WHICH MEAN DO YOU MEAN? AN EXPOSITION ON MEANS

Size: px
Start display at page:

Download "WHICH MEAN DO YOU MEAN? AN EXPOSITION ON MEANS"

Transcription

1 WHICH MEAN DO YOU MEAN? AN EXPOSITION ON MEANS A Thesis Submitted to the Graduate Faculty of the Louisiaa State Uiversity ad Agricultural ad Mechaical College i partial fulfillmet of the requiremets for the degree of Master of Sciece i The Departmet of Mathematics by Mabrouck K. Faradj B.S., L.S.U., 1986 M.P.A., L.S.U., 1997 August, 004

2 Ackowledgmets This work was motivated by a upublished paper writte by Dr. Madde i 000. This thesis would ot be possible without cotributios from may people. To every oe who cotributed to this project, my deepest gratitude. It is a pleasure to give special thaks to Professor James J. Madde for helpig me complete this work. This thesis is dedicated to my wife Mariaa for sacrificig so much of her self so that I may realize my dreams. It would ot have bee doe without her support. ii

3 Table of Cotets Ackowledgmets List of Tables List of Figures Abstract ii iv v vi Chapter 1. Itroductio The Origis of the Term Mea Atique Meas Geometric Iterpretatio of the Atique Meas Atique Meas Iequality Chapter. Classical Meas The History of Classical Meas The Developmet of Classical Meas Theory Nicomachus List of Meas Pappus List of Meas A Moder Recostructio of the Classical Meas Other Meas of the Aciet Greeks Chapter 3. Biary Meas The Theory of Biary Meas Classical Meas as Biary Mea Fuctios Biary Power Meas The Logarithmic Biary Mea Represetatio of Liks betwee Biary Meas Other Biary Meas Chapter 4. -ary Meas Historical Overview The Axiomatic Theory of -ary Meas Traslatio Ivariace Property of -ary Meas Iequality Amog -ary Meas Chapter 5. Coclusio Refereces Vita iii

4 List of Tables.1 Nicomachus Meas Pappus Equatios for Meas iv

5 List of Figures 1.1 Demostratio of Atique Meas usig a circle Proof Without Words: A Truly Algebraic Iequality Biary Meas as Parts of a Trapezoid Sigle Variable Fuctio Associated With Biary Meas v

6 Abstract The objective of this thesis is to give a brief expositio o the theory of meas. I Greek mathematics, meas are itermediate values betwee two extremes, while i moder mathematics, a mea is a measure of the cetral tedecy for a set of umbers. We begi by explorig the origi of the atique meas ad list the classical meas. Next, we preset a overview of the theories of biary meas ad -ary meas. We iclude a geeral discussio o axiomatic systems for meas ad preset theorems o properties that characterize the most commo types of meas. vi

7 Chapter 1. Itroductio I the this chapter we give a brief itroductio to the origis of the arithmetic, geometric, ad harmoic meas. 1.1 The Origis of the Term Mea Accordig to "Webster s New Uiversal Dictioary", the term mea is used to refer to a quatity that is betwee the values of two or more quatities. The term mea is derived from the Frech root word mie whose origi is the Lati word medius, a term used to refer to a place, time, quatity, value, kid, or quality which occupies a middle positio.the most commo usage of the term mea is to express the average of a set of values. The term average, from the Frech word averie, is itself rich i history ad has exteded usage. The term average was used i medieval Europe to refer to a taxig system levied by a liege lord o a vassal or a peasat. The word average is derived from the Arabic awariyah, which traslates as goods damaged i shippig. I the late middle ages, average was used i Frace ad Italy to refer to fiacial loss resultig from damaged goods, where it came to specify the portio of the loss bore by each of the may people who ivested i the ship or its cargo. I this usage, it is the amout idividually paid by each of the ivestors whe a loss is divided equally amog them. The otio of a average is very useful i commerce, sciece, ad legal pursuits; thus, it is ot surprisig that several possible kids of averages have bee iveted so that a wide array of choices of a itermediate value for a give set of values is available to the user to select from. 1. Atique Meas The earliest documeted usage of a mea was i coectio with arithmetic, geometry, ad music. I the 5th cetury B.C., the Greek mathematicia Archytas gave a defiitio of the three commoly used meas of his time i his treatise o music: we have the arithmetic mea whe, of three terms, the first exceeds the secod by the same amout as the secod exceeds the third; the geometric mea whe the first is to the 1

8 secod as the secod is to the third; the harmoic mea whe the three terms are such that by what ever part of itself the first exceeds the secod, the secod exceeds the third by the same part of the third. (Thomas, 1939, p. 36) This ca be traslated to moder terms as follows. Let a ad b be two whole umbers such that a > b ad A, G, ad H are the arithmetic, geometric, ad harmoic meas of a ad b respectively. The (i) a A = A b = A = a+b, (ii) a G = G b = G = ab, a H a (iii) = H b b = H = a+b ab. The origis of the ames give to the atique meas are obscured by time. The first of these meas, ad probably the oldest, is the arithmetic mea. To the aciet Greeks, the term αριθµητικς refers to the art of coutig, ad so, fittigly, they referred to what we commoly call the average as the arithmetic mea sice it pertais to fidig a umber that is itermediate to a give pair of atural umbers. As for the ame give to the geometric mea, it appears that the Pythagorea school coied the term mea proportioal, i.e., the geometric mea, to refer to the measure of a altitude draw from the right agle to the hypoteuse of a right triagle. The measure of such a altitude is betwee the measures of the two segmets of the hypoteuse. The source of the ame give to the harmoic mea ca oly be foud i legeds. The Roma Boethius (circa 5 A.D.) tells us of a leged about Pythagoras who o passig a blacksmith shop was struck by the fact that the souds caused by the beatig of differet hammers o the avil formed a fairly musical whole. This observatio motivated Pythagoras to ivestigate the relatio betwee the legth of a vibratig strig ad the musical toe it produced. He observed that differet harmoic musical toes are produced by particular ratios of the legth of the vibratig strig to its whole. He cocluded, accordig to the leged, that the musical harmoy produced was to be foud i particular ratios of the legth of the vibratig strig. Thus to the Pythagoreas, who believed that all kowledge ca be reduced to relatios betwee umbers, musical harmoy

9 occurred because certai ratio of umbers that lie betwee two extremes are harmoic, ad thus the term harmoic mea was give to that value. Propositio 1. Suppose 0 < a b. Let A := a+b, G := ab, ad H := a+b ab. The a A b, a G b, ad a H b Proof. Sice 0 < a b, the a + b b; therefore, a+b b. Similarly, a a + b; therefore a a+b a+b. Therefore, a b. Thus, a A b. Hece, b 1 a+b 1 ab a. Therefore, a a+b b, ad a H b. If a b, a ab b ; therefore, a ab b. Hece a G b. 1.3 Geometric Iterpretatio of the Atique Meas Sice geometry is the aciet Greeks preferred veue of scietific ivestigatio, Greek mathematicias produced umerous geometric treatises that related the three atique meas to each other by usig straight edge ad compass costructio. A excellet example ca be foud i Schild (1974) ad reproduced here: Example 1.1. Suppose a ad b are two whole umbers. Let A, G, ad H be the arithmetic, geometric, ad harmoic meas respectively of a ad b. The by usig a straight edge ad compass we ca illustrate that A = a+b, G = ab, ad H = a+b ab. Draw the lie segmet LMN with LM = a ad MN = b (see figure 1.1). With LN as diameter, draw a semi circle with ceter O ad fix P o its circumferece. Draw MQ perpedicular to OP ad MP perpedicular to LN. The OP = A, MP = G, ad QP = H. To show this is true, we give the followig argumet. Sice OP is the radius of the circle whose diameter is LN, the OP = 1 (a + b) = A, ad sice (MP) = (LM)(MN) = ab, the MP = ab = G. Let α = POM. Observe that QMP = α ad POM is similar to PMQ; thus, PQ PM = PM PO. Therefore, PQ = (PM) PO = ab a+b = ab a+b = H. I figure 1.1, observe what happes if (a + b) remais fixed, i.e., segmet LN is fixed, ad M is allowed to move. As M moves toward N, both G ad H decrease. As M moves towards O, both G ad H icrease. If M coicides with O, i.e., a = b, the A = G = H. This may have bee the motivatio for ivestigatig the iequality betwee the three meas. 3

