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

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

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

Transcription

1 THE ARITHMETIC OF INTEGERS - multiplicatio, expoetiatio, divisio, additio, ad subtractio What to do ad what ot to do.

2 THE INTEGERS Recall that a iteger is oe of the whole umbers, which may be either positive, egative, or zero. I that sese, a iteger is simply either a atural umber with a positive or egative sig attached to it or the umber 0 which may also cosidered a siged umber. The iteger umber system has five arithmetical operatios. We will study all five i detail i this tutorial. Expoetiatio (a special case of multiplicatio) will also be covered as a separate topic because of its importace i sciece-related applicatios. Siged Numbers Each iteger is said to be a siged umber, which has either a positive sig or a egative sig. Oly the umber 0 may be cosidered as beig either positive or egative.

3 Covetios Ay siged umber be positive. THE INTEGERS + with o visible sig is assumed to For example: Remember that is a exceptio because it may be cosidered as either positive or egative depedig o the cotext. Therefore: The mius sig i frot of a umber may be regarded as the product of (-1) times the umber or: 3. where is a atural umber: For example: - 11 (-1). ( +11) or: - 11 (-1). ( 11) (-1). ( ) The plus sig i frot of a umber may be regarded as the product of (+1) times the umber or: For example: + 11 ( +1). ( 11) + ( +1).( )

4 THE INTEGERS The Absolute Value of a Siged Number The absolute value of a siged umber is the umber itself without the sig. By covetio 1, it is always a positive umber. Sometimes, it is regarded as the positive part of the siged umber. Notatio: where is a atural umber. For example: by covetio 1 Note that the absolute value is always positive.

5 Multiplyig Siged Numbers What to do ad what ot to do

6 THE INTEGERS Multiplyig Siged Numbers What to do The product of two siged umbers is foud by multiplyig the umbers (without the sigs) together ad the applyig the Rule of Sigs to the aswer. The Rule of Sigs for Products Ay product of positive umbers is also a positive umber. The product of a eve umber of egative umbers is a positive umber. The product of a odd umber of egative umbers is a egative umber. EXAMPLE 1 This is the justificatio for covetio 2. ( 4).( -1 ) ( +4). (-1) - ( 4. 1 ) - 4 Each of these umbers is a factor of the etire product.

7 Expoetials of Siged Numbers What to do ad what ot to do

8 THE INTEGERS The Expoetial Operatio of Siged Numbers The expoetial operatio is a special case of the operatio of multiplicatio. A iteger, or a siged umber, whe multiplied together with itself times, is said to be take to the expoet (or power). The otatio for the expoetial operatio is as follows: expoet a : a. a..... a iteger times a What to do atural umber siged umber All rules for multiplyig siged umbers apply to evaluatig expoetials of itegers. Multiplyig Siged Numbers The product of two siged umbers is foud by multiplyig the umbers (without the sigs) together ad the applyig the Rule of Sigs to the aswer.

9 THE INTEGERS The Expoetial Operatio of Siged Numbers EXAMPLE 1 What to do (-4) 3 (-4).(-4).(-4) iteger expoet 3 times - 64 A odd umber of egative factors meas a egative product. EXAMPLE 2 What to do (-4) 4 (-4). (-4). (-4). (-4) iteger expoet 4 times A eve umber of egative factors meas a positive product.

10 Dividig Siged Numbers What to do ad what ot to do

11 THE INTEGERS Dividig Siged Numbers What to do At this iitial stage of divisio, simply determie the sig of the umerator ad the sig of the deomiator separately. The apply the Rule of Sigs for Quotiets. The Rule of Sigs for Quotiets Ay quotiet of positive umbers is also a positive umber. The quotiet of two egative umbers is a positive umber. The quotiet of oe egative umber ad oe positive umber is a egative umber. EXAMPLE 1 11 _ _ 6 Oe egative sig i both umerator ad deomiator meas a egative quotiet.

