19. The FermatEuler Prime Number Theorem


 Linette Adams
 2 years ago
 Views:
Transcription
1 19. The FermtEuler Prime Number Theorem Every prime number of the form 4n 1 cn be written s sum of two squres in only one wy (side from the order of the summnds). This fmous theorem ws discovered bout 1660 by Pierre de Fermt ( ), the gretest French mthemticin of the seventeenth century. It ws not published, however, until 1670, when it ppered, unfortuntely without proof, in the notes of the works of Diophntus, edited by Fermt s son. It is not certin whether or not Fermt hd obtined proof. The first proof of the theorem ws presented lmost 100 yers lter by Leonhrd Euler in his tretise "Demonstrtio theoremtis Fermtini, omnem numerum primum forme 4n 1 esse summm duorum qudrtorum" (Novi Commentrii Acdemie Petropolitne d nnos , vol. V), fter yers of fruitless ttempts t its proof. Tody there re severl proofs of the theorem. The following one is noted for its simplicity. It does however use fir number of results from number theory, some of which will be need in No. s well. In the following, ll vribles represent integers (whole numbers). Definition Two numbers nd b (ccording to Guss), re congruent mod m, m being positive integer, written Notes Theorem 1. q bmod m nd red is congruent to b mod m, if their difference is divisible by m, i.e., m Ÿ"b. Every number is congruent to its reminder, or residue, when divided by m. For exmple 65 q mod 7, but lso 65 q 9mod 7, thinking of Conventionl or common residues re nonnegtive integers less thn or equl to m. The set 0, 1,,...,m is complete residue system modm, becuse it hs m elements no two of which re congruent mod m, (nd every integer is congruent modm to one of its members). A miniml (or lest) residue mod m is residue whose bsolute vlue is less thn or equl to m. For instnce " is lest residue of 89mod 13, since. The set of lest residues mod13 is 89 q "mod13 nd " 13 "6,"5,...,,0, 1,...,5, 6. A set of lest residues mod6 is ",,0,1,, 3 s is "3,",,0, 1,. A set of lest residues mod m is complete residue system. 1. q mod m for ll.. If q bmod m, then b q modm. 3. If q bmod m nd b q c mod m, then q c mod m. 1
2 4. If two numbers re congruent to third, they re lso congruent to ech other. (This follows from nd 3.) 5. If q bmod m nd c q dmod m, then c q b dmodm, " c q b " dmodm, nd c q bdmod m. [If b gm nd c dhm, then c bd Ÿbhcgghm m.] 6. If q bmod m, then g q bg mod m for ny integer g, i.e., congruence cn be multiplied by ny number. 7. If g, g b nd gcdÿg, m 1, i.e., g nd m re reltively prime, then we cn divide the congruence q b modm by g resulting in g q b g modm. For exmple from 49 q 14mod 5, it follows tht 7 q mod If S 1,,..., m is complete residue system mod m, nd gcdÿ,m 1, then x q bmod m hs unique solution (or root) in S. [gcdÿ, m 1 there re integers s nd t such tht s mt 1 or s q 1mod m. Then sx q sbmod m, nd x q sbmodm. Furthermore sb is congruent to just one element of S.] 9. If S 1,,..., m is complete residue system mod m, nd gcdÿ,m 1, then so is T 1,,..., m. [ i q j modm i q j mod m by 7. Thus the elements of T re distinct nd no two re congruent mod m. Ech i is congruent to some j mod m since x q i mod m hs unique solution j by 8. Hence every integer n is congruent to some element in S nd then lso in T.] We lso need some results bout qudrtic residues. Definition. is qudrtic residue (QR)modm if gcdÿ, m 1 nd x q modm for some integer x. If there is no such x, then is qudrtic nonresidue (QNR). For exmple, 1 is QR mod 13, since 8 q 1mod 13, while is QNR mod 3, since x q mod3 hs no solution. Ÿ 1 if is QR mod p nd Ÿ p if is Nottion. If gcdÿ, p 1, p prime, p QNR mod p. Ÿ p is the Legendre symbol , 3. Throughout the following, p denotes n odd prime number. Theorem. There re totl of P mutully incongruent QRs nd just s mny mutully incongruent QNRs mod p. The QRs re 1,,...,P mod p. No two of (the QRs) 1,,...,P re congruent modp, becuse with x, y 1,,...,P, x q y mod p p Ÿxy Ÿx " y, but this cn t hppen since 0 xy, x " y p. This give us P mutully incongruent QRs. No new QRs re obtined going beyond P. Indeed, consider ŸPh mod p. Let k t P be such tht P h q k modp (i.e., k is the lest residue of Phmod p). Then ŸPh q k mod p,
