Number Conversion. Roland Backhouse March 5, 2001
|
|
- Moses Wilcox
- 5 years ago
- Views:
Transcription
1 1 Number Coversio Rolad Backhouse March 5, 2001
2 Outlie 2 We discuss the mathematical properties of covertig real umbers to itegers. I particular, we show how to implemet roudig up (of ratioal umbers) i terms of roudig dow. This provides a mechaism for implemetig roudig up i Java (which defies iteger divisio as roudig towards zero). The discussio illustrates the use of equivaleces i defiig importat mathematical otios.
3 Castig 3 Castig is the ame give i laguages like C ad Java for the operatio of covertig a value of oe type to aother. The cast from real umbers to itegers occurs whe evaluatig a iteger divisio. The text 1/2 (for example) meas a iteger divisio ad evaluates to zero. So, if the real value is iteded, oe has to write, say, 1.0/2.0. Addig decimal poits does t help, however, whe the values beig divided are expressios. So if, for example, m ad have bee declared to be itegers the we have to write somethig like (real)m/(real) to force the compiler to compute the real value of m/.
4 The Floor Fuctio 4 Defiitio: For all real x, x is a iteger such that, for all itegers, x x.
5 The Floor Fuctio 5 Defiitio: For all real x, x is a iteger such that, for all itegers, x x. O the right side of the equivalece, the at-most relatio ( ) is betwee reals whereas o the left side it is betwee itegers. Makig all coversios explicit, temporarily adoptig a Java-like otatio, is illumiatig. Doig so the defiitio becomes that, for all real x, (floor)x is a iteger such that for all itegers, (floor)x (real) x. So the floor of x is defied by coectig it to the coversio from itegers to reals i a simple equivalece.
6 Properties of Floor 6 Defiitio: For all real x, x is a iteger such that, for all itegers, x x. NB. x ca be istatiated to a iteger, but caot be istatiated to a real.
7 Properties of Floor 7 Defiitio: For all real x, x is a iteger such that, for all itegers, x x. NB. x ca be istatiated to a iteger, but caot be istatiated to a real. 1. x is by defiitio a iteger. So we ca istatiate to x. We get x x x x. Simplifyig, x x.
8 Properties of Floor 8 Defiitio: For all real x, x is a iteger such that, for all itegers, x x. NB. x ca be istatiated to a iteger, but caot be istatiated to a real. 1. x is by defiitio a iteger. So we ca istatiate to x. We get x x x x. Simplifyig, x x. 2. Istatiate x to.we get. Simplifyig,.
9 Properties of Floor 9 Defiitio: For all real x, x is a iteger such that, for all itegers, x x. NB. x ca be istatiated to a iteger, but caot be istatiated to a real. 1. x is by defiitio a iteger. So we ca istatiate to x. We get x x x x. Simplifyig, x x. 2. Istatiate x to.we get. Simplifyig,.
10 3. Istatiate x i 1. to. We get Combiig 2. ad 3., for all itegers =.
11 Properties of Floor Summary 11 For all reals x, For all itegers, x x. =.
12 Properties of Floor (Cotiued) 12 Recall the rule of cotrapositio: p q p q. The cotrapositive of the defiitio of the floor fuctio is, for all itegers ad real x, But ( m) m <. So ( x ) ( x). x < x <.
13 Properties of Floor (Cotiued) 13 Recall the rule of cotrapositio: p q p q. The cotrapositive of the defiitio of the floor fuctio is, for all itegers ad real x, But ( m) m <. So ( x ) ( x). x < x <. Istatiatig with x +1 ad simplifyig we deduce: x < x +1. Recallig that x x, we have established x x < x +1.
14 Idirect Equality 14 More complex properties use the rule of idirect equality: x = y z : z x z y. Example: x +m = { arithmetic } m x = { defiitio of floor } m x = { arithmetic } x+m = { defiitio of floor } Hece, x+m. x +m = x+m.
15 Roudig Off 15 Ceilig Fuctio Defiitio: For all real x, x is a iteger such that, for all itegers, x x. Exercise: derive properties of the ceilig fuctio dual to the properties of the floor fuctio.
16 The Problem 16 Roudig dow a iteger divisio of positive umbers m ad is expressed by m where m is the real divisio of m ad. Dually, roudig up is expressed by. Implemetig roudig up give a implemetatio of roudig dow amouts to fidig suitable values p ad q so that p =. q The values p ad q should be expressed as arithmetic fuctios of m ad (that is, fuctios ivolvig additio ad multiplicatio, but ot ivolvig the floor or ceilig fuctios).