10 P Q α α L O M N a b FIGURE 1.1. Demostratio of Atique Meas usig a circle. 1.4 Atique Meas Iequality I this sectio we will preset several proofs of the iequality: H G A (1.1) Of the umerous useful iequalities i mathematics, the arithmetic-geometric mea iequality occupies a special positio, ot oly from a historical stadpoit, but also o accout of its frequet usage i differet mathematical proofs. We will give a more i-depth discussio about this iequality i Chapters 3 ad 4. At this poit, it suffices to say that there have bee umerous proofs give for the above iequality over the ceturies. We begi our discussio by presetig a iformal argumet of the iequality. Referrig back to Figure 1.1, we ote that siα 1. From OPM, we have siα = MP OP, ad from PQM, we have siα = QP MP. Therefore MP = OPsiα G = Asiα. Hece G A, (1.) ad QP = MPsiα H = Gsiα. Hece H G. (1.3) From 1. ad 1.3, we get 1.1. However, sice the above argumet uses trigoometry, it does ot reflect the spirit of the aciet proofs for this iequality. I Figure 1., we preset a illustratio that captures the fudametal 4

11 character of this iequality i mathematics, which may have motivated the aciet mathematicias to establish proofs of the arithmetic-geometric mea iequality (Gallat 1977). The iequality as illustrated by Figure 1. requires oly rudimetary kowledge of geometry to prove. Now we give a more moder algebraic proof for the geometric-arithmetic mea iequality. ab a b FIGURE 1.. Proof Without Words: A Truly Algebraic Iequality. Theorem 1. For ay oegative umbers a ad b, ab a+b, with equality holdig if ad oly if a = b. Proof. Let a = c ad b = d. The a+b ab becomes c +d cd, or equivaletly, c +d cd 0. This is equivalet to c cd + d 0 which is i tur equivalet to (c d) 0. Sice the square of ay real umber is oegative, we see that the iequality stated i the theorem is ideed true. Equality holds if ad oly if c d = 0, that is c = d, or equivaletly, if ad oly if, a = b. We use the result from theorem 1 to establish a iequality betwee the harmoic ad geometric meas of ay two oegative umbers. Corollary 1.. For ay oegative umbers a ad b, ab a+b ab, with equality holdig if ad oly if a = b. Proof. Sice ab a+b, the ab (a + b). Therefore, ab (a + b) ab, ad ab a+b ab. 5

12 From theorem 1 ad corollary 1., we have H G A. (1.4) 6

13 Chapter. Classical Meas I this chapter we will explore the origis of the theory of biary meas. The chapter icludes two lists of the classical biary meas as give by Greek mathematicias. The followig list gives the ames of Greek mathematicia ad the approximate dates of their work o meas. It is helpful to the uderstadig of the historical developmet of the theory meas i the aciet Greek world (Smith 1951). Thales, 600 B.C. Pythagoras, 540 B.C. Archytas, 400 B.C. Plato, 380 B.C. Eudoxus, 370 B.C. Eudemus, 335 B.C. Euclid, 300 B.C. Archimedes, 30 B.C. Hero, 50 A.D. Nicomachus, 100 A.D. Theo, 15 A.D. Porphyrius, 75 A.D. Pappus, 300 A.D. Iamblichus, 35 A.D. Proclus, 460 A.D. Boethius, 510 A.D..1 The History of Classical Meas I this sectio we will give a brief discussio o what motivated Greek mathematicias to study ad develop a doctrie for meas by presetig the ratioale give by promiet Greek mathematicias who touched o the history of the theory of meas i their work ad the opiios of Greek mathematics scholars o this matter. Accordig to Gow (193), by Plato s time umbers were grouped ito two geeral categories. First, as sigle umbers categorized by their attributes such as odd, eve, triagular, perfect, excessive, defective, amicable etc. Secod, umbers were viewed as groups comprised of umbers that are either i series or proportios. The aciet Greeks viewed meas as a special case of proportios (Allema 1877, Thomas 1939, Gow 193). Smith (1951) writes, " Early [Greek] writers spoke of a arithmetic proportio, meaig b a = d c as i,3,4,5, ad of geometric proportio, meaig a : b = c : d as i, 4, 5, 10, ad a harmoic proportio, meaig 1 b a 1 = d 1 1 c as i 1, 1 3, 4 1; 1 5." I his commets o paradigms of aciet Greek mathematics, Allema (1877) says, "whe two quatities were compared [i Greek mathematics], the basis for the compariso seems to be either how much the oe is greater tha the other, i.e., a arithmetic 7

14 ratio, or how may times is the oe cotaied i the other, i.e., their geometrical ratio." Allema (1877) claims that this type of compariso of ratios would aturally lead to the theory of meas because for ay three positive magitudes, be it lies or umbers, a, b, ad c, if a b = b c, the three magitudes are i arithmetical proportio, but if a : b :: b : c, they are i geometrical proportio. Allema s claim seems to be supported by the work of Nicomachus i "Itroductio to Arithmetic". I this work, Nicomachus bega his discourse o meas by givig the defiitio that distiguished a ratio from a proportio. He referred to the latter as the compositio of two ratios. He the stated that whe oe term appears o both sides of a proportio, as i b a = b c, the proportio is kow as a cotiued proportio. The proportio is called disjuct whe the middle terms are differet. The highest term i a cotiued proportio is called the cosequet, the least is called the atecedet, ad the middle term is the mea, µεστητες, which is medius whe traslated ito Lati ad from which the word mea is derived (Gow 193). As we have oted above, Greek mathematics viewed meas as a special proportio ivolvig three magitudes; therefore, it is appropriate that we begi our review of the history of developmet of meas by metioig that Proclus attributed to Thales the begiig of the doctrie of proportios (Allema 1877). Thales established the theorem that equiagular triagles have proportioal sides (Allema 1877). I "Itroductio to Arithmetic", Nicomachus writes, "the kowledge of proportios is particularly importat for the study of aciet mathematicias." This ca be take to mea that the doctrie of proportios played a importat role i the developmet of Greek mathematics. Maziarz (1968) commets o the atural developmet of the theory of proportioals i Greek mathematics by sayig, "If a poit is a uit i a positio, the a lie is made of poits. Cosequetly, the ratio of two give segmets is merely the ratio of the umber of poits i each. Moreover, because ay magitude ivolves a ratio betwee the umber of uits it cotais ad the uit itself, ad, thus, the compariso of two magitudes implies either or 4 ratios." By poits, Maziarz seems to imply the tick marks that would be made if the segmets were divided ito may small equal uits. 8