3 one of the QRs 1,,...,P mod p. Since there re (side from 0mod p) P mutully incongruent numbers modp, there must be totl of P mutully incongruent QNRs mod p. R Theorem 3. The product of two QRs nd the product of two QNRs is QR; the product of QR nd QNR is QNR. Let r 1 nd r be QRs, nd n 1 nd n be QNRs mod p. 1. From 1 q r 1, q r, we obtin Ÿ 1 q r 1 r mod p, nd thus r 1 r is QR.. The P numbers 1,,...,P,n 1 1,n 1,...,n 1 P re mutully incongruent modp. Since the first P of these numbers re QRs mod p, nd since only P QRs exist, the P numbers n 1 1, n 1,...,n 1 P must be QNRs, i.e., n i r j is QNR. 3. The P numbers n 1 1, n 1,...,n 1 P, n 1 n 1, n 1 n,...,n 1 n P re mutully incongruent mod p. The first P of them, by, re QNRs; thus the others must be QRs, mong them n 1 n. R Theorem 4. if if q P mod p. In terms of the Legendre symbol Let gcdÿ,p 1. Then is QR mod p if q P 1 modp, nd is QNR mod p Ÿ p q mod p. For ny x S 1,,...,p " 1, there is unique y S such tht xy q mod p. Pick x 1 rbitrrily in S, nd let y 1 S be tht number such tht x 1 y 1 q mod p. Then pick x in S different from x 1 nd y 1, nd let y be tht number so tht x y q mod p. Continue in this mnner until ll the numbers in S hve been used. If is QR, then for some v, x v y v, i.e. x v q mod p. The sme is true for x6 p " x v, nd x v nd x6 re the only solutions to x q modp in S. Furthermore x v x6 x v p " x v q " modp. Multiply ll the P " 1 congruences xy q mod p with this lst one to get Ÿ! q " P mod p. Note tht when 1 (clerly QR), we hve Wilson s Theorem Ÿ! q mod p. By Wilson s Theorem, we conclude tht if is QR, then q P 1 modp. If is QNR, then there re exctly P congruences xy q modp, nd x nd y re never equl. Multiply them ll together to get Ÿ! q P mod p, nd by Wilson s Theorem, q P mod p. R Corollry. p p Ÿ Ÿ Ÿ q Ÿ equl (since p 4 ).. R mod p, nd since both sides re o1, it follows tht they re in fct Theorem 5. (Euler) is QR mod p if nd only if p q 1mod 4. If p q 1 mod4, then p 1 4n, n is even, nd p Ÿ Ÿ n 1. 3
4 Ifp q 3 mod4, then is odd, nd p Ÿ Ÿ. R Thus, x 1 q 0mod p hs solution if nd only if p is on the form 4n 1. Theorem 6. If p Ÿ b, but p 4 nd p 4 b, then p c d for some integers c nd d. (This with Theorem 5 shows tht only those primes of the form 4n 1 cn be written s sums of squres.) Let b pf. If f 1, we re done, so ssume f 1. Next, without loss of generlity, we my ssume tht f p. [If this is not the cse, simply replce nd b by their lest residues 0 nd b 0 mod p. Then 0 b 0 pf 0, nd since 0, b 0 p, pf 0 p p p, nd f 0 p. For exmple , but 50 q 1mod 61, nd Ÿ with 61.] If ) nd * re lest residues for nd b modf respectively, then ) * ff 1 where f 1 t 1 f, nd then Ÿ b Ÿ) * Ÿpf Ÿff 1 pf f 1, or Ÿ) b* Ÿ* " b) pf f 1. Since ) b* q b q 0 modf, nd * " b) q b " b q 0 modf, we cn divide this lst equlity through by f to get 1 b 1 pf 1, where f 1 t 1 f. Now f 1 p 0, for otherwise ) * 0, nd f nd f b, sy mf,b nf, nd then b Ÿmf Ÿnf pf, whence p Ÿm n f, nd f 1, contrry to f 1. If f 1 1, 1 b 1 p provides representtion of p s sum of squres. If f 1 1, repet this procedure strting with 1 b 1 pf 1 to get b pf with 0 f t 1 f 1, etc. This method of constructing new equtions with ever decresing fs continues until 1 ppers (which it must). This lst eqution gives representtion of p s sum of two squres. R For exmple: Ÿ Ÿ Ÿ Ÿ Theorem A prime number q of the form 4n 3 cnnot be written s sum of two squres.. Every prime number p of the form 4n 1 cn be written s sum of two squres in exctly one wy (up to the order in which the summnds re written). 4
5 1. Suppose tht b q. Then b q " modq. b is certinly QR mod q (since it s the squre of b). On the other hnd is QNR by Theorem 5, is certinly QR, nd Theorem 3 implies tht " is QNR. This mkes b both QR nd QNR, contrdiction.. In this cse, Theorem 5 gurntees the existence of x so tht p Ÿx 1. Then Theorem 6 implies tht p b for some positive integers nd b. Assume tht there is second representtion p A B. Then p Ÿ b ŸA B ŸAoBb ŸAb#B. Since p divides A p " b p A Ÿ b " b ŸA B A " B b ŸA Bb ŸA"Bb, p ŸABb or p ŸA " Bb. Since ABb 0 nd AbB 0, we conclude tht either or ABb p nd t the sme time Ab"B 0 AbB p nd t the sme time A"Bb 0 Note. A nd either A b B or A B b. The first of these equtions implies tht A A nd B b while the second implies tht A b B b A B B b A B b 1, nd 1, nd A b nd B. Thus the representtion of p s sum of two squres is unique up to the order in which the squres re written. R B b A B B b A kb nd kb for some k (not necessrily n integer). Then Ÿk1 b Ÿk1 B b. 5
