# The Field Q of Rational Numbers

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Chapter 3 The Field Q of Ratioal Numbers I this chapter we are goig to costruct the ratioal umber from the itegers. Historically, the positive ratioal umbers came first: the Babyloias, Egyptias ad Grees ew how to wor with fractios, but egative umbers were itroduced by the Hidus hudreds of years later. It is possible to reflect this i the build-up of the ratioals from the atural umbers by first costructig the positive ratioal umbers from the aturals, ad the itroducig egatives (Ladau proceeds lie this i his Foudatios of Aalysis). While beig closer to history, this has the disadvatage of gettig a rig structure oly at the ed. 3. The Ratioal Numbers Let Z deote the rig of itegers ad cosider the set V {(r, s) : r, s Z, s 0} of pairs of itegers. Let us defie a equivalece relatio o V by puttig (r, s) (t, u) ru st. It is easily see that this is a equivalece relatio, ad we ow let [r, s] {(x, y) V : (x, y) (r, s)} deote the equivalece class of (r, s). Such a equivalece class [r, s] is called a ratioal umber, ad we ofte write r s istead of [r, s]. We deote by Q the set of all equivalece classes [r, s] with (r, s) V. We start studyig Q by realizig Z as a subset of Q via the map ι : Z Q defied by ι(r) [r, ]. The ι is ijective; i fact, assume that x, y Z are 3

2 such that ι(x) ι(y). The [x, ] [y, ], i.e., (x, ) (y, ), ad by defiitio of equivalece i V this meas x y, hece x y. We wat to have r s + t u ru+st su, so we are led to defie [r, s] [t, u] [ru + st, su] (3.) for r, s, t, u Z with s, u > 0. This is well defied ad agrees with additio o Z uder the idetificatio ι: i fact, ι(x) ι(y) [x, ] [y, ] [x + y, ] [x + y, ] ι(x + y). Thus it does ot matter whether we add i Z ad the idetify the result with a ratioal umber, or first view the itegers as elemets of Q ad add there. Next we defie multiplicatio of fractios by [r, s] [t, u] [rt, su]. (3.) This is motivated by r s t u rt su. Agai, multiplicatio is well defied ad agrees with multiplicatio o the subset Z Q: we have ι(x) ι(y) ι(xy) because ι(x) ι(y) [x, ] [y, ] by defiitio of ι [xy, ] by defiitio (3.) ι(xy) by defiitio of ι Remar. The map ι : Z Q from the rig Z to the rig of fractios Q satisfies ι(x) ι(y) ι(x + y), ι(x) ι(y) ι(xy), Maps R S betwee rigs with these properties (we say that they respect the rig structure ) are called rig homomorphisms if they map the uit elemet of R to the uit elemet of S. I particular, our idetificatio map ι is a rig homomorphism. Usig these defiitios, we ca prove associativity, commutativity, distributivity, thereby verifyig that Q is a rig. I fact, Q is eve a field! A field F is a commutative rig i which, iformally speaig. we ca divide by ozero elemets: thus F is a field if F satsifies the rig axioms (i particular we have 0), ad if i additio F For every r F \ {0} there is a s F such that rs. Observe that F holds if ad oly if F F \ {0}. This is a strog axiom: together with some other rig axioms it implies that fields are itegral domais: 4