15 From the historical perspective, the aciet sources of Greek mathematics history that we have refereced do ot metio whe the arithmetic mea was first developed. However, they offer various explaatios as to whe the geometric ad harmoic meas were first itroduced. Allema (1877) states that aciet sources (Iamblichus, Nicomachus, Proclus) poit to Eudoxus as the oe who established the harmoic mea ad to Pythagoras as the oe who established the otio of a mea proportioal betwee two give lies. It is iterestig to ote that some facets of the theory of meas appear i various aciet Greek texts. Some of these were iteded as mathematics treatises, such as the collectio of books that costitute Euclid s work kow as the "Elemets", but others did ot have a apparet mathematical purpose. Oe such example, oted by Maziarz (1968), ca be foud i passages of "Timaeus" kow as "The Costructio the world-soul." I this sectio of the book, Plato attempts to costruct the arithmetical cotiuum usig two geometric progressios 1,,4,8 ad 1,3,9,7; the fillig i the itervals betwee these umbers with the arithmetic ad harmoic meas. By successive duplicatio of the two progressios ad fillig i with the appropriate combiatio of arithmetic ad harmoic meas, all umbers ca be geerated, but ot i their atural order. Aother example ca be foud i Aristotle s "Metaphysics". I this work, Aristotle describes Plato s otio of distributive justice as, " The just i this sese is a mea betwee two extremes that are disproportioate, sice the proportioate is a mea, ad the just is proportioate. This kid of proportio is termed by mathematicias geometrical proportio." From the above examples, oe gets the sese that to the aciet Greeks, the theory of meas ad proportios may ot have bee just a mere mathematical cocept sice some aspects of the theory of meas was also reflected i their literature, philosophy, ad religio.. The Developmet of Classical Meas Theory It appears that the classical meas were developed over a log period of time by the gradual additio of seve more meas to the first three (Heath 1963). I all his work, Euclid oly uses the three atique meas (Allema 1887, Gow 193). However, by first cetury A.D., we kow that 9

16 Greek mathematicias referred to te meas. All the sources reviewed (Allema 1887, Bema 1910, Heath 191, Gow 193, Thomas 1939, Smith 1951) suggest that Greek mathematicias geerated these meas by cosiderig three quatities a, b, ad c, such that a > b > c. They assumed b to be the mea ad formed three positive differeces with the a, b, ad c: (a b), (b c), ad (a c). The they formed a proportio by equatig a ratio of two of these differeces to a ratio of two of the origial magitudes, a, b, ad c. For example, b is the harmoic mea of a ad c whe a b b c = a c. Nicomachus i "Itroductio to Arithmetic" (Gow 193) goes o to say: "Pythagoras, Plato, ad Aristotle kew oly six kids of [cotiued] proportios: the arithmetic, geometric, ad harmoic meas, ad their subcotraries, which have o ames. Later writers added four more." Greek mathematicias referred to certai classical meas as cotrary ad subcotrary meas because these meas were see to be i a cotrary (opposite) order from the arithmetic mea whe compared to the geometric or harmoic meas (Oxford Eglish Dictioary 004). I his work "I Nicomachus" (Heath 191), Iamblichus says, "the first three [atique meas] oly were kow to Pythagoras, the secod three were iveted by Eudoxus." The remaiig four, Iamblichus attributed to the later Pythagoreas. He adds that all te were treated i the Euclidea maer by Pappus. Gow (193) states that the umber of cotiued proportios was raised to te ad kept at that umber because the umber te was held by the aciet Greek mathematicias to be the most perfect umber. He adds, "how else ca we explai the fact that the golde mea, which Nicomachus calls the most perfect ad embracig of all proportios, was left out from the list of meas." All these testimoies poit to the coclusio that the theory of meas i Greek mathematics was well established by the First Cetury. Our mai complete source for aciet Greek mathematics theory of meas is Boethius commetary o the works of Pappus ad Nicomachus. I this work, 10

17 Boethius credits Nicomachus ad Pappus as the mai Greek mathematicias who dealt with meas from a theoretical perspective (Smith 1951)..3 Nicomachus List of Meas The earliest kow treatmet of classical meas as a idepedet body of kowledge was give by Nicomachus i "Itroductio to Arithmetic" (Allema 1887, Heath 191, Gow 193, Thomas 1939, Smith 1951). Allema, Gow, Heath, ad Thomas cocluded (seemigly idepedet of each other) that Nicomachus proceeded to develop his list as follows: He bega his list by commetig o the cotiued arithmetical proportio a b = b c. This suggests that a b : b c :: a : a, which allows us to make a coectio to other meas. Gow (193) remarks, "I a cotiued geometric proportio, a : b :: b : c, he otices that a b : b c :: a : b. Fially, the three magitudes, a, b, c, are i harmoic proportio if a b : b c :: a : c." A similar approach was used by Archytas (as cited by Porphyrius i his commetary o Ptolemy s "Harmoics") whe discussig the three atique meas i terms of three magitudes i cotiued arithmetic, geometric, ad harmoic proportios (Thomas 1939). Gow (193) also poits out that Nicomachus failed to metio that the arithmetic, geometric, ad harmoic meas of two umbers are i geometric proportio: a+b : ab : a+b ab. I Thomas traslatio of Nicomachus "Itroductio to Arithmetic" (Thomas 1939), Nicomachus itroduces the seve other meas usig the same treatmet as the oe metioed above. (The reader may wish to refer to Table.3 for a compact summary of the followig.) The fourth mea, which is also called the subcotrary by reaso of its beig reciprocal ad atithetical to the harmoic, comes about whe of the three terms the greatest bears the same ratio to the least as the differece of the lesser terms bears to the differece of the greater, as i the case of 3, 5; 6 (Thomas, 1939, p. 119). Nicomachus itroduces the fifth mea as the subcotrary mea to the geometric mea, The fifth [mea] exists whe of the three terms, the middle bears to the least the same ratio as their differece bears to the differece betwee the greatest ad the middle 11

18 terms, as i the case of, 4; 5, for 4 is double, the middle term is double the least, ad is double 1, that is the differece of the least terms is double the differece of the greatest. What makes it subcotrary to the geometric mea is this property, that i the case of the geometric mea the middle term bears to the lesser the same ratio as the excess of the greater term over the middle bears to that of the middle term over the lesser, while i the case of this mea a cotrary relatio holds (Thomas, 1939, p. 11). Nicomachus itroduces the sixth mea as, The sixth mea comes about whe of the three terms the greatest bears the same ratio to the middle as the excess of the middle term over the least bears to the excess of the greatest term over the middle as i the case of 1, 4; 6, for i each case the ratio is sesquialter [3 : ]. No doubt, it is called subcotrary to the geometric mea because the ratios are reversed, as i the case of the fifth mea (Thomas, 1939, p. 11). Nicomachus itroduces the last 4 meas by sayig, By playig about with the terms ad their differeces certai me discovered four other meas which do ot fid a place i the writigs of the aciets, but which evertheless ca be treated briefly i some fashio, although they are superfluous refiemets, i order ot to appear igorat. The first of these, or the seveth i the complete list, exists whe the greatest term bears the same relatio to the least as their differece bears to the differece of the lesser terms, as i the case of 6, 8; 9, for the ratio of each is see by compoudig the terms to be the sesquialter. The eighth mea, or the secod of these, comes about whe the greatest term bears to the least the same ratio as the differece of the extreme bears to the differece of the greater terms, as i the case of 6, 7; 9, for here the two ratios are the sesquialter. The ith mea i the complete series, ad the third i the umber of those more recetly discovered, comes about whe there are three terms ad the middle bears to the least the same ratio as the differece betwee the extremes bears to the differece betwee the least terms, as 4, 6; 7. Fially, the teth i the 1

