a Lesson from the forthcoming every. ) textbook Mathematics: Building 3. a, b, and c, a + (b + c) = (a + b) + c.

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1 There are eleve basic rules that gover all work i elemetary algebra. Five of these rules relate to additio, five relate to multiplicatio, ad the fial rule coects additio to multiplicatio. These rules ecapsulate all of the basic properties of real umbers. Because the structure of the set of real umbers reflects all of these rules, mathematicias call the real umbers a field. Here they are i symbolic form:. a, b a + b. ( meas implies. ) Note: This essay is extracted from. a, b, a + b = b + a. ( meas for all or for a Lesso from the forthcomig every. ) textbook Mathematics: Buildig. a, b, ad c, a + (b + c) = (a + b) + c. o Foudatios. 4. a, a elemet i called a = a. Remember, meas such that ad meas there exists. 5. For each pair a ad b, there is exactly oe x a + x = b. 6. a, b ab. 7. a, b, ab = ba. 8. a, b, ad c, a(bc) = (ab)c. 9. a, a elemet i called a = a. 0. For each pair a ad b (where a 0), there is exactly oe x ax = b.. a, b, ad c, a(b + c) = ab + ac. The closure property for real umbers uder additio ad multiplicatio coect to their respective iverses. The iverse operatio of additio is subtractio ad the iverse operatio of multiplicatio is divisio. Oe iverse operatio of expoetiatio is extractio of roots. For a review, the followig table illustrates these operatios ad their iverses. Operatio Iverse Additio (7 + = 0) Subtractio (0 = 7) Multiplicatio (6 5 = 0) Divisio (0/5 = 6) Expoetiatio ( = 4) Extractio of roots ( 4= ) The iverse of raisig ay umber x to the secod power, i.e., x = a, is called extractig the square (from ) root, i.e., a x. The iverse of raisig ay umber x to the third power, i.e., x = a, is called extractig the cube (from ) root, i.e., a x. The iverse of raisig ay umber to the fourth power, i.e., x 4 = a, is called extractio the fourth root i.e., 4 a x. I symbols, these operatios look as follows: Raisig to the th power Extractig the th root = 4 4 = = = b b ifiseve There is actually a distictio betwee what mathematicias call a field ad a ordered field, but we will ot embrace such miutiae i this essay.

2 Raisig to the th power Extractig the th root b ifisodd Before we cotiue, ote the odd/eve priciple. Whe you are extractig the th root of a positive umber ad is eve, you will always get two aswers, oe positive ad the other egative. This is because a egative umber times a egative umber equals a positive umber; i.e., (-)(-) = 4. Note also that (-)(-)(-)(-) = 6. If you are multiplyig a eve umber of egative umbers, your aswer will also be positive. If you are multiplyig a odd umber of egative umbers, the aswer will always be egative. For example, (-)(-)(-) = -8 but ()()() = 8. Therefore, 8,but 8. Before the advet of the electroic had held calculator (i the late 960s), all arithmetical operatios had to be worked out by had. By the early 7 th cetury, great advaces had bee made both i astroomy ad i explorig the world through sea voyages. Both these advaces ecessitated performig arithmetical calculatios usig large umbers. A Scottish mathematicia, Joh Napier (550-67) oted these difficulties ad echoed the bae of may studets of arithmetic, Seeig there is othig that is so troublesome to mathematical practice, or that doth more molest ad hider calculators, tha the multiplicatio, divisios, square ad cubical extractios of great umbers. I bega therefore to cosider Joh Napier (Public i my mid by what certai ad ready art I might remove those hidraces. Domai) Napier removed these difficulties by developig a system of arithmetic that replaced multiplicatio by additio ad divisio by subtractio. Cosider the table below (powers of ): Napier first oted that if he multiplied ay umber i row by ay other umber i row, his aswer was a umber i row. Secod, he oted that whe you multiplied ay two umbers i row, the the aswer correlated to a additio problem i row (he made use of the sum law of expoets). Iversely, if you divided ay two umbers i row, the the aswer correlated to a subtractio problem i row (the iverse of the sum law of expoets). Row Row 4 = 8 + = (read 8 i row ) 8 = 6 + = 4 (read 6 i row ) 8 6 = = 7 (read 8 i row ) 6 64 = = 0 (read,04 i row ) = 8 4 = 7 (read 8 i row ) 64 8 = 8 6 = (read 8 i row ) By this method, Napier arrived at aother iverse of expoetiatio. Give x =, recall that we deote as the base ad x as the expoet. I expoetiatio, we are give the base ad the expoet. From this, we Joh Napier, Mirifici logarithmorum caois descriptio (64). Cited i George A. Gibso, Napier ad the Ivetio of Logarithms, i Hadbook of the Napier Terceteary Celebratio, or Moder Istrumets ad Methods of Calculatios, ed. E. M. Horsburgh (Los Ageles: Tomash Publishers, [94] 98), p. 9.