Solutions to Section 1
Solutions to Section Exercise. Show tht nd. This follows from the fct tht mx{, } nd mx{, } Exercise. Show tht = { if 0 if < 0 Tht is, the bsolute vlue function is piecewise defined function. Grph this
More informationMATH 150 HOMEWORK 4 SOLUTIONS
MATH 150 HOMEWORK 4 SOLUTIONS Section 1.8 Show tht the product of two of the numbers 65 1000 8 2001 + 3 177, 79 1212 9 2399 + 2 2001, nd 24 4493 5 8192 + 7 1777 is nonnegtive. Is your proof constructive
More informationPolynomial Functions. Polynomial functions in one variable can be written in expanded form as ( )
Polynomil Functions Polynomil functions in one vrible cn be written in expnded form s n n 1 n 2 2 f x = x + x + x + + x + x+ n n 1 n 2 2 1 0 Exmples of polynomils in expnded form re nd 3 8 7 4 = 5 4 +
More informationMath 135 Circles and Completing the Square Examples
Mth 135 Circles nd Completing the Squre Exmples A perfect squre is number such tht = b 2 for some rel number b. Some exmples of perfect squres re 4 = 2 2, 16 = 4 2, 169 = 13 2. We wish to hve method for
More informationand thus, they are similar. If k = 3 then the Jordan form of both matrices is
Homework ssignment 11 Section 7. pp. 24925 Exercise 1. Let N 1 nd N 2 be nilpotent mtrices over the field F. Prove tht N 1 nd N 2 re similr if nd only if they hve the sme miniml polynomil. Solution: If
More informationExponentiation: Theorems, Proofs, Problems Pre/Calculus 11, Veritas Prep.
Exponentition: Theorems, Proofs, Problems Pre/Clculus, Verits Prep. Our Exponentition Theorems Theorem A: n+m = n m Theorem B: ( n ) m = nm Theorem C: (b) n = n b n ( ) n n Theorem D: = b b n Theorem E:
More informationExample 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers.
2 Rtionl Numbers Integers such s 5 were importnt when solving the eqution x+5 = 0. In similr wy, frctions re importnt for solving equtions like 2x = 1. Wht bout equtions like 2x + 1 = 0? Equtions of this
More informationGeneralized Inverses: How to Invert a NonInvertible Matrix
Generlized Inverses: How to Invert NonInvertible Mtrix S. Swyer September 7, 2006 rev August 6, 2008. Introduction nd Definition. Let A be generl m n mtrix. Then nturl question is when we cn solve Ax
More informationChapter 6 Solving equations
Chpter 6 Solving equtions Defining n eqution 6.1 Up to now we hve looked minly t epressions. An epression is n incomplete sttement nd hs no equl sign. Now we wnt to look t equtions. An eqution hs n = sign
More informationFactoring Polynomials
Fctoring Polynomils Some definitions (not necessrily ll for secondry school mthemtics): A polynomil is the sum of one or more terms, in which ech term consists of product of constnt nd one or more vribles
More informationQuadratic Equations. Math 99 N1 Chapter 8
Qudrtic Equtions Mth 99 N1 Chpter 8 1 Introduction A qudrtic eqution is n eqution where the unknown ppers rised to the second power t most. In other words, it looks for the vlues of x such tht second degree
More informationSPECIAL PRODUCTS AND FACTORIZATION
MODULE  Specil Products nd Fctoriztion 4 SPECIAL PRODUCTS AND FACTORIZATION In n erlier lesson you hve lernt multipliction of lgebric epressions, prticulrly polynomils. In the study of lgebr, we come
More informationnot to be republished NCERT POLYNOMIALS CHAPTER 2 (A) Main Concepts and Results (B) Multiple Choice Questions
POLYNOMIALS (A) Min Concepts nd Results Geometricl mening of zeroes of polynomil: The zeroes of polynomil p(x) re precisely the xcoordintes of the points where the grph of y = p(x) intersects the xxis.
More informationBinary Representation of Numbers Autar Kaw
Binry Representtion of Numbers Autr Kw After reding this chpter, you should be ble to: 1. convert bse rel number to its binry representtion,. convert binry number to n equivlent bse number. In everydy
More informationNet Change and Displacement
mth 11, pplictions motion: velocity nd net chnge 1 Net Chnge nd Displcement We hve seen tht the definite integrl f (x) dx mesures the net re under the curve y f (x) on the intervl [, b] Any prt of the
More information11. Fourier series. sin mx cos nx dx = 0 for any m, n, sin 2 mx dx = π.