3 Propositio 3.. If F is a field ad if xy 0 for x, y F, the x 0 or y 0. Proof. I fact, assume that xy 0 ad y 0. Sice the ozero elemets of F form a group, y has a iverse, that is, there is a z F such that yz. But ow 0 xy implies 0 0z (xy)z x(yz) x x; here we have used associativity of multiplicatio. We have proved Theorem 3.. The set Q of ratioal umbers forms a field with respect to additio ad multiplicatio. We ca also defie powers of ratioal umbers: if a Q is ozero, we put a 0 ad a + a a. This defies a for all N; if is egative, we put a /a. We ow ca prove the well ow set of rules a a m a +m, a m (a m ), a b (ab) etc. Biomial Theorem The ext result is called the Biomial Theorem. Before we ca state it, we have to itroduce the biomial coefficiets. These are defied i terms of factorials, so we have to defie these first. To this ed, we put 0! ad (+)!! (+) for N. Now we set ( )!!( )! for 0 ad ( ) 0 if < 0 or >. Lemma 3.3. The biomial coefficiets are itegers. I fact, we have ( ) + ( ) ( + + +) for 0 ad. Proof. This is a simple computatio: ( ) ( )! { + +!( )! + } +! +!( )! ( )( + ) +. + This calculatio is valid for 0; for, we have ( ) ( 0, +) ( + +), ad the claim holds. Now we have Theorem 3.4 (Biomial Theorem). For a, b Q ad N, we have (a + b) 0 a b. 5

4 Proof. This is doe by iductio o. For, we have to prove (a + b) ( 0 ) a b ( ) 0 a b 0 + ( ) a 0 b, which is true sice ( ( 0) ). Now assume that the claim holds for some iteger ; the ( (a + b) + (a + b) (a + b) a b ) (a + b) 0 a + b + a b ( ) + a + b + a + l b l l 0 l ( ( ) ( ) a + ) + + a + b ( ) ( ) + + a + + a + b a + b, 0 which is exactly what we wated to prove. 3. Q as a ordered field b + b + Observe that every ratioal umber ca be writte as [r, s] with s (if s, recall that [r, s] [ r, s]). From ow o, we will assume that all our ratioal umbers are preseted lie this. We defie a order relatio < o Q by puttig [r, s] < [t, u] ru < st (recall that s, u N). This is well defied: if [r, s] [r, s ] ad [t, u] [t, u ], the rs r s ad tu t u. Now [r, s] < [t, u] ru < st by defiitio rus u < sts u sice s u > 0 r suu < ss t u sice rs r s ad tu t u r u < s t sice su > 0 [r, s ] < [t, u ] by defiitio Now we have Theorem 3.5. Q is a ordered domai (eve field). Proof. Sice exactly oe of the relatios ru < st, ru st or ru > st is true by the trichotomy law for itegers, exactly oe of x < y, x y or x > y is true for x [r, s] ad y [t, u]. 6

5 Next assume that x < y ad y < z, where z [v, w]. The ru < st ad tw < uv, hece ruw < stw ad stw < suv sice w, s > 0; trasitivity for the itegers gives ruw < suv, ad sice u > 0, this is equivalet to rw < sv, i.e., x < z. This shows that Q is simply ordered. The rest of the proof that Q is a ordered domai is left as a exercise. Thus everythig proved for geeral ordered domais holds for the ratioals; i particular, x 0 for all x Q, ad x + y x + y for x, y Q. Now let us collect a few simple results that will tur out to be useful. Lemma 3.6. We have x < y if ad oly if x < y for some N. Proof. Exercise. Propositio 3.7. Let x, y Q ad assume that for every ratioal ε > 0 we have x y < ε; the x y. Proof. Assume that this is false, i.e. that x y 0. The ε x y is a positive ratioal umber, so by assumptio we have x y < ε. This implies ε < ε, which is a cotradictio. Propositio 3.8. Let 0 < x < y be ratioal umbers. The there is a N such that x > y. Proof. Write x r s ad y s t with r, s, t, u N (here we have used that x, y > 0). The x < y is equivalet to ru < st; by the Archimedea property of the atural umbers there is a N such that (ru) > st. But the last iequality is equivalet to x > y. Divisio with remaider i Z allows us to itroduce the floor fuctio i Q: for ratioal umbers x a b with b > 0, we put x q if a bq+r with q, r Z ad 0 r < b. Note that this is well defied: if x c d with d > 0, c dq + r ad 0 r < d, the ad bc, hece ad bdq +rd, bc bdq +br, ad therefore 0 bd(q q ) + rd r b. We may assume without loss of geerality that q q ; if q q, the q q +, hece bd > r b bd(q q ) + rd bd + rd bd: cotradictio. Propositio 3.9. For x Q, the iteger x is the uique iteger satisfyig x < x x. Proof. First, there is exactly oe iteger m satisfyig x < m x because m < for itegers implies m 0, hece m. It is therefore sufficiet to prove that x < x x. To this ed, recall that q x is defied for x a b by 0 a bq < b. Dividig through by b ad addig x we get x < q x as claimed. For ay ratioal umber x, we call x x x the fractioal part of x. Note that 0 x < for all ratioal umbers x Q. 7