12 Dividig Siged Numbers EXAMPLE 2-7 _ -4 / THE INTEGERS What ot to do Do t assume that a egative sig outside the fractio affects both umerator ad deomiator.. (-1). (-4) - (-1) (-7) + _ 7 4 The egative sig may be applied to either umerator or deomiator but NOT BOTH. - (-1) (-7) (-7) - _ 7-7 _ -4. (-4) What to do (-1). (-4) 4

13 THE INTEGERS Dividig Siged Numbers What to do Alterate Rule of Sigs for Quotiets If the umber of egative factors i both umerator ad deomiator is eve, the the quotiet is positive. If the umber of egative factors i both umerator ad deomiator is odd, the the quotiet is egative. EXAMPLE 3 Three egative sigs i both umerator ad deomiator meas a egative quotiet _ - _ 7-7 _ -4 (- 1 ). -4 4

14 Addig Siged Numbers What to do ad what ot to do

15 Addig Siged Numbers THE INTEGERS What to do Case 1: whe both umbers have the same sig The sum of two siged umbers with the same sig is foud by addig the umbers (without the sigs) together ad the applyig the Rule of Sigs to the aswer. Case 2: whe the umbers have opposite sigs The sum of two siged umbers havig opposite sigs is foud by subtractig the smaller umber (without the sig) from the larger umber (without the sig) ad the applyig the Rule of Sigs to the aswer. The Rule of Sigs for Sums Ay sum of positive umbers is also a positive umber. Ay sum of egative umbers is also a egative umber. The sum of a positive umber with a egative umber will have the same sig as the larger (i absolute value) of the two umbers.

16 THE INTEGERS Addig Siged Numbers What to do Case 1: whe both umbers have the same sig The sum of two siged umbers with the same sig is foud by addig the umbers (without the sigs) together ad the applyig the Rule of Sigs to the aswer. The Rule of Sigs for Sums Ay sum of positive umbers is also a positive umber. Ay sum of egative umbers is also a egative umber. EXAMPLE 1 ( 4) + ( 17) + ( ) + 21 (-9) + (-9) - ( 9 + 9) - 18 (-3) + (-1) + ( 0) + (-8) -( ) - 12 Each of these umbers is called a term of the etire sum.

17 Addig Siged Numbers THE INTEGERS What to do Case 2: whe the umbers have opposite sigs The sum of two siged umbers havig opposite sigs is foud by subtractig the smaller umber (without the sig) from the larger umber (without the sig) ad the applyig the Rule of Sigs to the aswer. The Rule of Sigs for Sums The sum of a positive umber with a egative umber will have the same sig as the larger (i absolute value) of the two umbers. EXAMPLE ( 3) + (-7) ( 7 ) - 4 (-33) + ( 75 ) +( 75-33) + 42

18 Addig Siged Numbers THE INTEGERS Case 2: whe the umbers have opposite sigs EXAMPLE 3 What to do Whe there are more tha two terms i a sum, add the umbers pairwise from left to right. (-33) + ( 75 ) + (- 43) + (- 43) (- 43)

19 Subtractig Siged Numbers What to do ad what ot to do

20 THE INTEGERS Subtractig Siged Numbers What to do Covert the subtractio problem ito a additio of siged umbers as follows: - ( m) ( ) ( m) + (- ) m where, are siged umbers. The add the resultig siged umbers together accordig to the appropriate rules. EXAMPLE 1 - (-33) ( 75 ) What ot to do - What to do (-33) (- 75) + -( ) After brigig the mius sig ito the bracket of the secod term, do ot forget to place a plus sig i betwee the two terms. Otherwise, the subtractio problem has icorrectly tured ito a multiplicatio problem. / (-33) ( 75 ) (-33) (- 75)

21 Subtractig Siged Numbers THE INTEGERS EXAMPLE 2 What to do Whe there are two subtractios, oe followig the other, subtract the umbers pairwise from left to right. (-33) - ( 75 ) - (-10) (-33) + (- 75) - (-10) (-108) - (-10) (-108)+ ( +10) -( ) - 98

22 THE ARITHMETIC OF FRACTIONS - multiplicatio, divisio - additio, subtractio What to do ad what ot to do.

23 THE RATIONAL NUMBERS REVIEW: Recall that a ratioal umber is simply a well defied fractio, meaig oe havig a o-zero deomiator. Moreover, two or more fractios may have the same umerical value eve though their umerators ad deomiators are ot idetical. For example, 4/8 ad 5/10 are both equal to ½, but their umerators ad deomiators do ot match. I this case, they are said to be equivalet. The Fudametal Law of Fractios The value of a fractio is NOT chaged if BOTH umerator ad deomiator are either multiplied or divided by the same o-zero umber.

24 THE RATIONAL NUMBERS The ratioal umber system has five arithmetical operatios. Four of them - multiplicatio, divisio, additio, ad subtractio - we will study i detail i this tutorial. The fifth, that of expoetiatio (a special case of multiplicatio), will be covered as a topic i its ow right. Divisio ad Subtractio of Ratioal Numbers as Special Cases of Multiplicatio ad Additio We are goig to discover that divisio of fractios is simply a special case of multiplicatio ad that subtractio of fractios, as i the case of whole umbers, may be iterpreted as the additio of siged umbers.