19 complete series, ad the fourth i the list set out by the moders, is see whe i three terms the middle term bears to the least the same ratio as the differece betwee the extremes bears to the differece of the greater terms, as i the case of 3, 5; 8, for the ratio i each couple is the super-bi-partiet [5 : 3] (Thomas, 1939, p. 11). TABLE.1. Nicomachus Meas Mea Proportio Numbers Exhibitig the Mea Arithmetic a b : b c :: a : a, 4, 6 Geometric a b : b c :: a : b 4,, 1 Harmoic a b : b c :: a : c 6, 3, Cot. Harmoic b c : a b :: a : c 3, 5, 6 Cot. Geometric b c : a b :: b : c, 4, 5 Subco. Geometric b c : a b :: a : b 1,4,6 Seveth a c : b c :: a : c 6, 8, 9 Eighth a c : a b :: a : c 6, 7, 9 Nith a c : b c :: b : c 4, 6, 7 Teth a c : a b :: b : c 3, 5, 8 (Thomas 1939).4 Pappus List of Meas Pappus used a differet approach tha Nicomachus whe presetig his list of meas (Heath 191, Thomas 1939). Both Heath ad Thomas state that the meas o Pappus list are similar to those preseted by Nicomachus, but i a differet order after the sixth mea. Meas umber 8, 9, ad 10 i Nicomachus list are respectively umbers 9, 10, ad 7 o Pappus list. Moreover, Pappus omits mea umber 7 o Nicomachus list ad gives as umber 8 a additioal mea equivalet to the proportio c : b :: c a : c b. Therefore, the two lists combied give five additioal meas to the first six. I Thomas traslatio (1939) of Pappus work kow as "Collectios III", Pappus itroduces his discussio o meas as a respose to a questio posed by a uiformed geometer. He 13

20 demostrates his aswer by the costructio of the three meas i a semicircle (see figure 1.1). Pappus shows, i a series of propositios, that give three terms α, β, ad γ i geometrical progressio (Heath 191 uses "i geometric proportio"), it is possible to form from them three other terms a, b, ad c which are itegral liear combiatio of α, β, ad γ such that b is oe of the classical meas. The solutios to Pappus s equatios are show i Table.. The liear (Heath 191, Thomas 1939) TABLE.. Pappus Equatios for Meas Mea a, b, c Numbers exhibitig the mea Arithmetic a = α + 3β + γ 6, 4, b = α + β + γ c = β + γ Geometric a = α + β + γ 4,, 1 b = β + γ c = γ Harmoic a = α + 3β + γ 6, 3, b = β + γ c = β + γ Subcotrary a = α + 3β + γ 6, 5, b = α + β + γ c = β + γ Fifth a = α + 3β + γ 5, 4, b = α + β + γ c = β + γ Sixth a = α + 3β + γ 6, 4, 1 b = α + β + γ c = α + β γ Seveth a = α + β + γ 3,, 1 b = β + γ c = γ Eighth a = α + 3β + γ 6, 4, 3 b = α + β + γ c = β + γ Nith a = α + β + γ 4, 3, b = α + β + γ c = β + γ Teth a = α + β + γ 3,, 1 b = β + γ c = γ equatios show i Table. are moder equivalets of the literal traslatio of the Greek versio of Pappus. For example (Thomas 1939), i the case of the geometric mea metioed i Table., the literal traslatio of Pappus words would be, "To form a take α oce, β twice, ad γ oce; ad to form b we have to take β oce ad γ oce; ad to form c we take γ oce." Notice also that the examples give by Pappus for the proportios formed by his equatios sometimes differ 14

21 from those give by Nicomachus. For example for the fourth mea, Nicomachus gave 3, 5, ad 6 as a example for a solutio, while Pappus gave, 5, ad 6 as a solutio. Pappus expositio o meas by usig equatios may be better uderstood from the perspective that proportios were used i those days to solve equatios. Usig Proclus commetary o Euclid as a referece, Klei (1966) states, " Greek mathematics usage of proportios ca be compared to the moder sese of costructio of a equatio, ad a equatio may be viewed as a solutio of a proportio. This may be due to the uderstadig of ratios, proportios, ad harmoy o the basis of a commo mathematical property." Bema (1910) claims that the mathematicias of Alexadria uderstood equatios of secod degree mostly i the form of proportios. If we express Pappus method i moder terms, Pappus is parmeterizig meas by quadratics ad, equivaletly, givig quadratic polyomials to illustrate the relatio amog terms i the various meas. For example, to calculate the harmoic mea, usig three quatities i geometric progressio is equivalet to usig α = 1, β = x, ad γ = x ; thus, give a = + 3x + x, b = x + x, ad c = x + x, we have ac a+c = (+3x+x )(x+x ) = x(x+1)(x +3x+) = x(x +3x+) +3x+x +x+x (x+1) x+1 = x(x + ) = x + x = b..5 A Moder Recostructio of the Classical Meas I this sectio, we will use a similar approach to the oe used by Nicomachus to geerate the classical meas by cosiderig three positive quatities a, b, ad c such that a > b > c, ad we wish to make b the mea of a ad c. We will form three positive differeces with these quatities: (a b), (b c), ad (a c). The we will form a proportio by equatig a ratio of two of these differeces to a ratio of two of the origial quatities (ot ecessarily distict). For example, if we set the ratio a b b c equal to the ratio a b, the result is b = ac, which represets the geometric mea. If you look at all the possible ways of doig this, several of them are automatically ruled out by the assumed iequality of a, b, ad c. The oes that are ot (ecessarily) ruled out are the eleve meas summarized below (Madde 000, Heath 1963): 1. (a b) (b c) = a a = b b = c c (a+c), we have the arithmetic mea b =. 15

22 . (a b) (b c) = b c = a b, we have the geometric mea b = ac (a b) (b c) = a c ; we have the harmoic mea b = 1 a +. 1 c (a b) (b c) = c a ; we have the cotra-harmoic mea b = a +c a+c. (a b) (b c) = c b ; we have the first cotra-geometric mea b = a c+ a ac+5c. (a b) (b c) = b a ; we have the secod cotra-geometric mea b = c a+ 5a ac+c. (b c) (a c) = a c; b = ac c a. This mea is o Nicomachus list but ot Pappus list. (b c) (a c) = c b ; b = c+ 4ac 3c. (a b) (a c) = c a ; b = a ac+c a. (a b) (a c) = b a ; b = a c a. This mea is o Pappus list but ot Nicomachus list. (a b) (a c) = c b ; b = a c. Note that some of these meas are ot very robust defiitios of meas. For example, if oe uses the 11 th mea o our list to fid the mea of 5 ad 4, the M(5,4) = 1, which is ot betwee 5 ad 4. Note also that usig the 5 th mea o our list to fid the mea of 1 ad, we obtai the celebrated golde umber Φ = However, as we will show i the ext sectio, the above list does ot exhaust all the meas kow to the aciet Greek world..6 Other Meas of the Aciet Greeks I this sectio, we poit out that Greek mathematicias cotiued to develop ew meas which were ever icluded amog the classical meas. Nicomachus referred to a special mea obtaied by the divisio of a segmet ito what he called "the most perfect proportios". This mea, which we will call b, ca be expressed by the divisio of a segmet of magitude a ito two parts: A greater part, b, ad a lesser part, a b, i such a fashio that the ratio of a to b is equal to the ratio 16

Trigonometric 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

Trigonometric 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 information

Convexity, Inequalities, and Norms

Convexity, Inequalities, and Norms Covexity, Iequalities, ad Norms Covex Fuctios You are probably familiar with the otio of cocavity of fuctios. Give a twicedifferetiable fuctio ϕ: R R, We say that ϕ is covex (or cocave up) if ϕ (x) 0 for

More information

Sequences and Series

Sequences and Series CHAPTER 9 Sequeces ad Series 9.. Covergece: Defiitio ad Examples Sequeces The purpose of this chapter is to itroduce a particular way of geeratig algorithms for fidig the values of fuctios defied by their

More information

In nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008

In nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008 I ite Sequeces Dr. Philippe B. Laval Keesaw State Uiversity October 9, 2008 Abstract This had out is a itroductio to i ite sequeces. mai de itios ad presets some elemetary results. It gives the I ite Sequeces

More information

Lecture 13. Lecturer: Jonathan Kelner Scribe: Jonathan Pines (2009)

Lecture 13. Lecturer: Jonathan Kelner Scribe: Jonathan Pines (2009) 18.409 A Algorithmist s Toolkit October 27, 2009 Lecture 13 Lecturer: Joatha Keler Scribe: Joatha Pies (2009) 1 Outlie Last time, we proved the Bru-Mikowski iequality for boxes. Today we ll go over the

More information

Department of Computer Science, University of Otago

Department of Computer Science, University of Otago Departmet of Computer Sciece, Uiversity of Otago Techical Report OUCS-2006-09 Permutatios Cotaiig May Patters Authors: M.H. Albert Departmet of Computer Sciece, Uiversity of Otago Micah Colema, Rya Fly

More information

I. Chi-squared Distributions

I. Chi-squared Distributions 1 M 358K Supplemet to Chapter 23: CHI-SQUARED DISTRIBUTIONS, T-DISTRIBUTIONS, AND DEGREES OF FREEDOM To uderstad t-distributios, we first eed to look at aother family of distributios, the chi-squared distributios.

