This enables us to also express rational numbers other than natural numbers, for example:

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1 Overview Study Mteril Business Mthemtis Alger The Rel Numers The si numers re,,3,4, these numers re nturl numers nd lso lled positive integers. The positive integers, together with the negtive integers 0,-,-,-3,, mke up the integers. The integers n e represented on numer line showed elow), where the rrow provides the diretion in whih the numers inrese. Rtionl numers re ll numers tht re written in the form /d where nd d re oth integers). They n lso e presented on numer line. An integer sy n) is lso rtionl numer, euse n = n/. There re lso Irrtionl Numers like 3, this is euse there re no integers p nd q suh tht 3 = p/q. The wy people write numers tody is lled the deiml system or se 0 system. This is positionl system with 0 s the se numer, where every nturl numer n e written y using only the symols:,,3,4 ) 9, whih re lled digits. An emple of how the deiml system works: This enles us to lso epress rtionl numers other thn nturl numers, for emple: 0 3 4, /0 7/0 3/0 5/0 4 Rtionl numers tht n e written etly, y using only finite numer of deiml ples, re lled finite deiml frtions. If this is not the se, we ll them infinite deiml frtions. An emple of n infinite deiml frtion is: 0/6 = 3,3333, where the three dots indite tht the digit 3 is repeted indefinitely. This is lso lled rel numer. When pplied to rel numer, the four rithmeti opertions ddition, sutrtion, multiplition nd division) lwys result in rel numer. The only eeption is tht they nnot devide y 0. Integer Powers When is ny numer nd n is ny nturl numer, the following pplies: n =... n ftors This epression is lled the nth power of : where is the se nd n the eponent. 0 A speil se is, where n e ny numer eept 0. Powers with negtive eponents n e defined s: n n / Twitter.om/SlimStuderen Feook.om/SlimStuderenVUBKIBAEBE

2 Overview Study Mteril Business Mthemtis Properties of Powers There re severl rules tht pply to Powers. The most importnt two re: k l k l ) kl The division of two powers with the sme se goes like: k / Finlly, note tht: k k k ) One of the most ommon errors ommitted is: k ) is NOT k + k in generl). Compound Interest Suppose ompny deposits n mount of K euros in nk ount pying p% of interest per yer. After t yers the mount of money in the ount will e: is the growth ftor for growth of p%. If you wnt to lulte the originl mount of money deposited in the nk, t yers go. Where the nk pys n interest rte of p% per yer, use the following formul, where K is the mount of money now nd X is the originl mount: This formul n lso e used for lulting the depreition of n sset. Suppose r worth K euros, with depreition rte of p%. After t yers, the vlue of the r will e: Here l / ) kl kl k k / p t K ) 00 p Where 00 p X K / ) 00 p t K ) 00 p 00 k t k k is lled the growth ftor for deline of p%. Rules of Alger If,, nd re ritrry numers, the following rules re the most importnt: ) ) 0 ) 0 ) ) These rules n e omined. voor 0 ) ) ) ) ) ) Twitter.om/SlimStuderen Feook.om/SlimStuderenVUBKIBAEBE

3 Overview Study Mteril Business Mthemtis Qudrti Identities There re 3 importnt qudrti identities, whih re very importnt to rememer: ) ) ) ) The lst one is lled differenes-of-squres-formul. This is the proof for the formul: ) ) Algeri Epressions The epression is n lgeri epression, euse it involves letters. We ll the terms in the epression. The numers 5, 3 nd 7 re the numeril oeffiients of the terms. Terms where only the numeril oeffiient is different, re lled terms of the sme type. For emple: z en 6 z Ftoring When we write 8 s 99, we hve ftored this numer. Algeri epressions n e ftored in similr wy: ) With Ftoring, we men: to epress it s produt of simpler ftors. However, most lgeri epressions n not e ftored. Frtions In the eqution, is the numertor nd d is the denomintor. The frtion is proper frtion euse the numertor is smller thn the denomintor. An improper frtion is: ,3 nd7 3 d frtion n e re-written s:, euse the numertor is lrger or equl to) the denomintor. This whih is lled mied numer. 4 7 Twitter.om/SlimStuderen Feook.om/SlimStuderenVUBKIBAEBE 3