3 determie that result; i.e.,. For example, 4 = 6. I Napier s iverse, we are give the result (i.e., ) ad the base i.e., ). From this, we determie the expoet (i.e., x). He called this process fidig the logarithm. For example, if x = 6, the x (the logarithm of 6) = 4. The logarithmic table looks as follows: Number Logarithm This table has oe serious limitatio. What happes if we wat to multiply by? What is the logarithm of? What is the logarithm of? From the table, we ote that the logarithm of must be betwee ad ad the logarithm of must be betwee 4 ad 5. To fid the logarithm of, we must determie x such that x =. We kow for certai that x will ot be a positive iteger; it will be a fractio or its decimal equivalet. Fractios do cause some icoveiece i calculatios. Our table does ot yet cotai ay fractios (the smallest is 0 = ). We must ow itroduce powers of that are less tha if we wat to express fractios i our table. Hece, we must cosider -, -, -, etc. Whe we raise ay umber to a egative expoet, it meas you take its reciprocal. I symbols, a. Hece, 0.5, 0.5, ad a We ca add these umbers to our table: 8 Number Logarithm Logarithm is derived from two Greek words, logos (reaso or proportio) ad arithmos (umber), ad it literally meas proportio umber. Although, we itroduced expoets (Lesso.5) before logarithms, accordig to mathematics historia Howard Eves (9-004), Oe of the aomalies i the history of mathematics is the fact that logarithms were discovered before expoets were i use. See Howard Eves, A Itroductio to the History of Mathematics (New York: Holt, Rhiehart ad Wisto, [95, 964, 969] 976, p. 50.

4 This helps us somewhat with fidig the logarithm of umbers less tha. But, there are still gaps i the table. How do we fid the logarithm of ; i.e. determie x such that x =? As oted before, < x < (that is, x must be betwee ad ). x, the logarithm of, must be a fractioal expoet. We must be able to determie x where x = a b. Let s cosider how to calculate x whe x = square this umber ad apply the product law of expoets. 6 x 8.. We ca rewrite Sice 8,the as. We ow cosider. Let s Note that 8 is a irratioal umber ad that we rouded it off to the earest teth. Sice.5, the we ca add aother etry to our table: Number Logarithm We have t quite got the logarithm of, but we are gettig close. It ca be show that the solutio to x = is a irratioal umber (x caot be writte as a ratio of two itegers) ad that we ca approximate x to ay desired precisio usig fractioal powers. The process that I have led you through is exactly the pathway that Napier took i costructig his logarithm tables (it took him twety years to do it i the pre-computer ad pre-calculator days). I our example, we are costructig logarithms to the base. This is sigified as follows: log 8 = or, i geeral, log b x = y. The iverse operatio is sigified as = 8 or, i geeral, b y = x. Hece we have this equivalecy: y b x logbx y With this equivalecy, we ca techically ote the two iverses of expoetiatio: Expoetiatio Iverse b x = y Computig logarithms x logby x b = y Extractio of roots x = b y = y b The miraculous powers of moder calculatio are due to three ivetios: the Arabic Notatio, Decimal Fractios, ad Logarithms. Floria Cajori, A History of Mathematics (89). The list below eumerates all the essetial properties of logarithms (for ow, just take ote of the third ad fourth property; we shall prove them as a exercise):. Additio property: log b (xy) = log b x + log b y x. Subtractio property: logb = log b x log b y y