6 3.3 Irratioal Numbers The irratioality of, at least i its geometric form (the side ad the diagoal of a square are icommesurable) seems to have bee discovered by the Pythagoreas. Although by the time of Euclid it was ow that square roots of osquares are irratioal, Euclid s elemets oly cotai the proof that is ot ratioal; i this case, a proof ca be give depedig oly o the theory of the odd ad the eve, as the Grees called the most elemetary parts of umber theory. Theorem 3.0. If N is ot the square of a iteger, the it is ot the square of a ratioal umber. Proof. I fact, if is ot a square of a iteger, the it lies betwee two squares, that is, we ca fid a iteger a such that a < < (a + ). Assume that p q with q > 0 miimal. The p q, hece p(p aq) p apq q apq q(q ap), so p q ap q p aq. But a < p q < a + implies 0 < p aq < q: this cotradicts the miimality of the deomiator q. Proof. Assume that A/B with B > 0 miimal; the A/B B/A, hece both of these fractios have the same fractioal part, say b/b A/B B/A a/a with 0 < a < A ad 0 < b < B (ote that e.g. a 0 would imply that is a iteger). But the A/B a/b, ad 0 < b < B cotradicts the miimality of B > 0. Proof 3. Sice is ot a square, at least oe prime p divides to a odd power. If we had b a, squarig ad clearig deomiators would give b a, ad p would divide a to a odd power, cotradictig uique factorizatio. 3.4 Historical Remars Log before maid discovered 0 ad the egative umbers, they started worig with positive ratioal umbers. The Babyloias had sexagesimal fractios, ad the Egyptias essetially wored with pure fractios, that is, those that ca be writte i the form. Every fractio was represeted by a sum of differet pure fractios: for example, they would have writte 5 ot as 5 + 5, but as Fractios were ot regarded as umbers i Euclid s elemets; Eudoxos theory of proportios dealt with magitudes, that is, legths ad areas etc. Archimedes ad Diophatus, o the other had, wored freely with positive ratioal umbers. Due to Joh Coway. 8

7 Exercises 3. For a, b Q, we have a a + b b. The ratioal umber a+b is called the arithmetic mea of a ad b. 3. For a, b Q, we have The ratioal umber a + b 3.3 Prove that for all a, b Q, we have a a + b b. is called the harmoic mea of a ad b. a + b a + b. This is called the iequality betwee harmoic ad arithmetic mea. Show that equality holds if ad oly if a b. 3.4 Which of the proofs of the irratioality of for osquares geeralizes to m-th roots of itegers? 9

### 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

### Homework 1 Solutions

Homewor 1 Solutios Math 171, Sprig 2010 Please sed correctios to herya@math.staford.edu 2.2. Let h : X Y, g : Y Z, ad f : Z W. Prove that (f g h = f (g h. Solutio. Let x X. Note that ((f g h(x = (f g(h(x

### HW 1 Solutions Math 115, Winter 2009, Prof. Yitzhak Katznelson

HW Solutios Math 5, Witer 2009, Prof. Yitzhak Katzelso.: Prove 2 + 2 2 +... + 2 = ( + )(2 + ) for all atural umbers. The proof is by iductio. Call the th propositio P. The basis for iductio P is the statemet