4 Overview Study Mteril Business Mthemtis Twitter.om/SlimStuderen Feook.om/SlimStuderenVUBKIBAEBE 4 The most importnt properties of frtions re listed elow: Frtionl Powers An emple of power with frtionl eponent is, when nd = ½. We define s, the squre root of. is nonnegtive numer euse when multiplied y itself, the nswer gives. If nd re nonnegtive numers, then two rules pply: i) ii) For the seond rule, 0. In ddition: It is known tht nd hve the sme solution, whih is 4. When solving the eqution, the solution must e written like. Note however, tht 4 mens only, not -. Nth Roots The following rule pplies to : We n epnd this rule: Here must e positive >0), in ddition p must e n integer nd q must e nturl numer. 0, 0 ) ) ) ) ) ) d d d d d d d d 0 / 4 4 /n n n / p q p q q p ) ) / /

5 Overview Study Mteril Business Mthemtis Inequlities Properties: >0 nd >0, imply +>0 nd > 0; > mens tht ->0; mens tht - 0; If >, then + > + for ll. To del with more omplited inequlities involves using the following properties: > 0 nd > 0; + > 0; > 0 > then +> + > nd > > > nd > 0 > > nd < 0 < > nd > d +> + d Aording to these properties, the following rules pply: If the two sides of the inequlity re multiplied y positive numer, the diretion of the inequlity is preserved; If the two sides of n inequlity re multiplied y negtive numer, the diretion of the inequlity is reversed. Sign Digrms For the solution of n eqution, for emple, sign digrm might e useful. For n emple of sign digrm, review the ompulsory study mteril. Doule Inequlities An emple of doule inequlity is: d en d It is nturl to write: d Intervls nd Asolute Vlues Let y nd z e ny two numers on the rel line. All the numers tht lie etween y nd z, re lled n intervl. There re four different intervls tht ll hve nd s endpoints see the figure elow). Nottion Nme Intervl, ) Open intervl [, ],] [,) Closed intervl Hlf-open intervl Hlf-open intervl All four intervls hve the sme length ), only the endpoints differ. The open intervl for emple uses neither of the endpoints. All the intervls mentioned ove re ounded intervls. An emple of n unounded intervl is: [, ) = for ll vlues of, with 5) ) 0 Here, stnds for infinity. This symol is not numer t ll, the symol only indites tht we re onsidering the olletion of ll numers lrger thn, without ny upper ound on the size of the numer. Similrly, -,) hs no lower ound. When ll rel numers re permitted, we sy: -, ). Twitter.om/SlimStuderen Feook.om/SlimStuderenVUBKIBAEBE 5

6 Overview Study Mteril Business Mthemtis Asolute vlue The solute vlue of is denoted, nd: { } = if ³ 0 nd -if < 0 Suppose two numers, en. The distne etween nd on the numer line is - if nd -) if <. Therefore, we hve the distne etween nd on the numer line defined s: Asolute vlues n lso e prt of n inequlity: < mens- < < mens- Twitter.om/SlimStuderen Feook.om/SlimStuderenVUBKIBAEBE 6

7 Overview Study Mteril Business Mthemtis A. Equtions How to solve simple equtions To solve n eqution mens to find ll vlues of the vriles for whih the eqution is stisfied. If ny vlue of vrile mkes n epression in n eqution undefined, tht vlue is not llowed. An emple is: z/z-5), here z n not e 5 euse it mkes the epression undefined, euse it is 5/0. When two equtions hve etly the sme solution, we ll them equivlent. To get equivlent equtions, you hve to do the following on oth sides of the equlity sign =): Add or sutrt the sme numer; Multiply or divide y the sme numer 0). Equtions with Prmeters These equtions hve ommon struture, whih mkes it possile to write down generl eqution overing ll the speil ses: y Here nd re rel numers, lso lled prmeters. The vriles in the eqution re nd y. Consider the si mroeonomi model: Y C I C Y Where Y is the GDP Gross Domesti Produt), C is onsumption, nd is totl investment whih is treted s fied). Here, nd re the positive prmeters, with <. These equtions represent the struturl form of the model. Sustituting C = + Y + I gives us: Y Y I Net, rerrnge the eqution so tht ll the terms ontinting Y re on the left-hnd side: Y Y I The left-hnd side is equl to Y-), so devide oth sides y - so tht the oeffiient Y eomes. This gives us the nswer: I Y I This is one prt of the redued form, tht epresses endogenous vriles s funtions of eogenous vriles. Qudrti Equtions The generl qudrti eqution lso lled seond-degree eqution) hs the form: 0 0) Where is the unknown vrile, nd,, nd re given onstnts. Twitter.om/SlimStuderen Feook.om/SlimStuderenVUBKIBAEBE 7