17 The Strategy 17 We ca derive suitable expressios for p ad q usig the rule of idirect equality. For arbitrary iteger k, we aim to elimiate the ceilig fuctio from the iequality k obtaiig a iequality of the form k e where e is a arithmetic expressio i m ad. We may the coclude that e =.
18 The Solutio k 18
19 The Solutio k 19 = { iteger arithmetic } k 1 < = { cotrapositive of defiitio of ceilig } k 1 < m
20 The Solutio k 20 = { iteger arithmetic } k 1 < = { cotrapositive of defiitio of ceilig } k 1 < m = { arithmetic, > 0 } (k 1) < m = { iteger iequalities } (k 1) + 1 m = { arithmetic, > 0 } k m+ 1
21 The Solutio k 21 = { iteger arithmetic } k 1 < = { cotrapositive of defiitio of ceilig } k 1 < m = { arithmetic, > 0 } (k 1) < m = { iteger iequalities } (k 1) + 1 m = { arithmetic, > 0 } k m+ 1 = { defiitio of floor fuctio } m+ 1 k.
22 We have derived: k k m+ 1 Here k is arbitrary. So, by idirect equality, we get m+ 1 =. I Java, therefore, if it is required to roud up the result of dividig m by oe should compute (m+-1)/.. 22
Summary: Congruences. j=1. 1 Here we use the Mathematica syntax for the function. In Maple worksheets, the function
Summary: Cogrueces j whe divided by, ad determiig the additive order of a iteger mod. As described i the Prelab sectio, cogrueces ca be thought of i terms of coutig with rows, ad for some questios this
More informationLesson 10: Limits and Continuity
www.scimsacademy.com Lesso 10: Limits ad Cotiuity SCIMS Academy 1 Limit of a fuctio The cocept of limit of a fuctio is cetral to all other cocepts i calculus (like cotiuity, derivative, defiite itegrals
More informationRADICAL EXPRESSION. If a and x are real numbers and n is a positive integer, then x is an. n th root theorems: Example 1 Simplify
Example 1 Simplify 1.2A Radical Operatios a) 4 2 b) 16 1 2 c) 16 d) 2 e) 8 1 f) 8 What is the relatioship betwee a, b, c? What is the relatioship betwee d, e, f? If x = a, the x = = th root theorems: RADICAL
More informationSection 4.1. Properties of Exponents
Properties of Expoets Defiitio Defiitio: Expoet Defiitio of a Expoet For ay coutig umber, b = bbb b N factors of b We refer to b at the power; the th power of b, or b raised to the th power. We call b
More information6 Integers Modulo n. integer k can be written as k = qn + r, with q,r, 0 r b. So any integer.
6 Itegers Modulo I Example 2.3(e), we have defied the cogruece of two itegers a,b with respect to a modulus. Let us recall that a b (mod ) meas a b. We have proved that cogruece is a equivalece relatio
More information6.3 Testing Series With Positive Terms
6.3. TESTING SERIES WITH POSITIVE TERMS 307 6.3 Testig Series With Positive Terms 6.3. Review of what is kow up to ow I theory, testig a series a i for covergece amouts to fidig the i= sequece of partial
More informationP.3 Polynomials and Special products
Precalc Fall 2016 Sectios P.3, 1.2, 1.3, P.4, 1.4, P.2 (radicals/ratioal expoets), 1.5, 1.6, 1.7, 1.8, 1.1, 2.1, 2.2 I Polyomial defiitio (p. 28) a x + a x +... + a x + a x 1 1 0 1 1 0 a x + a x +... +
More informationMath 140A Elementary Analysis Homework Questions 1
Math 14A Elemetary Aalysis Homewor Questios 1 1 Itroductio 1.1 The Set N of Natural Numbers 1 Prove that 1 2 2 2 2 1 ( 1(2 1 for all atural umbers. 2 Prove that 3 11 (8 5 4 2 for all N. 4 (a Guess a formula
More information62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 +
62. Power series Defiitio 16. (Power series) Give a sequece {c }, the series c x = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + is called a power series i the variable x. The umbers c are called the coefficiets of
More information11. FINITE FIELDS. Example 1: The following tables define addition and multiplication for a field of order 4.
11. FINITE FIELDS 11.1. A Field With 4 Elemets Probably the oly fiite fields which you ll kow about at this stage are the fields of itegers modulo a prime p, deoted by Z p. But there are others. Now although
More informationIf a subset E of R contains no open interval, is it of zero measure? For instance, is the set of irrationals in [0, 1] is of measure zero?