### 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

### Continued Fractions continued. 3. Best rational approximations

Cotiued Fractios cotiued 3. Best ratioal approximatios We hear so much about π beig approximated by 22/7 because o other ratioal umber with deomiator < 7 is closer to π. Evetually 22/7 is defeated by 333/06

### Linear Algebra II. Notes 6 25th November 2010

MTH6140 Liear Algebra II Notes 6 25th November 2010 6 Quadratic forms A lot of applicatios of mathematics ivolve dealig with quadratic forms: you meet them i statistics (aalysis of variace) ad mechaics

### 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

### Module 4: Mathematical Induction

Module 4: Mathematical Iductio Theme 1: Priciple of Mathematical Iductio Mathematical iductio is used to prove statemets about atural umbers. As studets may remember, we ca write such a statemet as a predicate

### 1 n. n > dt. t < n 1 + n=1

Math 05 otes C. Pomerace The harmoic sum The harmoic sum is the sum of recirocals of the ositive itegers. We kow from calculus that it diverges, this is usually doe by the itegral test. There s a more

### Section 6.1 Radicals and Rational Exponents

Sectio 6.1 Radicals ad Ratioal Expoets Defiitio of Square Root The umber b is a square root of a if b The priciple square root of a positive umber is its positive square root ad we deote this root by usig

### 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

### 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

### Binet Formulas for Recursive Integer Sequences

Biet Formulas for Recursive Iteger Sequeces Homer W. Austi Jatha W. Austi Abstract May iteger sequeces are recursive sequeces ad ca be defied either recursively or explicitly by use of Biet-type formulas.

### 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,

### 2.7 Sequences, Sequences of Sets

2.7. SEQUENCES, SEQUENCES OF SETS 67 2.7 Sequeces, Sequeces of Sets 2.7.1 Sequeces Defiitio 190 (sequece Let S be some set. 1. A sequece i S is a fuctio f : K S where K = { N : 0 for some 0 N}. 2. For

### Ma/CS 6b Class 17: Extremal Graph Theory

//06 Ma/CS 6b Class 7: Extremal Graph Theory Paul Turá By Adam Sheffer Extremal Graph Theory The subfield of extremal graph theory deals with questios of the form: What is the maximum umber of edges that

### when n = 1, 2, 3, 4, 5, 6, This list represents the amount of dollars you have after n days. Note: The use of is read as and so on.

Geometric eries Before we defie what is meat by a series, we eed to itroduce a related topic, that of sequeces. Formally, a sequece is a fuctio that computes a ordered list. uppose that o day 1, you have

### 1 The Binomial Theorem: Another Approach

The Biomial Theorem: Aother Approach Pascal s Triagle I class (ad i our text we saw that, for iteger, the biomial theorem ca be stated (a + b = c a + c a b + c a b + + c ab + c b, where the coefficiets

### ARITHMETIC AND GEOMETRIC PROGRESSIONS

Arithmetic Ad Geometric Progressios Sequeces Ad ARITHMETIC AND GEOMETRIC PROGRESSIONS Successio of umbers of which oe umber is desigated as the first, other as the secod, aother as the third ad so o gives

### 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

### The Field of Complex Numbers

The Field of Complex Numbers S. F. Ellermeyer The costructio of the system of complex umbers begis by appedig to the system of real umbers a umber which we call i with the property that i = 1. (Note that

### A CHARACTERIZATION OF MINIMAL ZERO-SEQUENCES OF INDEX ONE IN FINITE CYCLIC GROUPS

INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 5(1) (2005), #A27 A CHARACTERIZATION OF MINIMAL ZERO-SEQUENCES OF INDEX ONE IN FINITE CYCLIC GROUPS Scott T. Chapma 1 Triity Uiversity, Departmet