More information

1. MATHEMATICAL INDUCTION

1. 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 information

Properties of MLE: consistency, asymptotic normality. Fisher information.

Properties of MLE: consistency, asymptotic normality. Fisher information. Lecture 3 Properties of MLE: cosistecy, asymptotic ormality. Fisher iformatio. I this sectio we will try to uderstad why MLEs are good. Let us recall two facts from probability that we be used ofte throughout

More information

Factors of sums of powers of binomial coefficients

Factors 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 information

Chapter 6: Variance, the law of large numbers and the Monte-Carlo method

Chapter 6: Variance, the law of large numbers and the Monte-Carlo method Chapter 6: Variace, the law of large umbers ad the Mote-Carlo method Expected value, variace, ad Chebyshev iequality. If X is a radom variable recall that the expected value of X, E[X] is the average value

More information

SAMPLE QUESTIONS FOR FINAL EXAM. (1) (2) (3) (4) Find the following using the definition of the Riemann integral: (2x + 1)dx

SAMPLE QUESTIONS FOR FINAL EXAM. (1) (2) (3) (4) Find the following using the definition of the Riemann integral: (2x + 1)dx SAMPLE QUESTIONS FOR FINAL EXAM REAL ANALYSIS I FALL 006 3 4 Fid the followig usig the defiitio of the Riema itegral: a 0 x + dx 3 Cosider the partitio P x 0 3, x 3 +, x 3 +,......, x 3 3 + 3 of the iterval

More information

.04. This means $1000 is multiplied by 1.02 five times, once for each of the remaining sixmonth

.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 information

Chapter 7 Methods of Finding Estimators

Chapter 7 Methods of Finding Estimators Chapter 7 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 011 Chapter 7 Methods of Fidig Estimators Sectio 7.1 Itroductio Defiitio 7.1.1 A poit estimator is ay fuctio W( X) W( X1, X,, X ) of

More information

The second difference is the sequence of differences of the first difference sequence, 2

The 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 information

Sequences II. Chapter 3. 3.1 Convergent Sequences

Sequences II. Chapter 3. 3.1 Convergent Sequences Chapter 3 Sequeces II 3. Coverget Sequeces Plot a graph of the sequece a ) = 2, 3 2, 4 3, 5 + 4,...,,... To what limit do you thik this sequece teds? What ca you say about the sequece a )? For ǫ = 0.,

More information

FOUNDATIONS OF MATHEMATICS AND PRE-CALCULUS GRADE 10

FOUNDATIONS OF MATHEMATICS AND PRE-CALCULUS GRADE 10 FOUNDATIONS OF MATHEMATICS AND PRE-CALCULUS GRADE 10 [C] Commuicatio Measuremet A1. Solve problems that ivolve liear measuremet, usig: SI ad imperial uits of measure estimatio strategies measuremet strategies.

More information

4.1 Sigma Notation and Riemann Sums

4.1 Sigma Notation and Riemann Sums 0 the itegral. Sigma Notatio ad Riema Sums Oe strategy for calculatig the area of a regio is to cut the regio ito simple shapes, calculate the area of each simple shape, ad the add these smaller areas

More information

A probabilistic proof of a binomial identity

A probabilistic proof of a binomial identity A probabilistic proof of a biomial idetity Joatho Peterso Abstract We give a elemetary probabilistic proof of a biomial idetity. The proof is obtaied by computig the probability of a certai evet i two

More information

Chapter 5: Inner Product Spaces

Chapter 5: Inner Product Spaces Chapter 5: Ier Product Spaces Chapter 5: Ier Product Spaces SECION A Itroductio to Ier Product Spaces By the ed of this sectio you will be able to uderstad what is meat by a ier product space give examples

More information

Soving Recurrence Relations

Soving 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 information

THE ARITHMETIC OF INTEGERS. - multiplication, exponentiation, division, addition, and subtraction

THE 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 information

3. Covariance and Correlation

3. Covariance and Correlation Virtual Laboratories > 3. Expected Value > 1 2 3 4 5 6 3. Covariace ad Correlatio Recall that by takig the expected value of various trasformatios of a radom variable, we ca measure may iterestig characteristics

More information

MARTINGALES AND A BASIC APPLICATION

MARTINGALES AND A BASIC APPLICATION MARTINGALES AND A BASIC APPLICATION TURNER SMITH Abstract. This paper will develop the measure-theoretic approach to probability i order to preset the defiitio of martigales. From there we will apply this

More information

Asymptotic Growth of Functions

Asymptotic 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 information

Case Study. Normal and t Distributions. Density Plot. Normal Distributions

Case Study. Normal and t Distributions. Density Plot. Normal Distributions Case Study Normal ad t Distributios Bret Halo ad Bret Larget Departmet of Statistics Uiversity of Wiscosi Madiso October 11 13, 2011 Case Study Body temperature varies withi idividuals over time (it ca

More information

where: T = number of years of cash flow in investment's life n = the year in which the cash flow X n i = IRR = the internal rate of return

where: T = number of years of cash flow in investment's life n = the year in which the cash flow X n i = IRR = the internal rate of return EVALUATING ALTERNATIVE CAPITAL INVESTMENT PROGRAMS By Ke D. Duft, Extesio Ecoomist I the March 98 issue of this publicatio we reviewed the procedure by which a capital ivestmet project was assessed. The

More information

CS103A Handout 23 Winter 2002 February 22, 2002 Solving Recurrence Relations

CS103A Handout 23 Winter 2002 February 22, 2002 Solving Recurrence Relations CS3A Hadout 3 Witer 00 February, 00 Solvig Recurrece Relatios Itroductio A wide variety of recurrece problems occur i models. Some of these recurrece relatios ca be solved usig iteratio or some other ad

More information

SEQUENCES AND SERIES

SEQUENCES 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 information

The analysis of the Cournot oligopoly model considering the subjective motive in the strategy selection

The analysis of the Cournot oligopoly model considering the subjective motive in the strategy selection The aalysis of the Courot oligopoly model cosiderig the subjective motive i the strategy selectio Shigehito Furuyama Teruhisa Nakai Departmet of Systems Maagemet Egieerig Faculty of Egieerig Kasai Uiversity

More information

CHAPTER 3 THE TIME VALUE OF MONEY

CHAPTER 3 THE TIME VALUE OF MONEY CHAPTER 3 THE TIME VALUE OF MONEY OVERVIEW A dollar i the had today is worth more tha a dollar to be received i the future because, if you had it ow, you could ivest that dollar ad ear iterest. Of all

More information

Lecture 4: Cauchy sequences, Bolzano-Weierstrass, and the Squeeze theorem

Lecture 4: Cauchy sequences, Bolzano-Weierstrass, and the Squeeze theorem Lecture 4: Cauchy sequeces, Bolzao-Weierstrass, ad the Squeeze theorem The purpose of this lecture is more modest tha the previous oes. It is to state certai coditios uder which we are guarateed that limits

More information

NUMBERS COMMON TO TWO POLYGONAL SEQUENCES

NUMBERS COMMON TO TWO POLYGONAL SEQUENCES NUMBERS COMMON TO TWO POLYGONAL SEQUENCES DIANNE SMITH LUCAS Chia Lake, Califoria a iteger, The polygoal sequece (or sequeces of polygoal umbers) of order r (where r is r > 3) may be defied recursively

More information

Distributions of Order Statistics

Distributions of Order Statistics Chapter 2 Distributios of Order Statistics We give some importat formulae for distributios of order statistics. For example, where F k: (x)=p{x k, x} = I F(x) (k, k + 1), I x (a,b)= 1 x t a 1 (1 t) b 1

More information

1 Computing the Standard Deviation of Sample Means

1 Computing the Standard Deviation of Sample Means Computig the Stadard Deviatio of Sample Meas Quality cotrol charts are based o sample meas ot o idividual values withi a sample. A sample is a group of items, which are cosidered all together for our aalysis.