2 Lebesgue Measure I Chapter 1 we defied the cocept of a set of measure zero, ad we have observed that every coutable set is of measure zero. Here are some atural questios: If a subset E of R cotais a
More informationn m CHAPTER 3 RATIONAL EXPONENTS AND RADICAL FUNCTIONS 3-1 Evaluate n th Roots and Use Rational Exponents Real nth Roots of a n th Root of a
CHAPTER RATIONAL EXPONENTS AND RADICAL FUNCTIONS Big IDEAS: 1) Usig ratioal expoets ) Performig fuctio operatios ad fidig iverse fuctios ) Graphig radical fuctios ad solvig radical equatios Sectio: Essetial
More informationUniversity of Manitoba, Mathletics 2009
Uiversity of Maitoba, Mathletics 009 Sessio 5: Iequalities Facts ad defiitios AM-GM iequality: For a, a,, a 0, a + a + + a (a a a ) /, with equality iff all a i s are equal Cauchy s iequality: For reals
More informationWORKING WITH NUMBERS
1 WORKING WITH NUMBERS WHAT YOU NEED TO KNOW The defiitio of the differet umber sets: is the set of atural umbers {0, 1,, 3, }. is the set of itegers {, 3,, 1, 0, 1,, 3, }; + is the set of positive itegers;
More informationProperties and Tests of Zeros of Polynomial Functions
Properties ad Tests of Zeros of Polyomial Fuctios The Remaider ad Factor Theorems: Sythetic divisio ca be used to fid the values of polyomials i a sometimes easier way tha substitutio. This is show by
More informationExponents. Learning Objectives. Pre-Activity
Sectio. Pre-Activity Preparatio Epoets A Chai Letter Chai letters are geerated every day. If you sed a chai letter to three frieds ad they each sed it o to three frieds, who each sed it o to three frieds,
More informationSeptember 2012 C1 Note. C1 Notes (Edexcel) Copyright - For AS, A2 notes and IGCSE / GCSE worksheets 1
September 0 s (Edecel) Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets September 0 Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets September 0 Copyright
More information2.4 - Sequences and Series
2.4 - Sequeces ad Series Sequeces A sequece is a ordered list of elemets. Defiitio 1 A sequece is a fuctio from a subset of the set of itegers (usually either the set 80, 1, 2, 3,... < or the set 81, 2,
More informationMath 299 Supplement: Real Analysis Nov 2013
Math 299 Supplemet: Real Aalysis Nov 203 Algebra Axioms. I Real Aalysis, we work withi the axiomatic system of real umbers: the set R alog with the additio ad multiplicatio operatios +,, ad the iequality
More informationWe are mainly going to be concerned with power series in x, such as. (x)} converges - that is, lims N n
Review of Power Series, Power Series Solutios A power series i x - a is a ifiite series of the form c (x a) =c +c (x a)+(x a) +... We also call this a power series cetered at a. Ex. (x+) is cetered at
More informationCh3. Asymptotic Notation
Ch. Asymptotic Notatio copyright 006 Preview of Chapters Chapter How to aalyze the space ad time complexities of program Chapter Review asymptotic otatios such as O, Ω, Θ, o for simplifyig the aalysis
More informationAPPENDIX F Complex Numbers
APPENDIX F Complex Numbers Operatios with Complex Numbers Complex Solutios of Quadratic Equatios Polar Form of a Complex Number Powers ad Roots of Complex Numbers Operatios with Complex Numbers Some equatios
More informationEssential Question How can you use properties of exponents to simplify products and quotients of radicals?
. TEXAS ESSENTIAL KNOWLEDGE AND SKILLS A.7.G Properties of Ratioal Expoets ad Radicals Essetial Questio How ca you use properties of expoets to simplify products ad quotiets of radicals? Reviewig Properties
More informationTopic 1 2: Sequences and Series. A sequence is an ordered list of numbers, e.g. 1, 2, 4, 8, 16, or
Topic : Sequeces ad Series A sequece is a ordered list of umbers, e.g.,,, 8, 6, or,,,.... A series is a sum of the terms of a sequece, e.g. + + + 8 + 6 + or... Sigma Notatio b The otatio f ( k) is shorthad
More information1 Approximating Integrals using Taylor Polynomials
Seughee Ye Ma 8: Week 7 Nov Week 7 Summary This week, we will lear how we ca approximate itegrals usig Taylor series ad umerical methods. Topics Page Approximatig Itegrals usig Taylor Polyomials. Defiitios................................................