25 Multiplyig Fractios What to do ad what ot to do

26 THE RATIONAL NUMBERS Multiplyig Ratioal Numbers What to do The umerator of the product of two fractios is foud by multiplyig the umerators of each of them together. The deomiator of the product of two fractios is foud by multiplyig the deomiators of each of them together. ( p. q _ ) m. q. p m,,, are, / p m q q 0 where itegers. EXAMPLE 1 ( 4 _ ). -1 _ (-1) _

27 THE RATIONAL NUMBERS Multiplyig Ratioal Numbers ( p. q _ ) m. q. p m,,, are, / p m q q 0 where itegers. The Fudametal Law of Fractios The value of a fractio is NOT chaged if BOTH umerator ad deomiator are either multiplied or divided by the same o-zero umber. EXAMPLE 2 ( _ 5 ). _ I this case, BOTH umerator ad deomiator are divided first by 5 ad the by 4.. (-8).(-5) \ 4 (-8) \.(-5) (-2) (-1) This techique of simplifyig fractios is called cacellig commo factors across the fractio lie. \ \

28 Dividig Fractios What to do ad what ot to do

29 THE RATIONAL NUMBERS Dividig Ratioal Numbers What to do Divisio of two fractios is simply the multiplicatio of the umerator fractio by the reciprocal of the deomiator fractio. ( m _) where p m q, : p q _ p q p q,, are, / m _ q, m 0 p.. ( _ ) m q. m itegers. EXAMPLE 1 11 _ -6 _ : ( 2 ) ( 11 _ ). ( _ 5 ) (-6). (2) 55 -_

30 THE RATIONAL NUMBERS Dividig Ratioal Numbers ( m _ : ) p q _ (_ p ) _ q m The Fudametal Law of Fractios.,,, are,, / p m q q m 0 where itegers. The value of a fractio is NOT chaged if BOTH umerator ad deomiator are either multiplied or divided by the same o-zero umber. EXAMPLE 2 To simplify the aswer, BOTH umerator ad deomiator are divided first by 3 ad the by (-7). _ _ ( _ 1 2 : -14) 3. (-14) 3 \\.(-14) \ (-49). 15 (-49). 15 ( _ ) \ _ 2 35

31 THE RATIONAL NUMBERS Dividig Ratioal Numbers Divisio of two fractios is simply the multiplicatio of the umerator fractio by the reciprocal of the deomiator fractio. EXAMPLE 3 What to do I the case of two (or more) divisios, whe brackets are preset: 1. divisios withi brackets are performed first. _ _ : : ( _ 4 ) _ ( 15 _ ).( _ 9 ) ( _ 3 ). ( (-56) : ) (-14) _ 135 (-56) 1-49 \ \ \ \ _ 8 315

32 THE RATIONAL NUMBERS Dividig Ratioal Numbers Divisio of two fractios is simply the multiplicatio of the umerator fractio by the reciprocal of the deomiator fractio. EXAMPLE 4 What to do I the case of two (or more) divisios with o bracketspreset: 1. perform the divisios i order from left to right. _ 3-49 ( 15 _ ) _ 3-49 ( 15 _ : ) :( _ 4 ) _ 3-49 : : _ ( -14 _ ) ( _ 4 ) _ 2. ( _ : 9 ) \ 4 2 \ _ 9 70

33 Addig Fractios What to do ad what ot to do

34 THE RATIONAL NUMBERS Addig Ratioal Numbers Let s thik a bit about how we collect thigs. If we collect stamps, the i order to add to our collectio, we must have aother stamp - meaig a object of the same type. This is also true i mathematics. I order to add two umbers together they must be of similar type or i some way like each other. This otio of likeess is defied differetly depedig o the cotext. For example, two ratioal umbers may be collected together, or added together, if they have the same deomiators. Why that is comes from the idea that a fractio is the relative magitude of a part of a whole. The deomiator is idicative of the value of the whole.