### 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

### x(x 1)(x 2)... (x k + 1) = [x] k n+m 1

1 Coutig mappigs For every real x ad positive iteger k, let [x] k deote the fallig factorial ad x(x 1)(x 2)... (x k + 1) ( ) x = [x] k k k!, ( ) k = 1. 0 I the sequel, X = {x 1,..., x m }, Y = {y 1,...,

### 7 b) 0. Guided Notes for lesson P.2 Properties of Exponents. If a, b, x, y and a, b, 0, and m, n Z then the following properties hold: 1 n b

Guided Notes for lesso P. Properties of Expoets If a, b, x, y ad a, b, 0, ad m, Z the the followig properties hold:. Negative Expoet Rule: b ad b b b Aswers must ever cotai egative expoets. Examples: 5

### The Euler Totient, the Möbius and the Divisor Functions

The Euler Totiet, the Möbius ad the Divisor Fuctios Rosica Dieva July 29, 2005 Mout Holyoke College South Hadley, MA 01075 1 Ackowledgemets This work was supported by the Mout Holyoke College fellowship

### Section IV.5: Recurrence Relations from Algorithms

Sectio IV.5: Recurrece Relatios from Algorithms Give a recursive algorithm with iput size, we wish to fid a Θ (best big O) estimate for its ru time T() either by obtaiig a explicit formula for T() or by

### Winter Camp 2012 Sequences Alexander Remorov. Sequences. Alexander Remorov

Witer Camp 202 Sequeces Alexader Remorov Sequeces Alexader Remorov alexaderrem@gmail.com Warm-up Problem : Give a positive iteger, cosider a sequece of real umbers a 0, a,..., a defied as a 0 = 2 ad =

### 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

### Lecture 7: Borel Sets and Lebesgue Measure

EE50: Probability Foudatios for Electrical Egieers July-November 205 Lecture 7: Borel Sets ad Lebesgue Measure Lecturer: Dr. Krisha Jagaatha Scribes: Ravi Kolla, Aseem Sharma, Vishakh Hegde I this lecture,

### {{1}, {2, 4}, {3}} {{1, 3, 4}, {2}} {{1}, {2}, {3, 4}} 5.4 Stirling Numbers

. Stirlig Numbers Whe coutig various types of fuctios from., we quicly discovered that eumeratig the umber of oto fuctios was a difficult problem. For a domai of five elemets ad a rage of four elemets,

### 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

### One-step equations. Vocabulary

Review solvig oe-step equatios with itegers, fractios, ad decimals. Oe-step equatios Vocabulary equatio solve solutio iverse operatio isolate the variable Additio Property of Equality Subtractio Property

### 1. Strong vs regular indiction 2. Strong induction examples: ! Divisibility by a prime! Recursion sequence: product of fractions

Today s Topics: CSE 0: Discrete Mathematics for Computer Sciece Prof. Miles Joes 1. Strog vs regular idictio. Strog iductio examples:! Divisibility by a prime! Recursio sequece: product of fractios 3 4

### Unit 8 Rational Functions

Uit 8 Ratioal Fuctios Algebraic Fractios: Simplifyig Algebraic Fractios: To simplify a algebraic fractio meas to reduce it to lowest terms. This is doe by dividig out the commo factors i the umerator ad

### Divide and Conquer, Solving Recurrences, Integer Multiplication Scribe: Juliana Cook (2015), V. Williams Date: April 6, 2016

CS 6, Lecture 3 Divide ad Coquer, Solvig Recurreces, Iteger Multiplicatio Scribe: Juliaa Cook (05, V Williams Date: April 6, 06 Itroductio Today we will cotiue to talk about divide ad coquer, ad go ito

### 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

### ON MINIMAL COLLECTIONS OF INDEXES. Egor A. Timoshenko

ON MINIMAL COLLECTIONS OF INDEXES Egor A. Timosheko We deote s [ +1 ], l [ ], M C s C; l idexes built for the case of colums (i.e., ordered subsets of the set {1,,..., }) will be called -idexes. The legth