More informationMini Lecture 10.1 Radical Expressions and Functions. 81x d. x 4x 4
Mii Lecture 0. Radical Expressios ad Fuctios Learig Objectives:. Evaluate square roots.. Evaluate square root fuctios.. Fid the domai of square root fuctios.. Use models that are square root fuctios. 5.
More informationBertrand s Postulate
Bertrad s Postulate Lola Thompso Ross Program July 3, 2009 Lola Thompso (Ross Program Bertrad s Postulate July 3, 2009 1 / 33 Bertrad s Postulate I ve said it oce ad I ll say it agai: There s always a
More informationC. Complex Numbers. x 6x + 2 = 0. This equation was known to have three real roots, given by simple combinations of the expressions
C. Complex Numbers. Complex arithmetic. Most people thik that complex umbers arose from attempts to solve quadratic equatios, but actually it was i coectio with cubic equatios they first appeared. Everyoe
More information1. By using truth tables prove that, for all statements P and Q, the statement
Author: Satiago Salazar Problems I: Mathematical Statemets ad Proofs. By usig truth tables prove that, for all statemets P ad Q, the statemet P Q ad its cotrapositive ot Q (ot P) are equivalet. I example.2.3
More informationAdvanced Course of Algorithm Design and Analysis
Differet complexity measures Advaced Course of Algorithm Desig ad Aalysis Asymptotic complexity Big-Oh otatio Properties of O otatio Aalysis of simple algorithms A algorithm may may have differet executio
More informationLecture 10: Mathematical Preliminaries
Lecture : Mathematical Prelimiaries Obective: Reviewig mathematical cocepts ad tools that are frequetly used i the aalysis of algorithms. Lecture # Slide # I this
More informationNICK DUFRESNE. 1 1 p(x). To determine some formulas for the generating function of the Schröder numbers, r(x) = a(x) =
AN INTRODUCTION TO SCHRÖDER AND UNKNOWN NUMBERS NICK DUFRESNE Abstract. I this article we will itroduce two types of lattice paths, Schröder paths ad Ukow paths. We will examie differet properties of each,
More information7 Sequences of real numbers
40 7 Sequeces of real umbers 7. Defiitios ad examples Defiitio 7... A sequece of real umbers is a real fuctio whose domai is the set N of atural umbers. Let s : N R be a sequece. The the values of s are
More informationProof of Goldbach s Conjecture. Reza Javaherdashti
Proof of Goldbach s Cojecture Reza Javaherdashti farzijavaherdashti@gmail.com Abstract After certai subsets of Natural umbers called Rage ad Row are defied, we assume (1) there is a fuctio that ca produce
More informationMa 530 Introduction to Power Series
Ma 530 Itroductio to Power Series Please ote that there is material o power series at Visual Calculus. Some of this material was used as part of the presetatio of the topics that follow. What is a Power
More informationMEI Casio Tasks for Further Pure
Task Complex Numbers: Roots of Quadratic Equatios. Add a ew Equatio scree: paf 2. Chage the Complex output to a+bi: LpNNNNwd 3. Select Polyomial ad set the Degree to 2: wq 4. Set a=, b=5 ad c=6: l5l6l
More informationOnce we have a sequence of numbers, the next thing to do is to sum them up. Given a sequence (a n ) n=1
. Ifiite Series Oce we have a sequece of umbers, the ext thig to do is to sum them up. Give a sequece a be a sequece: ca we give a sesible meaig to the followig expressio? a = a a a a While summig ifiitely
More informationComplex Numbers Primer
Complex Numbers Primer Complex Numbers Primer Before I get started o this let me first make it clear that this documet is ot iteded to teach you everythig there is to kow about complex umbers. That is
More informationComplex Numbers Primer
Before I get started o this let me first make it clear that this documet is ot iteded to teach you everythig there is to kow about complex umbers. That is a subject that ca (ad does) take a whole course
More informationThe natural exponential function
The atural expoetial fuctio Attila Máté Brookly College of the City Uiversity of New York December, 205 Cotets The atural expoetial fuctio for real x. Beroulli s iequality.....................................2
More informationAddition: Property Name Property Description Examples. a+b = b+a. a+(b+c) = (a+b)+c
Notes for March 31 Fields: A field is a set of umbers with two (biary) operatios (usually called additio [+] ad multiplicatio [ ]) such that the followig properties hold: Additio: Name Descriptio Commutativity
More informationTEACHER CERTIFICATION STUDY GUIDE
COMPETENCY 1. ALGEBRA SKILL 1.1 1.1a. ALGEBRAIC STRUCTURES Kow why the real ad complex umbers are each a field, ad that particular rigs are ot fields (e.g., itegers, polyomial rigs, matrix rigs) Algebra
More informationAppendix F: Complex Numbers
Appedix F Complex Numbers F1 Appedix F: Complex Numbers Use the imagiary uit i to write complex umbers, ad to add, subtract, ad multiply complex umbers. Fid complex solutios of quadratic equatios. Write
More information18.440, March 9, Stirling s formula
Stirlig s formula 8.44, March 9, 9 The factorial fuctio! is importat i evaluatig biomial, hypergeometric, ad other probabilities. If is ot too large,! ca be computed directly, by calculators or computers.
