
Chapter 9 SEQUENCES AND SERIES Natural umbers are the product of huma spirit. DEDEKIND 9.1 Itroductio I mathematics, the word, sequece is used i much the same way as it is i ordiary Eglish. Whe we say
More informationP.3 Polynomials and Factoring. P.3 an 1. Polynomial STUDY TIP. Example 1 Writing Polynomials in Standard Form. What you should learn
33337_0P03.qp 2/27/06 24 9:3 AM Chpter P Pge 24 Prerequisites P.3 Polynomils nd Fctoring Wht you should lern Polynomils An lgeric epression is collection of vriles nd rel numers. The most common type of
More information1 Numerical Solution to Quadratic Equations
cs42: introduction to numericl nlysis 09/4/0 Lecture 2: Introduction Prt II nd Solving Equtions Instructor: Professor Amos Ron Scribes: Yunpeng Li, Mrk Cowlishw Numericl Solution to Qudrtic Equtions Recll
More information