### 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

### 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...........................

### 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

### Math 114- Intermediate Algebra Integral Exponents & Fractional Exponents (10 )

Math 4 Math 4- Itermediate Algebra Itegral Epoets & Fractioal Epoets (0 ) Epoetial Fuctios Epoetial Fuctios ad Graphs I. Epoetial Fuctios The fuctio f ( ) a, where is a real umber, a 0, ad a, is called

### Lecture 17 Two-Way Finite Automata

This is page 9 Priter: Opaque this Lecture 7 Two-Way Fiite Automata Two-way fiite automata are similar to the machies we have bee studyig, except that they ca read the iput strig i either directio. We

### 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

### Chapter Suppose you wish to use the Principle of Mathematical Induction to prove that 1 1! + 2 2! + 3 3! n n! = (n + 1)! 1 for all n 1.

Chapter 4. Suppose you wish to prove that the followig is true for all positive itegers by usig the Priciple of Mathematical Iductio: + 3 + 5 +... + ( ) =. (a) Write P() (b) Write P(7) (c) Write P(73)

### + 1= x + 1. These 4 elements form a field.

Itroductio to fiite fields II Fiite field of p elemets F Because we are iterested i doig computer thigs it would be useful for us to costruct fields havig elemets. Let s costruct a field of elemets; we

### 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

### Chapter Gaussian Elimination

Chapter 04.06 Gaussia Elimiatio After readig this chapter, you should be able to:. solve a set of simultaeous liear equatios usig Naïve Gauss elimiatio,. lear the pitfalls of the Naïve Gauss elimiatio

### 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

### MATH 140A - HW 5 SOLUTIONS

MATH 40A - HW 5 SOLUTIONS Problem WR Ch 3 #8. If a coverges, ad if {b } is mootoic ad bouded, rove that a b coverges. Solutio. Theorem 3.4 states that if a the artial sums of a form a bouded sequece; b

### 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

### 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

### 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

### 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

### = 2, 3, 4, etc. = { FLC Ch 7. Math 120 Intermediate Algebra Sec 7.1: Radical Expressions and Functions

Math 120 Itermediate Algebra Sec 7.1: Radical Expressios ad Fuctios idex radicad = 2,,, etc. Ex 1 For each umber, fid all of its square roots. 121 2 6 Ex 2 1 Simplify. 1 22 9 81 62 8 27 16 16 0 1 180 22

### Measurable Functions

Measurable Fuctios Dug Le 1 1 Defiitio It is ecessary to determie the class of fuctios that will be cosidered for the Lebesgue itegratio. We wat to guaratee that the sets which arise whe workig with these

### Riemann Sums y = f (x)

Riema Sums Recall that we have previously discussed the area problem I its simplest form we ca state it this way: The Area Problem Let f be a cotiuous, o-egative fuctio o the closed iterval [a, b] Fid

### if A S, then X \ A S, and if (A n ) n is a sequence of sets in S, then n A n S,

Lecture 5: Borel Sets Topologically, the Borel sets i a topological space are the σ-algebra geerated by the ope sets. Oe ca build up the Borel sets from the ope sets by iteratig the operatios of complemetatio

### 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

### SUMS OF n-th POWERS OF ROOTS OF A GIVEN QUADRATIC EQUATION. N.A. Draim, Ventura, Calif., and Marjorie Bicknell Wilcox High School, Santa Clara, Calif.