More information

Building Blocks Problem Related to Harmonic Series

Building Blocks Problem Related to Harmonic Series TMME, vol3, o, p.76 Buildig Blocks Problem Related to Harmoic Series Yutaka Nishiyama Osaka Uiversity of Ecoomics, Japa Abstract: I this discussio I give a eplaatio of the divergece ad covergece of ifiite

More information

Discrete Mathematics and Probability Theory Spring 2014 Anant Sahai Note 13

Discrete Mathematics and Probability Theory Spring 2014 Anant Sahai Note 13 EECS 70 Discrete Mathematics ad Probability Theory Sprig 2014 Aat Sahai Note 13 Itroductio At this poit, we have see eough examples that it is worth just takig stock of our model of probability ad may

More information

Class Meeting # 16: The Fourier Transform on R n

Class Meeting # 16: The Fourier Transform on R n MATH 18.152 COUSE NOTES - CLASS MEETING # 16 18.152 Itroductio to PDEs, Fall 2011 Professor: Jared Speck Class Meetig # 16: The Fourier Trasform o 1. Itroductio to the Fourier Trasform Earlier i the course,

More information

Week 3 Conditional probabilities, Bayes formula, WEEK 3 page 1 Expected value of a random variable

Week 3 Conditional probabilities, Bayes formula, WEEK 3 page 1 Expected value of a random variable Week 3 Coditioal probabilities, Bayes formula, WEEK 3 page 1 Expected value of a radom variable We recall our discussio of 5 card poker hads. Example 13 : a) What is the probability of evet A that a 5

More information

3. Greatest Common Divisor - Least Common Multiple

3. 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 information

FIBONACCI NUMBERS: AN APPLICATION OF LINEAR ALGEBRA. 1. Powers of a matrix

FIBONACCI NUMBERS: AN APPLICATION OF LINEAR ALGEBRA. 1. Powers of a matrix FIBONACCI NUMBERS: AN APPLICATION OF LINEAR ALGEBRA. Powers of a matrix We begi with a propositio which illustrates the usefuless of the diagoalizatio. Recall that a square matrix A is diogaalizable if

More information

Notes on exponential generating functions and structures.

Notes on exponential generating functions and structures. Notes o expoetial geeratig fuctios ad structures. 1. The cocept of a structure. Cosider the followig coutig problems: (1) to fid for each the umber of partitios of a -elemet set, (2) to fid for each the

More information

SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES

SECTION 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 information

Basic Elements of Arithmetic Sequences and Series

Basic Elements of Arithmetic Sequences and Series MA40S PRE-CALCULUS UNIT G GEOMETRIC SEQUENCES CLASS NOTES (COMPLETED NO NEED TO COPY NOTES FROM OVERHEAD) Basic Elemets of Arithmetic Sequeces ad Series Objective: To establish basic elemets of arithmetic

More information

Modified Line Search Method for Global Optimization

Modified Line Search Method for Global Optimization Modified Lie Search Method for Global Optimizatio Cria Grosa ad Ajith Abraham Ceter of Excellece for Quatifiable Quality of Service Norwegia Uiversity of Sciece ad Techology Trodheim, Norway {cria, ajith}@q2s.tu.o

More information

hp calculators HP 12C Statistics - average and standard deviation Average and standard deviation concepts HP12C average and standard deviation

hp calculators HP 12C Statistics - average and standard deviation Average and standard deviation concepts HP12C average and standard deviation HP 1C Statistics - average ad stadard deviatio Average ad stadard deviatio cocepts HP1C average ad stadard deviatio Practice calculatig averages ad stadard deviatios with oe or two variables HP 1C Statistics

More information

Here 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.

Here 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 information

1 Correlation and Regression Analysis

1 Correlation and Regression Analysis 1 Correlatio ad Regressio Aalysis I this sectio we will be ivestigatig the relatioship betwee two cotiuous variable, such as height ad weight, the cocetratio of a ijected drug ad heart rate, or the cosumptio

More information

THE HEIGHT OF q-binary SEARCH TREES

THE HEIGHT OF q-binary SEARCH TREES THE HEIGHT OF q-binary SEARCH TREES MICHAEL DRMOTA AND HELMUT PRODINGER Abstract. q biary search trees are obtaied from words, equipped with the geometric distributio istead of permutatios. The average

More information

Taking DCOP to the Real World: Efficient Complete Solutions for Distributed Multi-Event Scheduling

Taking DCOP to the Real World: Efficient Complete Solutions for Distributed Multi-Event Scheduling Taig DCOP to the Real World: Efficiet Complete Solutios for Distributed Multi-Evet Schedulig Rajiv T. Maheswara, Milid Tambe, Emma Bowrig, Joatha P. Pearce, ad Pradeep araatham Uiversity of Souther Califoria

More information

Our aim is to show that under reasonable assumptions a given 2π-periodic function f can be represented as convergent series

Our aim is to show that under reasonable assumptions a given 2π-periodic 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 information

Example 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).

Example 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 information

Irreducible polynomials with consecutive zero coefficients

Irreducible polynomials with consecutive zero coefficients Irreducible polyomials with cosecutive zero coefficiets Theodoulos Garefalakis Departmet of Mathematics, Uiversity of Crete, 71409 Heraklio, Greece Abstract Let q be a prime power. We cosider the problem

More information

Engineering 323 Beautiful Homework Set 3 1 of 7 Kuszmar Problem 2.51

Engineering 323 Beautiful Homework Set 3 1 of 7 Kuszmar Problem 2.51 Egieerig 33 eautiful Homewor et 3 of 7 Kuszmar roblem.5.5 large departmet store sells sport shirts i three sizes small, medium, ad large, three patters plaid, prit, ad stripe, ad two sleeve legths log

More information

Gregory Carey, 1998 Linear Transformations & Composites - 1. Linear Transformations and Linear Composites

Gregory Carey, 1998 Linear Transformations & Composites - 1. Linear Transformations and Linear Composites Gregory Carey, 1998 Liear Trasformatios & Composites - 1 Liear Trasformatios ad Liear Composites I Liear Trasformatios of Variables Meas ad Stadard Deviatios of Liear Trasformatios A liear trasformatio

More information

AP Calculus BC 2003 Scoring Guidelines Form B

AP Calculus BC 2003 Scoring Guidelines Form B AP Calculus BC Scorig Guidelies Form B The materials icluded i these files are iteded for use by AP teachers for course ad exam preparatio; permissio for ay other use must be sought from the Advaced Placemet

More information

Section 11.3: The Integral Test

Section 11.3: The Integral Test Sectio.3: The Itegral Test Most of the series we have looked at have either diverged or have coverged ad we have bee able to fid what they coverge to. I geeral however, the problem is much more difficult

More information

Theorems About Power Series

Theorems About Power Series Physics 6A Witer 20 Theorems About Power Series Cosider a power series, f(x) = a x, () where the a are real coefficiets ad x is a real variable. There exists a real o-egative umber R, called the radius