. Fourier series Summry of the bsic ides The following is quick summry of the introductory tretment of Fourier series in MATH. We consider function f with period π, tht is, stisfying f(x + π) = f(x) for
More informationPROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY
MAT 0630 INTERNET RESOURCES, REVIEW OF CONCEPTS AND COMMON MISTAKES PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY Contents 1. ACT Compss Prctice Tests 1 2. Common Mistkes 2 3. Distributive
More informationFUNCTIONS AND EQUATIONS. xεs. The simplest way to represent a set is by listing its members. We use the notation
FUNCTIONS AND EQUATIONS. SETS AND SUBSETS.. Definition of set. A set is ny collection of objects which re clled its elements. If x is n element of the set S, we sy tht x belongs to S nd write If y does
More informationIn this section make precise the idea of a matrix inverse and develop a method to find the inverse of a given square matrix when it exists.
Mth 52 Sec S060/S0602 Notes Mtrices IV 5 Inverse Mtrices 5 Introduction In our erlier work on mtrix multipliction, we sw the ide of the inverse of mtrix Tht is, for squre mtrix A, there my exist mtrix
More informationSection 74 Translation of Axes
62 7 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY Section 74 Trnsltion of Aes Trnsltion of Aes Stndrd Equtions of Trnslted Conics Grphing Equtions of the Form A 2 C 2 D E F 0 Finding Equtions of Conics In the
More information4.11 Inner Product Spaces
314 CHAPTER 4 Vector Spces 9. A mtrix of the form 0 0 b c 0 d 0 0 e 0 f g 0 h 0 cnnot be invertible. 10. A mtrix of the form bc d e f ghi such tht e bd = 0 cnnot be invertible. 4.11 Inner Product Spces
More information9 CONTINUOUS DISTRIBUTIONS
9 CONTINUOUS DISTIBUTIONS A rndom vrible whose vlue my fll nywhere in rnge of vlues is continuous rndom vrible nd will be ssocited with some continuous distribution. Continuous distributions re to discrete
More informationExponents base exponent power exponentiation
Exonents We hve seen counting s reeted successors ddition s reeted counting multiliction s reeted ddition so it is nturl to sk wht we would get by reeting multiliction. For exmle, suose we reetedly multily
More informationSection A4 Rational Expressions: Basic Operations
A Appendi A A BASIC ALGEBRA REVIEW 7. Construction. A rectngulr opentopped bo is to be constructed out of 9 by 6inch sheets of thin crdbord by cutting inch squres out of ech corner nd bending the
More information4.0 5Minute Review: Rational Functions
mth 130 dy 4: working with limits 1 40 5Minute Review: Rtionl Functions DEFINITION A rtionl function 1 is function of the form y = r(x) = p(x) q(x), 1 Here the term rtionl mens rtio s in the rtio of two
More informationMathematics. Vectors. hsn.uk.net. Higher. Contents. Vectors 128 HSN23100
hsn.uk.net Higher Mthemtics UNIT 3 OUTCOME 1 Vectors Contents Vectors 18 1 Vectors nd Sclrs 18 Components 18 3 Mgnitude 130 4 Equl Vectors 131 5 Addition nd Subtrction of Vectors 13 6 Multipliction by
More informationSequences and Series
Centre for Eduction in Mthemtics nd Computing Euclid eworkshop # 5 Sequences nd Series c 014 UNIVERSITY OF WATERLOO While the vst mjority of Euclid questions in this topic re use formule for rithmetic
More informationArc Length. P i 1 P i (1) L = lim. i=1
Arc Length Suppose tht curve C is defined by the eqution y = f(x), where f is continuous nd x b. We obtin polygonl pproximtion to C by dividing the intervl [, b] into n subintervls with endpoints x, x,...,x
More informationIntroduction to Mathematical Reasoning, Saylor 111
Frction versus rtionl number. Wht s the difference? It s not n esy question. In fct, the difference is somewht like the difference between set of words on one hnd nd sentence on the other. A symbol is
More informationThe Chain Rule. rf dx. t t lim " (x) dt " (0) dx. df dt = df. dt dt. f (r) = rf v (1) df dx
The Chin Rule The Chin Rule In this section, we generlize the chin rule to functions of more thn one vrible. In prticulr, we will show tht the product in the singlevrible chin rule extends to n inner
More informationUniform convergence and its consequences
Uniform convergence nd its consequences The following issue is centrl in mthemtics: On some domin D, we hve sequence of functions {f n }. This mens tht we relly hve n uncountble set of ordinry sequences,
More informationLecture 3 Basic Probability and Statistics
Lecture 3 Bsic Probbility nd Sttistics The im of this lecture is to provide n extremely speedy introduction to the probbility nd sttistics which will be needed for the rest of this lecture course. The
More information5.2. LINE INTEGRALS 265. Let us quickly review the kind of integrals we have studied so far before we introduce a new one.