SUMS OF -th OWERS OF ROOTS OF A GIVEN QUADRATIC EQUATION N.A. Draim, Vetura, Calif., ad Marjorie Bickell Wilcox High School, Sata Clara, Calif. The quadratic equatio whose roots a r e the sum or differece

### Lecture 5: Span, linear independence, bases, and dimension

Lecture 5: Spa, liear idepedece, bases, ad dimesio Travis Schedler Thurs, Sep 23, 2010 (versio: 9/21 9:55 PM) 1 Motivatio Motivatio To uderstad what it meas that R has dimesio oe, R 2 dimesio 2, etc.;

### Repeating Decimals are decimal numbers that have number(s) after the decimal point that repeat in a pattern.

5.5 Fractios ad Decimals Steps for Chagig a Fractio to a Decimal. Simplify the fractio, if possible. 2. Divide the umerator by the deomiator. d d Repeatig Decimals Repeatig Decimals are decimal umbers

### 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.

### Factoring x n 1: cyclotomic and Aurifeuillian polynomials Paul Garrett <garrett@math.umn.edu>

(March 16, 004) Factorig x 1: cyclotomic ad Aurifeuillia polyomials Paul Garrett Polyomials of the form x 1, x 3 1, x 4 1 have at least oe systematic factorizatio x 1 = (x 1)(x 1

### NPTEL STRUCTURAL RELIABILITY

NPTEL Course O STRUCTURAL RELIABILITY Module # 0 Lecture 1 Course Format: Web Istructor: Dr. Aruasis Chakraborty Departmet of Civil Egieerig Idia Istitute of Techology Guwahati 1. Lecture 01: Basic Statistics

### MATH 361 Homework 9. Royden Royden Royden

MATH 61 Homework 9 Royde..9 First, we show that for ay subset E of the real umbers, E c + y = E + y) c traslatig the complemet is equivalet to the complemet of the traslated set). Without loss of geerality,

### ORDERS OF GROWTH KEITH CONRAD

ORDERS OF GROWTH KEITH CONRAD Itroductio Gaiig a ituitive feel for the relative growth of fuctios is importat if you really wat to uderstad their behavior It also helps you better grasp topics i calculus

### THE LEAST SQUARES REGRESSION LINE and R 2

THE LEAST SQUARES REGRESSION LINE ad R M358K I. Recall from p. 36 that the least squares regressio lie of y o x is the lie that makes the sum of the squares of the vertical distaces of the data poits from

### Math Discrete Math Combinatorics MULTIPLICATION PRINCIPLE:

Math 355 - Discrete Math 4.1-4.4 Combiatorics Notes MULTIPLICATION PRINCIPLE: If there m ways to do somethig ad ways to do aother thig the there are m ways to do both. I the laguage of set theory: Let

### 8.3 POLAR FORM AND DEMOIVRE S THEOREM

SECTION 8. POLAR FORM AND DEMOIVRE S THEOREM 48 8. POLAR FORM AND DEMOIVRE S THEOREM Figure 8.6 (a, b) b r a 0 θ Complex Number: a + bi Rectagular Form: (a, b) Polar Form: (r, θ) At this poit you ca add,

### 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

### B1. Fourier Analysis of Discrete Time Signals

B. Fourier Aalysis of Discrete Time Sigals Objectives Itroduce discrete time periodic sigals Defie the Discrete Fourier Series (DFS) expasio of periodic sigals Defie the Discrete Fourier Trasform (DFT)

### SUMS OF GENERALIZED HARMONIC SERIES. Michael E. Ho man Department of Mathematics, U. S. Naval Academy, Annapolis, Maryland

#A46 INTEGERS 4 (204) SUMS OF GENERALIZED HARMONIC SERIES Michael E. Ho ma Departmet of Mathematics, U. S. Naval Academy, Aapolis, Marylad meh@usa.edu Courtey Moe Departmet of Mathematics, U. S. Naval

### Lectures # 7: The Class Number Formula For Positive Definite Binary Quadratic Forms.

Lectures # 7: The Class Number Formula For Positive efiite Biary uadratic Forms. Noah Syder July 17, 00 1 efiitios efiitio 1.1. A biary quadratic form (BF) is a fuctio (x, y) = ax +bxy+cy (with a, b, c