35 Addig Ratioal Numbers ( p + q _ ) m _ p q _ THE RATIONAL NUMBERS What to do.. q. where p, m, q, are q, / 0 : p + m q formally m _ itegers. Step 1 is to fid a commo deomiator for both fractos. The smallest oe, called the lowest commo deomiator (LCD), is the best choice - but ot the oly choice. q. is oe possible choice for the above two fractios because both deomiators ad are preset. Recall that, to be able to add a uit whole _1 4 + _1 3 q, we would eed Step 2 is to fid equivalet fractios to the origial oes, BOTH havig the chose commo deomiator. _ p. q. NOTE q.. q _ m. q. q

36 Addig Ratioal Numbers ( p + q _ ) m _ The equivalet fractios: p q _ THE RATIONAL NUMBERS What to do.. q. p, m, q, are q / 0 : p + m q formally m _ itegers. Step 3 is to ADD the equivalet fractios together, by ADDING THE NUMERATORS ad placig the sum over the chose commo deomiator. _ p. q. NOTE Performig the additio: q.. q _ m. q. q p _ + p. m. )( m _) q q (_ p. ) q. + (_ m.q q ) +. q.. NOTE q. q

37 THE RATIONAL NUMBERS Addig Ratioal Numbers ( p + q _ ) m _ : p + m q. q.,,, are, / p m q q 0 where. itegers. EXAMPLE _ _ 9 What to do Step 1 is to fid a commo deomiator. The smallest oe, called the lowest commo deomiator (LCD), is the best choice - but ot the oly choice.. is oe possible choice for the above two fractios because both deomiators ad are preset To choose as the commo deomiator would be tatamout to dividig the pie ito pieces. 6 9 However, both ad fit ito a much smaller umber tha. I fact, both fit ito (or divide ito). Is this the smallest?

38 THE RATIONAL NUMBERS EXAMPLE cotiued What is the smallest umber ito which both 6 ad 9 fit? Fidig LCD s To fid the lowest commo deomiator (LCD) from we proceed as follows: 6 ad 9 1. Factor both deomiators ito primes (i.e. util they ca t be factored further) Collect all of the distict factors across both deomiators ad take the maximum umbers of each which appear. distict factors i both 2, 3 maximum umber of each: oe factor of 2 ; two factors of 3 3. Multiply together the maximum umbers of those distict factors which appear. This is the LCD. _5 6 LCD for ad is:.. -1 _

39 THE RATIONAL NUMBERS EXAMPLE cotiued _5 6 LCD for ad is:.. -1 _ Step 2 is to fid equivalet fractios to the origial oes, BOTH havig the chose commo deomiator. 5 _ 6 _ _ 9 _(-1) Step 3 is to ADD the equivalet fractios together, by ADDING THE NUMERATORS ad placig the sum over the chose commo deomiator. _ _ 9 _ _(-1) (-2)

40 THE RATIONAL NUMBERS Addig Ratioal Numbers EXAMPLE _ _ 5 What ot to do _ _ 5 \ 1 _ 5 4 \ + -3 _ 5 1 DO NOT cacel commo factors across a sig. + \ \ 5 + (-3) DO NOT add deomiators ad DO NOT add umerators without a commo deomiator.

41 Subtractig Fractios What to do ad what ot to do

42 p q _ - m _ THE RATIONAL NUMBERS Subtractig Ratioal Numbers p q _ : p - m q formally What to do. -. q. where p, m, q, are q, / 0 m _ itegers. Step 1 is to fid a commo deomiator for both fractos. The smallest oe, called the lowest commo deomiator (LCD), is the best choice - but ot the oly choice. q. is oe possible choice for the above two fractios because both deomiators ad are preset. Recall that, to be able to subtract a uit whole _1 3 - q _1 4, we eeded Step 2 is to fid equivalet fractios to the origial oes, BOTH havig the chose commo deomiator. _ p. q. NOTE q.. q _ m. q. q

43 p q _ THE RATIONAL NUMBERS Subtractig Ratioal Numbers ( m _) : p - - m q The equivalet fractios: p q _ formally. What to do -. q. p, m, q, are q / 0 Step 3 is to SUBTRACT the equivalet fractios together, by SUBTRACTING THE NUMERATORS ad placig the differece over the chose commo deomiator. _ p. q. NOTE Performig the subtractio: q.. q m m. q. q itegers. -. _ m.q q p. - m. - q p _ ) m q ( p ).. q NOTE q.. q q.

44 THE RATIONAL NUMBERS Subtractig Ratioal Numbers ( m _) : p - - m q p q _ formally. - q..,,, are / p m q q 0 itegers. EXAMPLE _ _ 7 What to do Step 1 is to fid a commo deomiator. The smallest oe, called the lowest commo deomiator (LCD), is the best choice - but ot the oly choice is oe possible choice for the above two fractios because both deomiators ad are preset Is this the lowest commo deomiator?