More information3 Gauss map and continued fractions
ICTP, Trieste, July 08 Gauss map ad cotiued fractios I this lecture we will itroduce the Gauss map, which is very importat for its coectio with cotiued fractios i umber theory. The Gauss map G : [0, ]
More informationSection 1 of Unit 03 (Pure Mathematics 3) Algebra
Sectio 1 of Uit 0 (Pure Mathematics ) Algebra Recommeded Prior Kowledge Studets should have studied the algebraic techiques i Pure Mathematics 1. Cotet This Sectio should be studied early i the course
More informationCHAPTER I: Vector Spaces
CHAPTER I: Vector Spaces Sectio 1: Itroductio ad Examples This first chapter is largely a review of topics you probably saw i your liear algebra course. So why cover it? (1) Not everyoe remembers everythig
More informationSequences. Notation. Convergence of a Sequence
Sequeces A sequece is essetially just a list. Defiitio (Sequece of Real Numbers). A sequece of real umbers is a fuctio Z (, ) R for some real umber. Do t let the descriptio of the domai cofuse you; it
More informationMath 155 (Lecture 3)
Math 55 (Lecture 3) September 8, I this lecture, we ll cosider the aswer to oe of the most basic coutig problems i combiatorics Questio How may ways are there to choose a -elemet subset of the set {,,,
More informationMath 257: Finite difference methods
Math 257: Fiite differece methods 1 Fiite Differeces Remember the defiitio of a derivative f f(x + ) f(x) (x) = lim 0 Also recall Taylor s formula: (1) f(x + ) = f(x) + f (x) + 2 f (x) + 3 f (3) (x) +...
More informationProofs. Indirect Proofs
Math 2326 Fall 2008 Proofs. Idirect Proofs 9fte, it is very difficult to give a direct approach to! x (P(x) " C(x))D the coectio betwee P!x) ad C!x) might ot be suitable to this approach. For example,
More informationSEQUENCES AND SERIES
Sequeces ad 6 Sequeces Ad SEQUENCES AND SERIES Successio of umbers of which oe umber is desigated as the first, other as the secod, aother as the third ad so o gives rise to what is called a sequece. Sequeces
More informationAlgebra II Notes Unit Seven: Powers, Roots, and Radicals
Syllabus Objectives: 7. The studets will use properties of ratioal epoets to simplify ad evaluate epressios. 7.8 The studet will solve equatios cotaiig radicals or ratioal epoets. b a, the b is the radical.
More information[ 11 ] z of degree 2 as both degree 2 each. The degree of a polynomial in n variables is the maximum of the degrees of its terms.