### arxiv:1012.1336v2 [cs.cc] 8 Dec 2010

Uary Subset-Sum is i Logspace arxiv:1012.1336v2 [cs.cc] 8 Dec 2010 1 Itroductio Daiel M. Kae December 9, 2010 I this paper we cosider the Uary Subset-Sum problem which is defied as follows: Give itegers

### 2.3. GEOMETRIC SERIES

6 CHAPTER INFINITE SERIES GEOMETRIC SERIES Oe of the most importat types of ifiite series are geometric series A geometric series is simply the sum of a geometric sequece, Fortuately, geometric series

### 1 Notes on Little s Law (l = λw)

Copyright c 29 by Karl Sigma Notes o Little s Law (l λw) We cosider here a famous ad very useful law i queueig theory called Little s Law, also kow as l λw, which asserts that the time average umber of

### 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

### π d i (b i z) (n 1)π )... sin(θ + )

SOME TRIGONOMETRIC IDENTITIES RELATED TO EXACT COVERS Joh Beebee Uiversity of Alaska, Achorage Jauary 18, 1990 Sherma K Stei proves that if si π = k si π b where i the b i are itegers, the are positive

### Math 475, Problem Set #6: Solutions

Math 475, Problem Set #6: Solutios A (a) For each poit (a, b) with a, b o-egative itegers satisfyig ab 8, cout the paths from (0,0) to (a, b) where the legal steps from (i, j) are to (i 2, j), (i, j 2),

### Measure Theory, MA 359 Handout 1

Measure Theory, M 359 Hadout 1 Valeriy Slastikov utum, 2005 1 Measure theory 1.1 Geeral costructio of Lebesgue measure I this sectio we will do the geeral costructio of σ-additive complete measure by extedig

### Solutions to Exercises Chapter 3: Subsets, partitions, permutations

Solutios to Exercises Chapter 3: Subsets, partitios, permutatios 1 A restaurat ear Vacouver offered Dutch pacaes with a thousad ad oe combiatios of toppigs. What do you coclude? Sice 1001 ( 14 4, it is

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

### 8.1 Arithmetic Sequences

MCR3U Uit 8: Sequeces & Series Page 1 of 1 8.1 Arithmetic Sequeces Defiitio: A sequece is a comma separated list of ordered terms that follow a patter. Examples: 1, 2, 3, 4, 5 : a sequece of the first

### 1 Introduction to reducing variance in Monte Carlo simulations

Copyright c 007 by Karl Sigma 1 Itroductio to reducig variace i Mote Carlo simulatios 11 Review of cofidece itervals for estimatig a mea I statistics, we estimate a uow mea µ = E(X) of a distributio by

### 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

### Infinite Sequences and Series

CHAPTER 4 Ifiite Sequeces ad Series 4.1. Sequeces A sequece is a ifiite ordered list of umbers, for example the sequece of odd positive itegers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29...

### 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

### 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.

### 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.

### 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

Advaced Probability Theory Math5411 HKUST Kai Che (Istructor) Chapter 1. Law of Large Numbers 1.1. σ-algebra, measure, probability space ad radom variables. This sectio lays the ecessary rigorous foudatio

### 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

### 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

### Math 115 HW #4 Solutions

Math 5 HW #4 Solutios From 2.5 8. Does the series coverge or diverge? ( ) 3 + 2 = Aswer: This is a alteratig series, so we eed to check that the terms satisfy the hypotheses of the Alteratig Series Test.

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

### Lecture 4: Cheeger s Inequality

Spectral Graph Theory ad Applicatios WS 0/0 Lecture 4: Cheeger s Iequality Lecturer: Thomas Sauerwald & He Su Statemet of Cheeger s Iequality I this lecture we assume for simplicity that G is a d-regular

### Using Excel to Construct Confidence Intervals

OPIM 303 Statistics Ja Stallaert Usig Excel to Costruct Cofidece Itervals This hadout explais how to costruct cofidece itervals i Excel for the followig cases: 1. Cofidece Itervals for the mea of a populatio