More information

Equation of a line. Line in coordinate geometry. Slope-intercept form ( 斜 截 式 ) Intercept form ( 截 距 式 ) Point-slope form ( 點 斜 式 )

Equation of a line. Line in coordinate geometry. Slope-intercept form ( 斜 截 式 ) Intercept form ( 截 距 式 ) Point-slope form ( 點 斜 式 ) Chapter : Liear Equatios Chapter Liear Equatios Lie i coordiate geometr I Cartesia coordiate sstems ( 卡 笛 兒 坐 標 系 統 ), a lie ca be represeted b a liear equatio, i.e., a polomial with degree. But before

More information

INVESTMENT PERFORMANCE COUNCIL (IPC)

INVESTMENT PERFORMANCE COUNCIL (IPC) INVESTMENT PEFOMANCE COUNCIL (IPC) INVITATION TO COMMENT: Global Ivestmet Performace Stadards (GIPS ) Guidace Statemet o Calculatio Methodology The Associatio for Ivestmet Maagemet ad esearch (AIM) seeks

More information

2-3 The Remainder and Factor Theorems

2-3 The Remainder and Factor Theorems - The Remaider ad Factor Theorems Factor each polyomial completely usig the give factor ad log divisio 1 x + x x 60; x + So, x + x x 60 = (x + )(x x 15) Factorig the quadratic expressio yields x + x x

More information

7. Sample Covariance and Correlation

7. Sample Covariance and Correlation 1 of 8 7/16/2009 6:06 AM Virtual Laboratories > 6. Radom Samples > 1 2 3 4 5 6 7 7. Sample Covariace ad Correlatio The Bivariate Model Suppose agai that we have a basic radom experimet, ad that X ad Y

More information

PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY AN ALTERNATIVE MODEL FOR BONUS-MALUS SYSTEM

PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY AN ALTERNATIVE MODEL FOR BONUS-MALUS SYSTEM PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY Physical ad Mathematical Scieces 2015, 1, p. 15 19 M a t h e m a t i c s AN ALTERNATIVE MODEL FOR BONUS-MALUS SYSTEM A. G. GULYAN Chair of Actuarial Mathematics

More information

CHAPTER 7: Central Limit Theorem: CLT for Averages (Means)

CHAPTER 7: Central Limit Theorem: CLT for Averages (Means) CHAPTER 7: Cetral Limit Theorem: CLT for Averages (Meas) X = the umber obtaied whe rollig oe six sided die oce. If we roll a six sided die oce, the mea of the probability distributio is X P(X = x) Simulatio:

More information

Biology 171L Environment and Ecology Lab Lab 2: Descriptive Statistics, Presenting Data and Graphing Relationships

Biology 171L Environment and Ecology Lab Lab 2: Descriptive Statistics, Presenting Data and Graphing Relationships Biology 171L Eviromet ad Ecology Lab Lab : Descriptive Statistics, Presetig Data ad Graphig Relatioships Itroductio Log lists of data are ofte ot very useful for idetifyig geeral treds i the data or the

More information

INFINITE SERIES KEITH CONRAD

INFINITE SERIES KEITH CONRAD INFINITE SERIES KEITH CONRAD. Itroductio The two basic cocepts of calculus, differetiatio ad itegratio, are defied i terms of limits (Newto quotiets ad Riema sums). I additio to these is a third fudametal

More information

5: Introduction to Estimation

5: Introduction to Estimation 5: Itroductio to Estimatio Cotets Acroyms ad symbols... 1 Statistical iferece... Estimatig µ with cofidece... 3 Samplig distributio of the mea... 3 Cofidece Iterval for μ whe σ is kow before had... 4 Sample

More information

1.3 Binomial Coefficients

1.3 Binomial Coefficients 18 CHAPTER 1. COUNTING 1. Biomial Coefficiets I this sectio, we will explore various properties of biomial coefficiets. Pascal s Triagle Table 1 cotais the values of the biomial coefficiets ( ) for 0to

More information

THE ABRACADABRA PROBLEM

THE ABRACADABRA PROBLEM THE ABRACADABRA PROBLEM FRANCESCO CARAVENNA Abstract. We preset a detailed solutio of Exercise E0.6 i [Wil9]: i a radom sequece of letters, draw idepedetly ad uiformly from the Eglish alphabet, the expected

More information

THE REGRESSION MODEL IN MATRIX FORM. For simple linear regression, meaning one predictor, the model is. for i = 1, 2, 3,, n

THE 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 information

NATIONAL SENIOR CERTIFICATE GRADE 12

NATIONAL SENIOR CERTIFICATE GRADE 12 NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P EXEMPLAR 04 MARKS: 50 TIME: 3 hours This questio paper cosists of 8 pages ad iformatio sheet. Please tur over Mathematics/P DBE/04 NSC Grade Eemplar INSTRUCTIONS

More information

Maximum Likelihood Estimators.

Maximum Likelihood Estimators. Lecture 2 Maximum Likelihood Estimators. Matlab example. As a motivatio, let us look at oe Matlab example. Let us geerate a radom sample of size 00 from beta distributio Beta(5, 2). We will lear the defiitio

More information

How Euler Did It. In a more modern treatment, Hardy and Wright [H+W] state this same theorem as. n n+ is perfect.

How Euler Did It. In a more modern treatment, Hardy and Wright [H+W] state this same theorem as. n n+ is perfect. Amicable umbers November 005 How Euler Did It by Ed Sadifer Six is a special umber. It is divisible by, ad 3, ad, i what at first looks like a strage coicidece, 6 = + + 3. The umber 8 shares this remarkable

More information

1. C. The formula for the confidence interval for a population mean is: x t, which was

1. C. The formula for the confidence interval for a population mean is: x t, which was s 1. C. The formula for the cofidece iterval for a populatio mea is: x t, which was based o the sample Mea. So, x is guarateed to be i the iterval you form.. D. Use the rule : p-value

More information

Definition. A variable X that takes on values X 1, X 2, X 3,...X k with respective frequencies f 1, f 2, f 3,...f k has mean

Definition. A variable X that takes on values X 1, X 2, X 3,...X k with respective frequencies f 1, f 2, f 3,...f k has mean 1 Social Studies 201 October 13, 2004 Note: The examples i these otes may be differet tha used i class. However, the examples are similar ad the methods used are idetical to what was preseted i class.

More information

Chapter 7 - Sampling Distributions. 1 Introduction. What is statistics? It consist of three major areas:

Chapter 7 - Sampling Distributions. 1 Introduction. What is statistics? It consist of three major areas: Chapter 7 - Samplig Distributios 1 Itroductio What is statistics? It cosist of three major areas: Data Collectio: samplig plas ad experimetal desigs Descriptive Statistics: umerical ad graphical summaries

More information

4 n. n 1. You shold think of the Ratio Test as a generalization of the Geometric Series Test. For example, if a n ar n is a geometric sequence then

4 n. n 1. You shold think of the Ratio Test as a generalization of the Geometric Series Test. For example, if a n ar n is a geometric sequence then SECTION 2.6 THE RATIO TEST 79 2.6. THE RATIO TEST We ow kow how to hadle series which we ca itegrate (the Itegral Test), ad series which are similar to geometric or p-series (the Compariso Test), but of

More information

Linear Algebra II. 4 Determinants. Notes 4 1st November Definition of determinant

Linear Algebra II. 4 Determinants. Notes 4 1st November Definition of determinant MTH6140 Liear Algebra II Notes 4 1st November 2010 4 Determiats The determiat is a fuctio defied o square matrices; its value is a scalar. It has some very importat properties: perhaps most importat is