[ 11 ] 1 1.1 Polyomial Fuctios 1 Algebra Ay fuctio f ( x) ax a1x... a1x a0 is a polyomial fuctio if ai ( i 0,1,,,..., ) is a costat which belogs to the set of real umbers ad the idices,, 1,...,1 are atural
More information3. Z Transform. Recall that the Fourier transform (FT) of a DT signal xn [ ] is ( ) [ ] = In order for the FT to exist in the finite magnitude sense,
3. Z Trasform Referece: Etire Chapter 3 of text. Recall that the Fourier trasform (FT) of a DT sigal x [ ] is ω ( ) [ ] X e = j jω k = xe I order for the FT to exist i the fiite magitude sese, S = x [
More informationPUTNAM TRAINING INEQUALITIES
PUTNAM TRAINING INEQUALITIES (Last updated: December, 207) Remark This is a list of exercises o iequalities Miguel A Lerma Exercises If a, b, c > 0, prove that (a 2 b + b 2 c + c 2 a)(ab 2 + bc 2 + ca
More informationComplex Numbers Solutions
Complex Numbers Solutios Joseph Zoller February 7, 06 Solutios. (009 AIME I Problem ) There is a complex umber with imagiary part 64 ad a positive iteger such that Fid. [Solutio: 697] 4i + + 4i. 4i 4i
More informationCSE 1400 Applied Discrete Mathematics Number Theory and Proofs
CSE 1400 Applied Discrete Mathematics Number Theory ad Proofs Departmet of Computer Scieces College of Egieerig Florida Tech Sprig 01 Problems for Number Theory Backgroud Number theory is the brach of
More informationMATH 112: HOMEWORK 6 SOLUTIONS. Problem 1: Rudin, Chapter 3, Problem s k < s k < 2 + s k+1
MATH 2: HOMEWORK 6 SOLUTIONS CA PRO JIRADILOK Problem. If s = 2, ad Problem : Rudi, Chapter 3, Problem 3. s + = 2 + s ( =, 2, 3,... ), prove that {s } coverges, ad that s < 2 for =, 2, 3,.... Proof. The
More informationThe Boolean Ring of Intervals
MATH 532 Lebesgue Measure Dr. Neal, WKU We ow shall apply the results obtaied about outer measure to the legth measure o the real lie. Throughout, our space X will be the set of real umbers R. Whe ecessary,
More information(A sequence also can be thought of as the list of function values attained for a function f :ℵ X, where f (n) = x n for n 1.) x 1 x N +k x N +4 x 3
MATH 337 Sequeces Dr. Neal, WKU Let X be a metric space with distace fuctio d. We shall defie the geeral cocept of sequece ad limit i a metric space, the apply the results i particular to some special
More information, then cv V. Differential Equations Elements of Lineaer Algebra Name: Consider the differential equation. and y2 cos( kx)
Cosider the differetial equatio y '' k y 0 has particular solutios y1 si( kx) ad y cos( kx) I geeral, ay liear combiatio of y1 ad y, cy 1 1 cy where c1, c is also a solutio to the equatio above The reaso
More information2 Geometric interpretation of complex numbers
2 Geometric iterpretatio of complex umbers 2.1 Defiitio I will start fially with a precise defiitio, assumig that such mathematical object as vector space R 2 is well familiar to the studets. Recall that
More informationRevision Topic 1: Number and algebra
Revisio Topic : Number ad algebra Chapter : Number Differet types of umbers You eed to kow that there are differet types of umbers ad recogise which group a particular umber belogs to: Type of umber Symbol
More informationInfinite Sequences and Series
Chapter 6 Ifiite Sequeces ad Series 6.1 Ifiite Sequeces 6.1.1 Elemetary Cocepts Simply speakig, a sequece is a ordered list of umbers writte: {a 1, a 2, a 3,...a, a +1,...} where the elemets a i represet
More informationMIXED REVIEW of Problem Solving
MIXED REVIEW of Problem Solvig STATE TEST PRACTICE classzoe.com Lessos 2.4 2.. MULTI-STEP PROBLEM A ball is dropped from a height of 2 feet. Each time the ball hits the groud, it bouces to 70% of its previous
More information4.3 Growth Rates of Solutions to Recurrences
4.3. GROWTH RATES OF SOLUTIONS TO RECURRENCES 81 4.3 Growth Rates of Solutios to Recurreces 4.3.1 Divide ad Coquer Algorithms Oe of the most basic ad powerful algorithmic techiques is divide ad coquer.
More informationThe log-behavior of n p(n) and n p(n)/n
Ramauja J. 44 017, 81-99 The log-behavior of p ad p/ William Y.C. Che 1 ad Ke Y. Zheg 1 Ceter for Applied Mathematics Tiaji Uiversity Tiaji 0007, P. R. Chia Ceter for Combiatorics, LPMC Nakai Uivercity
More informationMath 4400/6400 Homework #7 solutions
MATH 4400 problems. Math 4400/6400 Homewor #7 solutios 1. Let p be a prime umber. Show that the order of 1 + p modulo p 2 is exactly p. Hit: Expad (1 + p) p by the biomial theorem, ad recall from MATH
More informationA FIBONACCI MATRIX AND THE PERMANENT FUNCTION
A FIBONACCI MATRIX AND THE PERMANENT FUNCTION BRUCE W. KING Burt Hiils-Ballsto Lake High School, Ballsto Lake, New York ad FRANCIS D. PARKER The St. Lawrece Uiversity, Cato, New York The permaet of a -square
More informationSequences, Mathematical Induction, and Recursion. CSE 2353 Discrete Computational Structures Spring 2018
CSE 353 Discrete Computatioal Structures Sprig 08 Sequeces, Mathematical Iductio, ad Recursio (Chapter 5, Epp) Note: some course slides adopted from publisher-provided material Overview May mathematical
More informationLyman Memorial High School. Honors Pre-Calculus Prerequisite Packet. Name:
Lyma Memorial High School Hoors Pre-Calculus Prerequisite Packet 2018 Name: Dear Hoors Pre-Calculus Studet, Withi this packet you will fid mathematical cocepts ad skills covered i Algebra I, II ad Geometry.