8 Overview Study Mteril Business Mthemtis The Qudrti formul To solve qudrti eqution like: formul. If: Then: If nd only if:, it is possile to use the qudrti When < 0, the squre root of negtive numer ppers nd no rel solution eists. The solutions of the eqution re often lled the roots of the eqution. Beuse the Qudrti formul ontins ± sign, there re two solutions, nd, tht mke sure the qudrti eqution equls 0. This n e written s: ) ) 0 then: ) ) Liner Equtions in Two Unknowns This setion will review some methods for solving two liner equtions with two unknowns: We must find the vlues of nd y tht stisfy oth equtions. There re two methods for this: Method ) Solve one of the equtions for one of the vriles in terms of the other, then ) sustitute the result into the other eqution: ) 6y 36 4 ) The solutions should lwys e heked y diret sustitution. Method This method is sed on eliminting one of the vriles y dding or sutrting multiple of one eqution from the other. In the following eqution, we will eliminte. We multiply the seond eqution with -/3, then we will dd the trnsformed equtions. The term disppers nd we otin: y y 4 y 6 / / /3 4 / y 4 4 6y / 3y= 9 / 3 4 6y /3y 9 /3 0 /3y 45 /3 4 Hene, y = 4 whih gives us =3 through sustitution of 4 for y). To solve generl liner system, we use the following two equtions nd two unknowns: y Twitter.om/SlimStuderen Feook.om/SlimStuderenVUBKIBAEBE 8

9 Overview Study Mteril Business Mthemtis d ey f Here,,, d, e, nd f re ritrry given numers, wheres nd y re the unknowns. Using Method, we multiply the first eqution y e nd the seond y to otin the following, whih gives the vlue for : e f )/ e d) y f d)/ e d) y d ey By sustituting it k into or, we get y. f Nonliner Equtions An emple of nonliner eqution is: in the following wy: People tend to lulte the solution However, this is n esy wy to mke serious mistke, euse here ftor is nelled whih might e zero =0 is solution too!). A sfer method is =0 is lso solution. In generl: is equivlent to =0 of =. 6) 0. When the eqution is in this form, it is esier to see tht Twitter.om/SlimStuderen Feook.om/SlimStuderenVUBKIBAEBE 9

10 Overview Study Mteril Business Mthemtis B. Summtion The pitl Greek letter is used s summtion symol, nd the sum is written s: n N i i This reds the sum from i= to n of N N... N n N i. If there re n regions, then: The symol i is lled the inde of summtion. It is dummy vrile tht n e repled y ny other letter whih hs not lredy een used for something else). The lower nd upper limits of the summtion n oth vry, for emple: 5 i i en r iq i The solution to the first summtion: 5 i i In order to summrize the overll effet of prie hnges, numer of prie indies hve een suggested. A prie inde for yer t, in whih yer 0 s the se yer, is defined s: n i n i p p i) t i) 0 q q i) i) 00 Here qi) is the numer of goods in the sket), p0i) is the prie per unit of good i in yer 0, nd pti) is the prie per unit of good i in yer t. In the se where the quntities lled: the Lspeyres prie inde. But if the quntities in yer t, this is lled the Pshe prie inde. re levels of onsumption in the se yer 0, this is re levels of onsumption A few spets of logi Propositions re ssertions tht re either true or flse. This is n emple of mthemtil proposition:. i) q Implitions Suppose P nd Q re two propositions, suh tht whenever P is true, then Q is neessrily true. If this is the se, we usully write: A B 4 0 The rrow is n implition rrow nd it points in the diretion of the logil implition. i) q Twitter.om/SlimStuderen Feook.om/SlimStuderenVUBKIBAEBE 0