5.2. LINE INTEGRALS 265 5.2 Line Integrls 5.2.1 Introduction Let us quickly review the kind of integrls we hve studied so fr before we introduce new one. 1. Definite integrl. Given continuous relvlued
More information4. Greed. Algorithm Design by Éva Tardos and Jon Kleinberg Slides by Kevin Wayne (modified by Neil Rhodes)
4 Greed Greed is good Greed is right Greed works Greed clrifies, cuts through, nd cptures the essence of the evolutionry spirit  Gordon Gecko (Michel Dougls) Algorithm Design by Év Trdos nd Jon Kleinberg
More informationAlgorithms Chapter 4 Recurrences
Algorithms Chpter 4 Recurrences Outline The substitution method The recursion tree method The mster method Instructor: Ching Chi Lin 林清池助理教授 chingchilin@gmilcom Deprtment of Computer Science nd Engineering
More information1 Numerical Solution to Quadratic Equations
cs42: introduction to numericl nlysis 09/4/0 Lecture 2: Introduction Prt II nd Solving Equtions Instructor: Professor Amos Ron Scribes: Yunpeng Li, Mrk Cowlishw Numericl Solution to Qudrtic Equtions Recll
More informationMODULE 3. 0, y = 0 for all y
Topics: Inner products MOULE 3 The inner product of two vectors: The inner product of two vectors x, y V, denoted by x, y is (in generl) complex vlued function which hs the following four properties: i)
More informationWritten Homework 6 Solutions
Written Homework 6 Solutions Section.10 0. Explin in terms of liner pproximtions or differentils why the pproximtion is resonble: 1.01) 6 1.06 Solution: First strt by finding the liner pproximtion of f
More informationUse Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions.
Lerning Objectives Loci nd Conics Lesson 3: The Ellipse Level: Preclculus Time required: 120 minutes In this lesson, students will generlize their knowledge of the circle to the ellipse. The prmetric nd
More informationCurve Sketching. 96 Chapter 5 Curve Sketching
96 Chpter 5 Curve Sketching 5 Curve Sketching A B A B A Figure 51 Some locl mximum points (A) nd minimum points (B) If (x, f(x)) is point where f(x) reches locl mximum or minimum, nd if the derivtive of
More informationGraphs on Logarithmic and Semilogarithmic Paper
0CH_PHClter_TMSETE_ 3//00 :3 PM Pge Grphs on Logrithmic nd Semilogrithmic Pper OBJECTIVES When ou hve completed this chpter, ou should be ble to: Mke grphs on logrithmic nd semilogrithmic pper. Grph empiricl
More informationFormal Languages and Automata Exam
Forml Lnguges nd Automt Exm Fculty of Computers & Informtion Deprtment: Computer Science Grde: Third Course code: CSC 34 Totl Mrk: 8 Dte: 23//2 Time: 3 hours Answer the following questions: ) Consider
More informationAssuming all values are initially zero, what are the values of A and B after executing this Verilog code inside an always block? C=1; A <= C; B = C;
B26 Appendix B The Bsics of Logic Design Check Yourself ALU n [Arthritic Logic Unit or (rre) Arithmetic Logic Unit] A rndomnumer genertor supplied s stndrd with ll computer systems Stn KellyBootle,
More informationExample A rectangular box without lid is to be made from a square cardboard of sides 18 cm by cutting equal squares from each corner and then folding
1 Exmple A rectngulr box without lid is to be mde from squre crdbord of sides 18 cm by cutting equl squres from ech corner nd then folding up the sides. 1 Exmple A rectngulr box without lid is to be mde
More information10.5 Graphing Quadratic Functions
0.5 Grphing Qudrtic Functions Now tht we cn solve qudrtic equtions, we wnt to lern how to grph the function ssocited with the qudrtic eqution. We cll this the qudrtic function. Grphs of Qudrtic Functions
More informationTriangles, Altitudes, and Area Instructor: Natalya St. Clair
Tringle, nd ltitudes erkeley Mth ircles 015 Lecture Notes Tringles, ltitudes, nd re Instructor: Ntly St. lir *Note: This M session is inspired from vriety of sources, including wesomemth, reteem Mth Zoom,
More informationHomework 3 Solutions
CS 341: Foundtions of Computer Science II Prof. Mrvin Nkym Homework 3 Solutions 1. Give NFAs with the specified numer of sttes recognizing ech of the following lnguges. In ll cses, the lphet is Σ = {,1}.
More informationAppendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered:
Appendi D: Completing the Squre nd the Qudrtic Formul Fctoring qudrtic epressions such s: + 6 + 8 ws one of the topics introduced in Appendi C. Fctoring qudrtic epressions is useful skill tht cn help you
More informationMathematics Higher Level
Mthemtics Higher Level Higher Mthemtics Exmintion Section : The Exmintion Mthemtics Higher Level. Structure of the exmintion pper The Higher Mthemtics Exmintion is divided into two ppers s detiled below:
More information200506 Second Term MAT2060B 1. Supplementary Notes 3 Interchange of Differentiation and Integration
Source: http://www.mth.cuhk.edu.hk/~mt26/mt26b/notes/notes3.pdf 256 Second Term MAT26B 1 Supplementry Notes 3 Interchnge of Differentition nd Integrtion The theme of this course is bout vrious limiting
More informationSquare Roots Teacher Notes
Henri Picciotto Squre Roots Techer Notes This unit is intended to help students develop n understnding of squre roots from visul / geometric point of view, nd lso to develop their numer sense round this
More informationTreatment Spring Late Summer Fall 0.10 5.56 3.85 0.61 6.97 3.01 1.91 3.01 2.13 2.99 5.33 2.50 1.06 3.53 6.10 Mean = 1.33 Mean = 4.88 Mean = 3.
The nlysis of vrince (ANOVA) Although the ttest is one of the most commonly used sttisticl hypothesis tests, it hs limittions. The mjor limittion is tht the ttest cn be used to compre the mens of only
More informationChapter 9: Quadratic Equations
Chpter 9: Qudrtic Equtions QUADRATIC EQUATIONS DEFINITION + + c = 0,, c re constnts (generlly integers) ROOTS Synonyms: Solutions or Zeros Cn hve 0, 1, or rel roots Consider the grph of qudrtic equtions.
More information4: RIEMANN SUMS, RIEMANN INTEGRALS, FUNDAMENTAL THEOREM OF CALCULUS
4: RIEMA SUMS, RIEMA ITEGRALS, FUDAMETAL THEOREM OF CALCULUS STEVE HEILMA Contents 1. Review 1 2. Riemnn Sums 2 3. Riemnn Integrl 3 4. Fundmentl Theorem of Clculus 7 5. Appendix: ottion 10 1. Review Theorem
More informationLINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES
LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES DAVID WEBB CONTENTS Liner trnsformtions 2 The representing mtrix of liner trnsformtion 3 3 An ppliction: reflections in the plne 6 4 The lgebr of
More informationOn the Meaning of Regression Coefficients for Categorical and Continuous Variables: Model I and Model II; Effect Coding and Dummy Coding
Dt_nlysisclm On the Mening of Regression for tegoricl nd ontinuous Vribles: I nd II; Effect oding nd Dummy oding R Grdner Deprtment of Psychology This describes the simple cse where there is one ctegoricl
More informationOperations with Polynomials
38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: Write polynomils in stndrd form nd identify the leding coefficients nd degrees of polynomils Add nd subtrct polynomils Multiply
More informationLecture 3 Gaussian Probability Distribution
Lecture 3 Gussin Probbility Distribution Introduction l Gussin probbility distribution is perhps the most used distribution in ll of science. u lso clled bell shped curve or norml distribution l Unlike
More informationTHE RATIONAL NUMBERS CHAPTER
CHAPTER THE RATIONAL NUMBERS When divided by b is not n integer, the quotient is frction.the Bbylonins, who used number system bsed on 60, epressed the quotients: 0 8 s 0 60 insted of 8 s 7 60,600 0 insted
More informationIntegration by Substitution
Integrtion by Substitution Dr. Philippe B. Lvl Kennesw Stte University August, 8 Abstrct This hndout contins mteril on very importnt integrtion method clled integrtion by substitution. Substitution is
More informationDETERMINANTS. ] of order n, we can associate a number (real or complex) called determinant of the matrix A, written as det A, where a ij. = ad bc.