More informationMath 220A Fall 2007 Homework #2. Will Garner A
Math 0A Fall 007 Homewor # Will Garer Pg 3 #: Show that {cis : a o-egative iteger} is dese i T = {z œ : z = }. For which values of q is {cis(q): a o-egative iteger} dese i T? To show that {cis : a o-egative
More informationWrap of Number Theory & Midterm Review. Recall: Fundamental Theorem of Arithmetic
Wrap of Number Theory & Midterm Review F Primes, GCD, ad LCM (Sectio 3.5 i text) F Midterm Review Sectios.-.7 Propositioal logic Predicate logic Rules of iferece ad proofs Sectios.-.3 Sets ad Set operatios
More informationThe picture in figure 1.1 helps us to see that the area represents the distance traveled. Figure 1: Area represents distance travelled
1 Lecture : Area Area ad distace traveled Approximatig area by rectagles Summatio The area uder a parabola 1.1 Area ad distace Suppose we have the followig iformatio about the velocity of a particle, how
More informationComplex Numbers. Brief Notes. z = a + bi
Defiitios Complex Numbers Brief Notes A complex umber z is a expressio of the form: z = a + bi where a ad b are real umbers ad i is thought of as 1. We call a the real part of z, writte Re(z), ad b the
More informationMath 61CM - Solutions to homework 3
Math 6CM - Solutios to homework 3 Cédric De Groote October 2 th, 208 Problem : Let F be a field, m 0 a fixed oegative iteger ad let V = {a 0 + a x + + a m x m a 0,, a m F} be the vector space cosistig
More information6.046 Recitation 5: Binary Search Trees Bill Thies, Fall 2004 Outline
6.046 Recitatio 5: Biary Search Trees Bill Thies, Fall 2004 Outlie My cotact iformatio: Bill Thies thies@mit.edu Office hours: Sat 1-3pm, 36-153 Recitatio website: http://cag.lcs.mit.edu/~thies/6.046/
More informationChapter 6 Infinite Series
Chapter 6 Ifiite Series I the previous chapter we cosidered itegrals which were improper i the sese that the iterval of itegratio was ubouded. I this chapter we are goig to discuss a topic which is somewhat
More informationMA131 - Analysis 1. Workbook 9 Series III
MA3 - Aalysis Workbook 9 Series III Autum 004 Cotets 4.4 Series with Positive ad Negative Terms.............. 4.5 Alteratig Series.......................... 4.6 Geeral Series.............................
More informationJoe Holbrook Memorial Math Competition
Joe Holbrook Memorial Math Competitio 8th Grade Solutios October 5, 07. Sice additio ad subtractio come before divisio ad mutiplicatio, 5 5 ( 5 ( 5. Now, sice operatios are performed right to left, ( 5
More information(b) What is the probability that a particle reaches the upper boundary n before the lower boundary m?