11 Overview Study Mteril Business Mthemtis In some ses, where the previous implition is vlid, it my lso e possile to drw logil onlusion in the other diretion: A B In suh ses, we n write oth implitions together in single logil equivlene, s shown elow: AÛ B Û The symol is lled n equivlene rrow. Neessry nd Suffiient Conditions There re other wys of epressing tht proposition P implies proposition Q, or tht P is equivlent to Q. To emphsize this point, onsider the following two propositions: Brething is neessry ondition for person to e helthy; Brething is suffiient ondition for person to e helthy. The first proposition is oviously true. The seond proposition is flse, euse sik person is still rething. Twitter.om/SlimStuderen Feook.om/SlimStuderenVUBKIBAEBE

12 Overview Study Mteril Business Mthemtis C. Funtions of one vrile One vrile is funtion of nother if the first vrile depends upon the seond, for instne. Here, when is given, y n e determined, y is funtion of. In this sitution n e ny numer. However, mostly there re some restritions like: only the vlues to 0 re relevnt. Sometimes grph is preferle to formul, for etter overview of the sitution. Bsi Definitions A funtion of rel vrile with domin is rule tht ssigns unique rel numer to eh rel numer in D. As vries over the whole domin D) the set of ll possile vlues f is lled the rnge of. Funtions re given letter nmes like f, z, g, F, or φ. y If f is funtion, we often let y denote the vlue of f t, so: y f We ll the independent vrile or the rgument of f, wheres y is the dependent vrile, euse the vlue y depends on the vlue of in generl). The domin of f is then the set of ll possile vlues of the independent vrile, wheres the rnge is the set of orresponding vlues of the dependent vrile. In eonomis, is lled the eogenous vrile, whih is supposed to e fied OUTSIDE the eonomi model, wheres for eh given the eqution y = f serves to determined the endogenous vrile y INSIDE the eonomi model. Funtionl Nottion Consider the following funtion: f Now suppose ll vlues of inrese with. A ommon mistke is: f ) The right nswer is: f ) ) Sustituting for in the formul of f gives: f ) R f D f Wheres if + ): f ) f ) ) ) ) Domin nd Rnge The definition of funtion is not relly omplete unless its domin is either speified epliitly or ovious. So: if funtion is defined using n lgeri formul, the domin onsists of ll vlues of the independent vrile for whih the formul gives unique vlue unless nother domin is epliitly mentioned). Twitter.om/SlimStuderen Feook.om/SlimStuderenVUBKIBAEBE

13 Overview Study Mteril Business Mthemtis For emple, we re going to find the domin of: 3 f For =, the formul redues to meningless epression 6/0). For ll other vlues of, the formul mkes f well-defined numer. Thus, the domin onsists of ll numers eept =. Another emple, find the domin: 6 The funtion is uniquely defined for ll suh tht 6 + is nonnegtive. Thus, the domin of the funtion n e written s the intervl: [-6, ). The rnge of the funtion n lso e lulted. The rnge of f is ll the numers we get s output, using ll numers in the domin s imputs. The rnge of the funtion is in this se[0, ). A funtion f is lled inresing if implies. A funtion is stritly inresing if implies. Deresing nd stritly deresing re defined in the opposite wy. f ) f f ) ) f Grphs of Funtions A retngulr or Crtesin oordinte system is otined y first drwing two perpendiulr lines, lled oordinte es. The two es re respetively the y-is the vertil is) nd the -is the horizontl is). The intersetion point O is lled the orgin. This retngulr oordinte system is lso lled the y-plne. We mesure the rel numers long of eh line, lthough the unit distne on the -is is not neessrily the sme s on the y-is. Any point P in the plne n e represented y unique pir of rel numers,d). The point represented y, ) lies t the intersetion of = nd y=d. We ll, d) the oordintes of P,, d) is lso lled n ordered pir euse the order of the two numers in the pir is importnt. For instne, 4, 5) nd 5, 4) represent two ompletely different points. Eh funtion of one vrile n e represented y grph in suh retngulr oordinte system. The grph of funtion f is simply the visul representtion of the set of ll points, f ), where elongs to the domin of f. Liner Funtions Liner funtions our lot in eonomis, nd they re defined s: y where nd re onstnts. The grph of this eqution is stright line. Suppose inreses with, then: f ) f ) This shows tht mesures the hnge in the vlue of the funtion when inreses y unit, for this reson is the slope of the line or the funtion. Twitter.om/SlimStuderen Feook.om/SlimStuderenVUBKIBAEBE 3