5 . Power property: log b x = log b x 4. Root extractio property: logb x logbx Util the advet of the had held calculator, most high school ad college textbooks cotaied logarithmic tables as appedices. The base commoly used was 0 ad it was sigified without a subscript. It was called the commo logarithm. 4 If you have a scietific calculator, the commo logarithm key is sigified by log. For example, log 00 =. Try it out with your calculator. The iverse operatio is 0 = 00. Here is a table of logarithms to the base 0: Number Logarithm to the base ,000 4 Note the much loger gaps i the table betwee the umbers. log = 0 ad log 0 =. I cotrast, for logarithms to the base, log = 0 ad log =. It will take more effort to fill i the gaps for logarithms to the base 0. Note aother key ext to the log key o your scietific calculator. It is sigified by l. This logarithm is called the atural logarithm. The base of the atural logarithms is a very famous ad sigificat umber that Leohard Euler deoted by the letter e. 5 e is a irratioal umber ad it is approximately equal to Why would this umber be used as a base? Joh Napier came very close to discoverig this umber. As far as historias of mathematics ca ascertai, e first appeared i history i the cotext of fiacial calculatios. Deposit that thought i your metal bak for a momet. We shall withdraw it after we explore the costructio of atural logarithms. We oted that 0 is ot a very good base of logarithms because of the large gaps. is a better base but it still has may gaps. We ca try as the base but that really does o good sice all powers of are. It is ot good to try a base less tha because a fractio less tha raised to a power gets smaller ad smaller. Take ote of successive powers of i the table below: 4 The Eglish mathematicia Hery Briggs (56-60) developed the first base 0 logarithmic tables. Commo logarithms are sometimes called Briggsia logarithms i his hoor. 5 Euler may have chose this letter to stad for expoetial. Sice the letters a, b, c, ad d were already frequetly used i mathematics, Euler may have just choose the ext available letter i the alphabet to represet this uusual umber. It is ulikely that Euler choose this letter because of his ame. As a Christia (his father was a Calviist pastor), Euler uderstood ad practiced the Christia grace of humility. 5

6 So, let s try a umber betwee ad as a base, say.. Note the successive powers of. i the table below. There is a coectio to Pascal s triagle here. Do you see it? Pascal s Triagle Note immediately that these umbers grow slowly ad that there are may umbers betwee ad before we try to calculate those troublesome gaps. It seems that choosig a smaller umber would be a eve better choice. Let s try.00. I this case, we separate the terms i Pascal s triagle by pairs of zeroes (ote how the terms fit eatly ito the decimal expasio):

7 At first we may woder if we will ever reach. Evetually we will, yet very slowly. Mathematicias have prove that the powers of ay umber greater tha (eve slightly greater tha ) will coverge to ifiity. The closer the umber is to, the slower the rate of growth. Although this table cotais a excellet desity of umbers, the drawback is its slow rate of growth. Very large expoets are eeded to produce small umbers. For example, Take ote of this umber (Hit: compare it with the umerical value of e as we defied it earlier i this essay). So, if the expoets were,000 times as big, we could make the table more proportioate. So, let s cosider as a base. Now, we are goig to raise this base to a fractioal power as follows ad make use of the product law of expoets: etc. Note that we are successively raisig to a fractioal power i steps of fractios to decimals, we get: etc Note also that we get the same results as before. Ispect the table below: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Covertig these 000 Note the patter that has developed. The expoets ad the correspodig results do ot grow disproportioately. Yet, the desity is uimpaired. Therefore, the base of is a excellet choice. Ca we im-

8 prove o this choice? Yes. Cosider the base of Why stop there? What about the base of ? We could go o ad o ad ifiitum with this type of procedure util we coverge to the atural or best umber. What is this umber? It is e, the base of the atural logarithms. Note the table below: e Let s withdraw our previously deposited fiacial thought from the bak. What does e have to do with moey? Cetral to the cocept of moey is the cocept of iterest. Iterest is what baks charge you for borrowig moey from them. If you do ot have the moey to buy a item ad the bak does, the the bak loas you the purchasig power to buy the item (this is called the pricipal). I retur for this loa, the bak expects a repaymet of this pricipal over time. A extra amout is charged to you for the use of this pricipal. This amout is called iterest. It is a way to pay the bak back for its loss of purchasig power due to its loa to you. The Bible presupposes time ad iterest i its teachig about restitutio (see Exodus :-5; Leviticus 6:- 7; Numbers 5:5-8; Luke 9:-9). Restitutio meas to make ameds or to recompese a ijury. The Bible teaches two forms of restitutio (betwee ma ad God ad betwee ma ad ma). Ma s si agaist God is atoed for by the blood of Jesus Christ (ma is made right with God through faith i the perso ad work of Jesus Christ). Ma s si agaist ma is made right via restitutio. I the case of theft, the Bible requires paymet of iterest for the stole goods (cf. Leviticus 6:4-5; Exodus :). This iterest paymet recogizes that whe a ma steals aother ma s goods, he is stealig aother ma s capital. By robbig ma of his capital, a thief is deprivig ma of ay icrease o that capital durig the time that it is i the thief s possessio. Not oly must the property be restored by a thief, a thief must also restore its value over time. I the same sese (mius the theft cocept), this is what baks do whe they charge iterest for moey borrowed. There are two types of iterest, simple ad compoud. Both are govered by mathematical formulas. The simple iterest formula for derivig how much you must pay a bak back (called the maturity value) for a loa is: S = P( + rt) where S = maturity value, P = the pricipal (amout of the loa), t = iterest rate per time period, ad t = legth of time you take to repay the pricipal. Note that r ad t must be i the same time uits (e.g., iterest per year with a repaymet schedule of x years). Suppose you borrow $000 at % per year iterest for years. What must you pay back? S = 000[ + 0.()] S = 000 (.6) = 60 The maturity value is $60. Normally, with a bak, you pay the loa back o a mothly basis. Your paymets would be $7.78 per moth. I three years you would pay back the $000 plus a iterest charge of $60 for usig the moey. 8