Chpter 4 DETERMINANTS 4 Overview To every squre mtrix A = [ ij ] of order n, we cn ssocite number (rel or complex) clled determinnt of the mtrix A, written s det A, where ij is the (i, j)th element of
More informationMATLAB: Mfiles; Numerical Integration Last revised : March, 2003
MATLAB: Mfiles; Numericl Integrtion Lst revised : Mrch, 00 Introduction to Mfiles In this tutoril we lern the bsics of working with Mfiles in MATLAB, so clled becuse they must use.m for their filenme
More informationQuadratic Equations  1
Alger Module A60 Qudrtic Equtions  1 Copyright This puliction The Northern Alert Institute of Technology 00. All Rights Reserved. LAST REVISED Novemer, 008 Qudrtic Equtions  1 Sttement of Prerequisite
More informationThe Quadratic Formula and the Discriminant
99 The Qudrtic Formul nd the Discriminnt Objectives Solve qudrtic equtions by using the Qudrtic Formul. Determine the number of solutions of qudrtic eqution by using the discriminnt. Vocbulry discriminnt
More informationAe2 Mathematics : Fourier Series
Ae Mthemtics : Fourier Series J. D. Gibbon (Professor J. D Gibbon, Dept of Mthemtics j.d.gibbon@ic.c.uk http://www.imperil.c.uk/ jdg These notes re not identicl wordforword with my lectures which will
More informationSect 8.3 Triangles and Hexagons
13 Objective 1: Sect 8.3 Tringles nd Hexgons Understnding nd Clssifying Different Types of Polygons. A Polygon is closed twodimensionl geometric figure consisting of t lest three line segments for its
More informationLecture 15  Curve Fitting Techniques
Lecture 15  Curve Fitting Techniques Topics curve fitting motivtion liner regression Curve fitting  motivtion For root finding, we used given function to identify where it crossed zero where does fx
More informationa 2 + b 2 = c 2. There are many proofs of this theorem. An elegant one only requires that we know that the area of a square of side L is L 2
Pythgors Pythgors A right tringle, suh s shown in the figure elow, hs one 90 ngle. The long side of length is the hypotenuse. The short leg (or thetus) hs length, nd the long leg hs length. The theorem
More informationVectors 2. 1. Recap of vectors
Vectors 2. Recp of vectors Vectors re directed line segments  they cn be represented in component form or by direction nd mgnitude. We cn use trigonometry nd Pythgors theorem to switch between the forms
More informationSquare & Square Roots
Squre & Squre Roots Squre : If nuber is ultiplied by itself then the product is the squre of the nuber. Thus the squre of is x = eg. x x Squre root: The squre root of nuber is one of two equl fctors which
More information5.1 SecondOrder linear PDE
5.1 SecondOrder liner PDE Consider secondorder liner PDE L[u] = u xx + 2bu xy + cu yy + du x + eu y + fu = g, (x,y) U (5.1) for n unknown function u of two vribles x nd y. The functions,b nd c re ssumed
More informationSection 2.3. Motion Along a Curve. The Calculus of Functions of Several Variables
The Clculus of Functions of Severl Vribles Section 2.3 Motion Along Curve Velocity ccelertion Consider prticle moving in spce so tht its position t time t is given by x(t. We think of x(t s moving long
More informationRegular Sets and Expressions
Regulr Sets nd Expressions Finite utomt re importnt in science, mthemtics, nd engineering. Engineers like them ecuse they re super models for circuits (And, since the dvent of VLSI systems sometimes finite
More information2.4 Circular Waveguide
.4 Circulr Wveguide y x Figure.5: A circulr wveguide of rdius. For circulr wveguide of rdius (Fig..5, we cn perform the sme sequence of steps in cylindricl coordintes s we did in rectngulr coordintes to
More informationMath Review 1. , where α (alpha) is a constant between 0 and 1, is one specific functional form for the general production function.
Mth Review Vribles, Constnts nd Functions A vrible is mthemticl bbrevition for concept For emple in economics, the vrible Y usully represents the level of output of firm or the GDP of n economy, while
More informationRIGHT TRIANGLES AND THE PYTHAGOREAN TRIPLETS
RIGHT TRIANGLES AND THE PYTHAGOREAN TRIPLETS Known for over 500 yers is the fct tht the sum of the squres of the legs of right tringle equls the squre of the hypotenuse. Tht is +b c. A simple proof is
More informationEcon 4721 Money and Banking Problem Set 2 Answer Key
Econ 472 Money nd Bnking Problem Set 2 Answer Key Problem (35 points) Consider n overlpping genertions model in which consumers live for two periods. The number of people born in ech genertion grows in
More informationTwo special Righttriangles 1. The
Mth Right Tringle Trigonometry Hndout B (length of )  c  (length of side ) (Length of side to ) Pythgoren s Theorem: for tringles with right ngle ( side + side = ) + = c Two specil Righttringles. The
More informationMatrix Algebra CHAPTER 1 PREAMBLE 1.1 MATRIX ALGEBRA
CHAPTER 1 Mtrix Algebr PREAMBLE Tody, the importnce of mtrix lgebr is of utmost importnce in the field of physics nd engineering in more thn one wy, wheres before 1925, the mtrices were rrely used by the
More informationBasic Math Review. Numbers. Important Properties. Absolute Value PROPERTIES OF ADDITION NATURAL NUMBERS {1, 2, 3, 4, 5, }
ƒ Bsic Mth Review Numers NATURAL NUMBERS {1,, 3, 4, 5, } WHOLE NUMBERS {0, 1,, 3, 4, } INTEGERS {, 3,, 1, 0, 1,, } The Numer Line 5 4 3 1 0 1 3 4 5 Negtive integers Positive integers RATIONAL NUMBERS All