14 Overview Study Mteril Business Mthemtis If the slope of is positive, the line goes upwrd to the right, nd the lrger the vlue of, the steeper is the line. On the other hnd, if the vlue of is negtive, then the line slnts downwrd to the right, nd gin the solute vlue of mesures the steepness of the line. For emple, when is -6 the steepness is 6. In the speil se, when the slope) equls 0, the steepness is zero euse the line is horizontl. Algerilly, we hve y = + = for ll. Here, is lled the y- interept. The Point-Slope nd Point-Point Formuls Suppose point S, y ). Let us find the stright line eqution pssing through point P with slope. If formul: y y ) ) T, y ) where is ny other point on the line, then the slope is given y the Grphil solutions of Liner Equtions Erlier, we hve delt with lgeri methods for solving system of two liner equtions in two unknowns. The equtions re liner, so nturlly their grphs re stright lines. The oordintes of ny point on the line stisfy the eqution of tht line. Thus, the oordintes of ny point of the intersetion of these two lines will stisfy oth equtions. This mens tht ny point where these lines interset will stisfy the eqution system. Liner inequlities Suppose the liner inequlity 4 y 8. The inequlity n e written s y 8-4. The set of points, y) tht stisfy the inequlity y 8-4 must hve y-vlues elow those points on the line y = , euse y = is stright line. Liner Models In Keynesin mroeonomi theory, totl onsumption ependiture on goods nd servies) is C. C is ssumed to e funtion of ntionl inome Y: C f Y) C Y This is lled the Consumption Funtion. The slope of is lled the mrginl propensity to onsume. This tells us y how mny onsumption hnges if ntionl inome inreses with. C, Y, nd n lso e mesured in illions of dollrs for emple. The point where demnd is equl to supply, represents n equilirium. The prie P* t whih this ours is the equilirium prie nd the orresponding quntity Q* is the equilirium quntity. Consider the following simple emple of demnd nd supply funtions: D 00 P S 0 4P Equilirium ours t: P*= 36 nd Q*= 64. These re the generl liner demnd nd supply shedules: D P S P Here,,, nd re the positive prmeters of the funtions. Twitter.om/SlimStuderen Feook.om/SlimStuderenVUBKIBAEBE 4

15 Overview Study Mteril Business Mthemtis The formul for the equilirium Prie is: P* The formul for the equilirium Quntity is: Q* Qudrti Funtions The generl qudrti funtion is: f Where,, nd re onstnts nd 0. In generl, the grph of is lled prol. The shpe of prol roughly resemles n upside down U when < 0, nd U when > 0. Prols re symmetri out the is of symmetry, for emples review the ompulsory literture. To lulte for whih vlue of, f is equl to 0, you n use the qudrti formul. Suppose 0 nd, Where, 4 f 4 0 then: stnds for the two possile vlues where y=0. Furthermore: ) To find the mimum/minimum of the funtion use the following formuls: When >0, f hs minimum of t. When <0, f hs 4 mimum of t. Polynomils Polynomils or ui funtions hve generl form: f 3 d Here,,, nd d re onstnts nd 0. On the right is n emple of ui funtion. 4 Generl Polynomils Cui, qudrti nd liner funtions re ll emples of polynomils. The funtion P defined for ll y: Twitter.om/SlimStuderen Feook.om/SlimStuderenVUBKIBAEBE 5