9 Suppose that the iterest rate is % 0.0 per moth ad your paymet schedule is 6 moths 00 ( years). What is S, the maturity value the? S = 000[ + 0.0(6)] S = 000(.6) = 60 The maturity value would be the same. Compoud iterest is govered by this formula: S = P( + r) t where, as before, S = maturity value, P = the pricipal, r = iterest rate per time period, ad t = legth of time you take to repay the pricipal. Let s ow cosider the same situatio: a loa of $000 at % iterest per year with a repaymet schedule of years. What is S? S = 000( + 0.) S = 000 (.) = Note the icrease i the maturity value from $60 calculated at simple iterest to $404.9 calculated at compoud iterest ($9.0 per moth repaymet). What is the icrease? $44.9. Let s ow icrease the loa repaymet to 5 years. What happes the? S, uder simple iterest is: S = 000 [ + 0.(5)] S = 000 (.8) = It pays to pay off a loa early! Although your mothly paymets are oly $5.56 a moth 80, you are payig $800 iterest o a $000 loa. No woder why, i Scripture, a loa for emergecy eeds was limited to 6 years (see Deuteroomy 5:-6). The situatio is much worse with compoud iterest. S = 000( + 0.) 5 S = 000(.) 5 = I this case your mothly paymets are $0.4 a moth. But, you are payig $ iterest 80 o a $000 loa. Ouch! The lesso to be leared about borrowig moey: the higher the iterest rate ad the loger the repaymet schedule, the more you pay i iterest. Some baks compute compouded iterest ot oce but several times a year. If a aual iterest rate of % is compouded semiaually, the bak will use oe-half of the aual iterest rate as the rate per period. Hece, i oe year a pricipal of $000 will be compouded twice, each time at a rate of 6%. This will amout to or $.60, about $.60 cets more tha the same pricipal would yield if compouded aually at %. The bakig idustry uses all kids of compoudig periods from aual to semiaual to quarterly to mothly to weekly to eve daily. Suppose the compoudig is doe times a year. For each period, the bak uses the aual iterest divided by ; i.e., r. Sice i t years there are t periods, the formula for compoud iterest becomes: 9

10 t r SP If = (compouded aually), the formula returs to its origial state, S = P ( + r) t. Let s compare what happes to a $000 loa at % compouded for differet periods. Period t r S Aually 0. $0.00 Semiaually 0.06 $.60 Quarterly $5.5 Mothly 0.0 $6.8 Weekly $7.4 Daily $7.49 $7.49 is the extra that a bak receives from compoudig daily as agaist compoudig aually. It hardly makes a differece how the iterest is compouded. I this case, the differece lies i the pricipal. The larger the pricipal, the larger the iterest charged o a daily basis. Try the formula for the above periods with a pricipal of $,000,000 ad ote what happes. Note that the reverse of borrowig moey is ivestig moey. Ivestig moey at compoud iterest geerates, over the log term, sigificat earigs. What bites you oe way (borrowig) rewards you the other way (ivestig). Let s ow cosider a hypothetical case. Let s assume that the bak loas moey at a iterest rate of 00%. Of course, o oe i his or her right mid would take out a loa at this rate. Ad, if you could fid a ivestmet vehicle that returs 00% you would probably jump o it. Let s cosider what happes i this case. Istead of a pricipal of $000, let P = ad t =. Our equatio ow becomes: Let s see what happes as we vary. S , , , ,000, ,000, SP 0

11 Lo ad behold, the maturity value approaches e, the base of the atural logarithms. Mathematicias have sigified this relatioship as follows: lim e.788 Usig rhetorical algebra, these symbols are read as follows: The limit as approaches ifiity (or icreases without boud) of the sequece of umbers defied by coverges to e. This is almost too icredible to comprehed. Yet, it is aother woder of mathematics. The marvelous coectio betwee the base of the atural logarithms ad the calculatio of compoud iterest at 00% is too amazig to be just a coicidece. Is it ot oe of the astoudig particular details i the Creator s grad desig?