More informationBasic Analysis of Autarky and Free Trade Models
Bsic Anlysis of Autrky nd Free Trde Models AUTARKY Autrky condition in prticulr commodity mrket refers to sitution in which country does not engge in ny trde in tht commodity with other countries. Consequently
More informationBabylonian Method of Computing the Square Root: Justifications Based on Fuzzy Techniques and on Computational Complexity
Bbylonin Method of Computing the Squre Root: Justifictions Bsed on Fuzzy Techniques nd on Computtionl Complexity Olg Koshelev Deprtment of Mthemtics Eduction University of Texs t El Pso 500 W. University
More informationSolving Linear Equations  Formulas
1. Solving Liner Equtions  Formuls Ojective: Solve liner formuls for given vrile. Solving formuls is much like solving generl liner equtions. The only difference is we will hve severl vriles in the prolem
More informationLecture 1: Introduction to Economics
E111 Introduction to Economics ontct detils: E111 Introduction to Economics Ginluigi Vernsc Room: 5B.217 Office hours: Wednesdy 2pm to 4pm Emil: gvern@essex.c.uk Lecture 1: Introduction to Economics I
More informationDlNBVRGH + Sickness Absence Monitoring Report. Executive of the Council. Purpose of report
DlNBVRGH + + THE CITY OF EDINBURGH COUNCIL Sickness Absence Monitoring Report Executive of the Council 8fh My 4 I.I...3 Purpose of report This report quntifies the mount of working time lost s result of
More informationQUADRATURE METHODS. July 19, 2011. Kenneth L. Judd. Hoover Institution
QUADRATURE METHODS Kenneth L. Judd Hoover Institution July 19, 2011 1 Integrtion Most integrls cnnot be evluted nlyticlly Integrls frequently rise in economics Expected utility Discounted utility nd profits
More information9.3. The Scalar Product. Introduction. Prerequisites. Learning Outcomes
The Sclr Product 9.3 Introduction There re two kinds of multipliction involving vectors. The first is known s the sclr product or dot product. This is soclled becuse when the sclr product of two vectors
More informationAlgebra Review. How well do you remember your algebra?
Algebr Review How well do you remember your lgebr? 1 The Order of Opertions Wht do we men when we write + 4? If we multiply we get 6 nd dding 4 gives 10. But, if we dd + 4 = 7 first, then multiply by then
More informationBayesian Updating with Continuous Priors Class 13, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom
Byesin Updting with Continuous Priors Clss 3, 8.05, Spring 04 Jeremy Orloff nd Jonthn Bloom Lerning Gols. Understnd prmeterized fmily of distriutions s representing continuous rnge of hypotheses for the
More informationLectures 8 and 9 1 Rectangular waveguides
1 Lectures 8 nd 9 1 Rectngulr wveguides y b x z Consider rectngulr wveguide with 0 < x b. There re two types of wves in hollow wveguide with only one conductor; Trnsverse electric wves
More informationFor a solid S for which the cross sections vary, we can approximate the volume using a Riemann sum. A(x i ) x. i=1.
Volumes by Disks nd Wshers Volume of cylinder A cylinder is solid where ll cross sections re the sme. The volume of cylinder is A h where A is the re of cross section nd h is the height of the cylinder.
More informationTests for One Poisson Mean
Chpter 412 Tests for One Poisson Men Introduction The Poisson probbility lw gives the probbility distribution of the number of events occurring in specified intervl of time or spce. The Poisson distribution
More informationLecture 5. Inner Product
Lecture 5 Inner Product Let us strt with the following problem. Given point P R nd line L R, how cn we find the point on the line closest to P? Answer: Drw line segment from P meeting the line in right
More information4 Approximations. 4.1 Background. D. Levy
D. Levy 4 Approximtions 4.1 Bckground In this chpter we re interested in pproximtion problems. Generlly speking, strting from function f(x) we would like to find different function g(x) tht belongs to
More informationReasoning to Solve Equations and Inequalities
Lesson4 Resoning to Solve Equtions nd Inequlities In erlier work in this unit, you modeled situtions with severl vriles nd equtions. For exmple, suppose you were given usiness plns for concert showing
More informationLinear Equations in Two Variables
Liner Equtions in Two Vribles In this chpter, we ll use the geometry of lines to help us solve equtions. Liner equtions in two vribles If, b, ndr re rel numbers (nd if nd b re not both equl to 0) then
More informationModule Summary Sheets. C3, Methods for Advanced Mathematics (Version B reference to new book) Topic 2: Natural Logarithms and Exponentials
MEI Mthemtics in Ection nd Instry Topic : Proof MEI Structured Mthemtics Mole Summry Sheets C, Methods for Anced Mthemtics (Version B reference to new book) Topic : Nturl Logrithms nd Eponentils Topic
More informationLecture 2: Matrix Algebra. General
Lecture 2: Mtrix Algebr Generl Definitions Algebric Opertions Vector Spces, Liner Independence nd Rnk of Mtrix Inverse Mtrix Liner Eqution Systems, the Inverse Mtrix nd Crmer s Rule Chrcteristic Roots
More informationNotes for Thurs 8 Sept Calculus II Fall 2005 New York University Instructor: Tyler Neylon Scribe: Kelsey Williams
Notes for Thurs 8 Sept Clculus II Fll 00 New York University Instructor: Tyler Neylon Scribe: Kelsey Willims 8. Integrtion by Prts This section is primrily bout the formul u dv = uv v ( ) which is essentilly
More information