16 Overview Study Mteril Business Mthemtis n P n n n... Here, ll s re onstnts nd 0. n 0 This funtion is lled the generl polynomil of degree n with oeffiients,..., 0, n. Numerous prolems in mthemtis nd its pplitions involve polynomils. Often one is very interested in finding the numer of zeros in P. This mens, the vlues of suh tht P = 0. This eqution: is lled the generl eqution of degree n. This eqution hs t most n rel solutions or roots, ut it doesn't need to hve ny. Aording to the fundmentl theorem of lger, every polynomil of the form n n P n n... 0 n e written s produt of polynomils of degree or. For emple, we n re-write s. Ftoring Polynomils Let P nd Q e two polynomils for whih. Then there lwys eist unique polynomils q nd r suh tht: where the degree of r is less thn the degree of Q. This ft P q Q r is lled the reminder theorem. 3 ) ) P Q Q 0 P q Q r When is suh tht. Then n e written in the form: P r q Q Q An importnt nd speil se is when Q=-. Then Q is of the degree, so the reminder r must hve degree 0, nd is therefore onstnt. In this speil se, for ll : P q ) r In prtiulr, when =, P)=r. Hene, devides P if nd only if P) = 0.This n e formulted s: P ) 0 Polynomil division You n devide polynomils in muh the sme wy s you use long division to devide numers. It is possile tht devision gives solution ut leves reminder. For n emple onsider the ompulsory literture. Rtionl funtions A rtionl funtion is funtion: tht n e epressed s the rtio R P / Q of two polynomils P nd Q. This funtion is defined for ll where Q 0. The rtio is proper if the degree of P is less thn the degree of Q. When the degree of P is more thn the degree of Q, it is lled n improper rtionl funtion. Twitter.om/SlimStuderen Feook.om/SlimStuderenVUBKIBAEBE 6

17 Overview Study Mteril Business Mthemtis Power Funtions The generl power funtion f is defined y the formul: f A r Where >0 nd r nd A re onstnts). Grphs of Power Funtions vries nd the eponent is onstnt. The shpe of the grph depends ruilly on the vlue of r. See the emple elow: Eponentil Funtions A quntity tht inreses/dereses y fied ftor per unit of time is sid to inrese/derese eponentilly. If the ftor is, this leds to the following eponentil funtion: f t) A t A nd re positive onstnts) Bewre of the fundmentl differene etween: f en g. The first one is n eponentil funtion where the eponent vries. The seond one is n emple of power funtion where the se The douling time is the time it tkes for the result of n eponentil funtion to doule. Suppose the funtion: f t) A The vlue t t=0 is A. The douling time t* is given y the eqution fter nelling A; f t) t A t t*. Thus, the douling time of the eponentil funtion is the power to whih must e rised in order to get. f t*) A t * A or The Nturl Eponentil Funtion The most importnt se for n eponentil funtion is denoted y the letter e =, ). Given the se e the nturl eponentil funtion is: f e Logrithmi Funtions Like we sid efore, the douling time of n eponentil funtion defined s the time it tkes for ft) to eome twie s lrge: t* f t*) A t* ws To solve this, we use the nturl logrithm. Therefore, we introdue the following useful definition: If, we ll u the nturl logrithm of, nd we write u=ln. Hene, we hve the following definition of the symol ln : ln e e u Here, n e ll positive vlues. When, then is: ln 7. e 7 Twitter.om/SlimStuderen Feook.om/SlimStuderenVUBKIBAEBE 7

18 Overview Study Mteril Business Mthemtis There re ouple of rules for the nturl logrithmi funtion ln: nd y re positive); ln y) ln ln y ln ) ln ln y y ln p ) pln ln 0, ln e, e nd y re positive); is positive); ; There re no simple formuls for en. The Funtion g=ln For eh positive numer, the numer ln is defined y:. So, the u =ln is the solution to the eqution e^u=. We ll the resulting funtion the nturl logrithm of : >0) g ln The grph of the funtion n e found in the ompulsory study mteril. Logrithms with Bses other thn e Suppose >). If u log where, then we ll u the logrithm of to se, nd we write: is lso true. log By tking the ln on eh side of, we otin, so tht: log ln ln Log oeys the sme rules s ln: ; log ; ;. u log ln, ln e log y) log log y log ) log log y y log 0, log p ) p log ln y) ln y) ln e Twitter.om/SlimStuderen Feook.om/SlimStuderenVUBKIBAEBE 8

1 PYTHAGORAS THEOREM 1. Given a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

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