MATH 529 The Boudary Problem The drukard s walk (or boudary problem) is oe of the most famous problems i the theory of radom walks. Oe versio of the problem is described as follows: Suppose a particle
More informationExercises 1 Sets and functions
Exercises 1 Sets ad fuctios HU Wei September 6, 018 1 Basics Set theory ca be made much more rigorous ad built upo a set of Axioms. But we will cover oly some heuristic ideas. For those iterested studets,
More informationPolynomial Functions and Their Graphs
Polyomial Fuctios ad Their Graphs I this sectio we begi the study of fuctios defied by polyomial expressios. Polyomial ad ratioal fuctios are the most commo fuctios used to model data, ad are used extesively
More informationNAME: ALGEBRA 350 BLOCK 7. Simplifying Radicals Packet PART 1: ROOTS
NAME: ALGEBRA 50 BLOCK 7 DATE: Simplifyig Radicals Packet PART 1: ROOTS READ: A square root of a umber b is a solutio of the equatio x = b. Every positive umber b has two square roots, deoted b ad b or
More informationProblem 4: Evaluate ( k ) by negating (actually un-negating) its upper index. Binomial coefficient
Problem 4: Evaluate by egatig actually u-egatig its upper idex We ow that Biomial coefficiet r { where r is a real umber, is a iteger The above defiitio ca be recast i terms of factorials i the commo case
More informationa 3, a 4, ... are the terms of the sequence. The number a n is the nth term of the sequence, and the entire sequence is denoted by a n
60_090.qxd //0 : PM Page 59 59 CHAPTER 9 Ifiite Series Sectio 9. EXPLORATION Fidig Patters Describe a patter for each of the followig sequeces. The use your descriptio to write a formula for the th term
More informationOptimally Sparse SVMs
A. Proof of Lemma 3. We here prove a lower boud o the umber of support vectors to achieve geeralizatio bouds of the form which we cosider. Importatly, this result holds ot oly for liear classifiers, but
More informationSubject: Differential Equations & Mathematical Modeling-III
Power Series Solutios of Differetial Equatios about Sigular poits Subject: Differetial Equatios & Mathematical Modelig-III Lesso: Power series solutios of differetial equatios about Sigular poits Lesso
More information14.1 Understanding Rational Exponents and Radicals
Name Class Date 14.1 Uderstadig Ratioal Expoets ad Radicals Essetial Questio: How are radicals ad ratioal expoets related? Resource Locker Explore 1 Uderstadig Iteger Expoets Recall that powers like are
More informationSail into Summer with Math!
Sail ito Summer with Math! For Studets Eterig Hoors Geometry This summer math booklet was developed to provide studets i kidergarte through the eighth grade a opportuity to review grade level math objectives
More informationINTEGRATION BY PARTS (TABLE METHOD)
INTEGRATION BY PARTS (TABLE METHOD) Suppose you wat to evaluate cos d usig itegratio by parts. Usig the u dv otatio, we get So, u dv d cos du d v si cos d si si d or si si d We see that it is ecessary
More information12.1 Sequences. Write the First Several Terms of a Sequence. PREPARING FOR THIS SECTION Before g etting started, review the following concept: ( )
SECTION. Sequeces 79. Sequeces PREPARING FOR THIS SECTION Before g ettig started, review the followig cocept: Fuctios (Sectio., pp. 5) Now Work the Are You Prepared? problems o page 799. OBJECTIVES Write
More informationAssignment 5: Solutions
McGill Uiversity Departmet of Mathematics ad Statistics MATH 54 Aalysis, Fall 05 Assigmet 5: Solutios. Let y be a ubouded sequece of positive umbers satisfyig y + > y for all N. Let x be aother sequece
More informationReal Variables II Homework Set #5
Real Variables II Homework Set #5 Name: Due Friday /0 by 4pm (at GOS-4) Istructios: () Attach this page to the frot of your homework assigmet you tur i (or write each problem before your solutio). () Please
More information( ) 2 + k The vertex is ( h, k) ( )( x q) The x-intercepts are x = p and x = q.
A Referece Sheet Number Sets Quadratic Fuctios Forms Form Equatio Stadard Form Vertex Form Itercept Form y ax + bx + c The x-coordiate of the vertex is x b a y a x h The axis of symmetry is x b a + k The
More informationIn number theory we will generally be working with integers, though occasionally fractions and irrationals will come into play.
Number Theory Math 5840 otes. Sectio 1: Axioms. I umber theory we will geerally be workig with itegers, though occasioally fractios ad irratioals will come ito play. Notatio: Z deotes the set of all itegers
More informationTest One (Answer Key)
CS395/Ma395 (Sprig 2005) Test Oe Name: Page 1 Test Oe (Aswer Key) CS395/Ma395: Aalysis of Algorithms This is a closed book, closed otes, 70 miute examiatio. It is worth 100 poits. There are twelve (12)
More informationPrinciple Of Superposition
ecture 5: PREIMINRY CONCEP O RUCUR NYI Priciple Of uperpositio Mathematically, the priciple of superpositio is stated as ( a ) G( a ) G( ) G a a or for a liear structural system, the respose at a give
More informationpage Suppose that S 0, 1 1, 2.
page 10 1. Suppose that S 0, 1 1,. a. What is the set of iterior poits of S? The set of iterior poits of S is 0, 1 1,. b. Give that U is the set of iterior poits of S, evaluate U. 0, 1 1, 0, 1 1, S. The
More information