More information

5.4 Amortization. Question 1: How do you find the present value of an annuity? Question 2: How is a loan amortized?

5.4 Amortization. Question 1: How do you find the present value of an annuity? Question 2: How is a loan amortized? 5.4 Amortizatio Questio 1: How do you fid the preset value of a auity? Questio 2: How is a loa amortized? Questio 3: How do you make a amortizatio table? Oe of the most commo fiacial istrumets a perso

More information

BINOMIAL EXPANSIONS 12.5. In this section. Some Examples. Obtaining the Coefficients

BINOMIAL EXPANSIONS 12.5. In this section. Some Examples. Obtaining the Coefficients 652 (12-26) 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 information

Ekkehart Schlicht: Economic Surplus and Derived Demand

Ekkehart Schlicht: Economic Surplus and Derived Demand Ekkehart Schlicht: Ecoomic Surplus ad Derived Demad Muich Discussio Paper No. 2006-17 Departmet of Ecoomics Uiversity of Muich Volkswirtschaftliche Fakultät Ludwig-Maximilias-Uiversität Müche Olie at http://epub.ub.ui-mueche.de/940/

More information

Chapter 5 O A Cojecture Of Erdíos Proceedigs NCUR VIII è1994è, Vol II, pp 794í798 Jeærey F Gold Departmet of Mathematics, Departmet of Physics Uiversity of Utah Do H Tucker Departmet of Mathematics Uiversity

More information

Elementary Theory of Russian Roulette

Elementary Theory of Russian Roulette Elemetary Theory of Russia Roulette -iterestig patters of fractios- Satoshi Hashiba Daisuke Miematsu Ryohei Miyadera Itroductio. Today we are goig to study mathematical theory of Russia roulette. If some

More information

Divide and Conquer. Maximum/minimum. Integer Multiplication. CS125 Lecture 4 Fall 2015

Divide and Conquer. Maximum/minimum. Integer Multiplication. CS125 Lecture 4 Fall 2015 CS125 Lecture 4 Fall 2015 Divide ad Coquer We have see oe geeral paradigm for fidig algorithms: the greedy approach. We ow cosider aother geeral paradigm, kow as divide ad coquer. We have already see a

More information

Output Analysis (2, Chapters 10 &11 Law)

Output Analysis (2, Chapters 10 &11 Law) B. Maddah ENMG 6 Simulatio 05/0/07 Output Aalysis (, Chapters 10 &11 Law) Comparig alterative system cofiguratio Sice the output of a simulatio is radom, the comparig differet systems via simulatio should

More information

3 Basic Definitions of Probability Theory

3 Basic Definitions of Probability Theory 3 Basic Defiitios of Probability Theory 3defprob.tex: Feb 10, 2003 Classical probability Frequecy probability axiomatic probability Historical developemet: Classical Frequecy Axiomatic The Axiomatic defiitio

More information

Incremental calculation of weighted mean and variance

Incremental calculation of weighted mean and variance Icremetal calculatio of weighted mea ad variace Toy Fich faf@cam.ac.uk dot@dotat.at Uiversity of Cambridge Computig Service February 009 Abstract I these otes I eplai how to derive formulae for umerically

More information

WHEN 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? 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 information

Key Ideas Section 8-1: Overview hypothesis testing Hypothesis Hypothesis Test Section 8-2: Basics of Hypothesis Testing Null Hypothesis

Key Ideas Section 8-1: Overview hypothesis testing Hypothesis Hypothesis Test Section 8-2: Basics of Hypothesis Testing Null Hypothesis Chapter 8 Key Ideas Hypothesis (Null ad Alterative), Hypothesis Test, Test Statistic, P-value Type I Error, Type II Error, Sigificace Level, Power Sectio 8-1: Overview Cofidece Itervals (Chapter 7) are

More information

Approximating Area under a curve with rectangles. To find the area under a curve we approximate the area using rectangles and then use limits to find

Approximating Area under a curve with rectangles. To find the area under a curve we approximate the area using rectangles and then use limits to find 1.8 Approximatig Area uder a curve with rectagles 1.6 To fid the area uder a curve we approximate the area usig rectagles ad the use limits to fid 1.4 the area. Example 1 Suppose we wat to estimate 1.

More information

Confidence Intervals for One Mean

Confidence Intervals for One Mean Chapter 420 Cofidece Itervals for Oe Mea Itroductio This routie calculates the sample size ecessary to achieve a specified distace from the mea to the cofidece limit(s) at a stated cofidece level for a

More information

Recursion and Recurrences

Recursion and Recurrences Chapter 5 Recursio ad Recurreces 5.1 Growth Rates of Solutios to Recurreces Divide ad Coquer Algorithms Oe of the most basic ad powerful algorithmic techiques is divide ad coquer. Cosider, for example,

More information

CS103X: Discrete Structures Homework 4 Solutions

CS103X: 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 six-figure salaries i whole dollar amouts are there that cotai at least

More information

Your organization has a Class B IP address of 166.144.0.0 Before you implement subnetting, the Network ID and Host ID are divided as follows:

Your organization has a Class B IP address of 166.144.0.0 Before you implement subnetting, the Network ID and Host ID are divided as follows: Subettig Subettig is used to subdivide a sigle class of etwork i to multiple smaller etworks. Example: Your orgaizatio has a Class B IP address of 166.144.0.0 Before you implemet subettig, the Network

More information

Chapter 5 Unit 1. IET 350 Engineering Economics. Learning Objectives Chapter 5. Learning Objectives Unit 1. Annual Amount and Gradient Functions

Chapter 5 Unit 1. IET 350 Engineering Economics. Learning Objectives Chapter 5. Learning Objectives Unit 1. Annual Amount and Gradient Functions Chapter 5 Uit Aual Amout ad Gradiet Fuctios IET 350 Egieerig Ecoomics Learig Objectives Chapter 5 Upo completio of this chapter you should uderstad: Calculatig future values from aual amouts. Calculatig

More information

Section 1.6: Proof by Mathematical Induction

Section 1.6: Proof by Mathematical Induction Sectio.6 Proof by Iductio Sectio.6: Proof by Mathematical Iductio Purpose of Sectio: To itroduce the Priciple of Mathematical Iductio, both weak ad the strog versios, ad show how certai types of theorems

More information

Chapter 7: Confidence Interval and Sample Size

Chapter 7: Confidence Interval and Sample Size Chapter 7: Cofidece Iterval ad Sample Size Learig Objectives Upo successful completio of Chapter 7, you will be able to: Fid the cofidece iterval for the mea, proportio, ad variace. Determie the miimum

More information

Permutations, the Parity Theorem, and Determinants

Permutations, the Parity Theorem, and Determinants 1 Permutatios, the Parity Theorem, ad Determiats Joh A. Guber Departmet of Electrical ad Computer Egieerig Uiversity of Wiscosi Madiso Cotets 1 What is a Permutatio 1 2 Cycles 2 2.1 Traspositios 4 3 Orbits

More information

0.7 0.6 0.2 0 0 96 96.5 97 97.5 98 98.5 99 99.5 100 100.5 96.5 97 97.5 98 98.5 99 99.5 100 100.5

0.7 0.6 0.2 0 0 96 96.5 97 97.5 98 98.5 99 99.5 100 100.5 96.5 97 97.5 98 98.5 99 99.5 100 100.5 Sectio 13 Kolmogorov-Smirov test. Suppose that we have a i.i.d. sample X 1,..., X with some ukow distributio P ad we would like to test the hypothesis that P is equal to a particular distributio P 0, i.e.

More information

Analysis Notes (only a draft, and the first one!)

Analysis Notes (only a draft, and the first one!) Aalysis Notes (oly a draft, ad the first oe!) Ali Nesi Mathematics Departmet Istabul Bilgi Uiversity Kuştepe Şişli Istabul Turkey aesi@bilgi.edu.tr Jue 22, 2004 2 Cotets 1 Prelimiaries 9 1.1 Biary Operatio...........................

More information