45 THE RATIONAL NUMBERS EXAMPLE cotiued What is the smallest umber ito which both 10 ad 7 fit? Fidig LCD s To fid the lowest commo deomiator (LCD) from we proceed as follows: 10 ad 7 1. Factor both deomiators ito primes (i.e. util they ca t be factored further) distict factors i both, 2 5 maximum umber of each: oe factor of oe factor of _3 10 LCD for ad is: 7-4 _ is a prime umber. 2. Collect all of the distict factors across both deomiators ad take the maximum umbers of each which appear., 2 ; oe factor of 5 3. Multiply together the maximum umbers of those distict factors which appear. This is the LCD. 7 ;

46 THE RATIONAL NUMBERS EXAMPLE cotiued _3 10 LCD for ad is: -4 _ Step 2 is to fid equivalet fractios to the origial oes, BOTH havig the chose commo deomiator. (_ 3 ) (_ 3. 7 ) -4 _ _(-4) Step 3 is to ADD the equivalet fractios together, by ADDING THE NUMERATORS ad placig the sum over the chose commo deomiator. (_ 3 ) 10 -( -4 _) 7 _ _(-4) (-40)

47 THE RATIONAL NUMBERS Subtractig Ratioal Numbers EXAMPLE _ _ 13 What ot to do _ _ 13 \ _ _ 13 \ \ - DO NOT cacel commo factors - across a sig. \ DO NOT subtract deomiators ad DO NOT subtract umerators without a commo deomiator.

48 THE RATIONAL NUMBERS Addedum: Mixed Numbers Mixed umbers are combiatios (actually additios) of whole umbers ad fractios. EXAMPLES 1 _1 is read as: 2 oe ad a half is iterpreted as: 1 + _1 2 is equal to: 1 + _1 _ 1 + _1 1 _ 2 + _1 _ _5 is read as: 8 mius (or egative) three ad five eighths - is iterpreted as: 3 + _5 8 is equal to: - ( _ _5 ) - ( _ 3 + ) - (_ _5 ) - _

49

Example 2 Find the square root of 0. The only square root of 0 is 0 (since 0 is not positive or negative, so those choices don t exist here).

Example 2 Find the square root of 0. The only square root of 0 is 0 (since 0 is not positive or negative, so those choices don t exist here). BEGINNING ALGEBRA Roots ad Radicals (revised summer, 00 Olso) Packet to Supplemet the Curret Textbook - Part Review of Square Roots & Irratioals (This portio ca be ay time before Part ad should mostly

More information

Laws of Exponents Learning Strategies

Laws of Exponents Learning Strategies Laws of Epoets Learig Strategies What should studets be able to do withi this iteractive? Studets should be able to uderstad ad use of the laws of epoets. Studets should be able to simplify epressios that

More information

SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES

SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,

More information

Trigonometric Form of a Complex Number. The Complex Plane. axis. ( 2, 1) or 2 i FIGURE 6.44. The absolute value of the complex number z a bi is

Trigonometric Form of a Complex Number. The Complex Plane. axis. ( 2, 1) or 2 i FIGURE 6.44. The absolute value of the complex number z a bi is 0_0605.qxd /5/05 0:45 AM Page 470 470 Chapter 6 Additioal Topics i Trigoometry 6.5 Trigoometric Form of a Complex Number What you should lear Plot complex umbers i the complex plae ad fid absolute values

More information

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

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

More information

1. MATHEMATICAL INDUCTION

1. MATHEMATICAL INDUCTION 1. MATHEMATICAL INDUCTION EXAMPLE 1: Prove that for ay iteger 1. Proof: 1 + 2 + 3 +... + ( + 1 2 (1.1 STEP 1: For 1 (1.1 is true, sice 1 1(1 + 1. 2 STEP 2: Suppose (1.1 is true for some k 1, that is 1

More information

Mathematical goals. Starting points. Materials required. Time needed

Mathematical goals. Starting points. Materials required. Time needed Level A1 of challege: C A1 Mathematical goals Startig poits Materials required Time eeded Iterpretig algebraic expressios To help learers to: traslate betwee words, symbols, tables, ad area represetatios

More information

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

BINOMIAL EXPANSIONS 12.5. In this section. Some Examples. Obtaining the Coefficients 652 (12-26) Chapter 12 Sequeces ad Series 12.5 BINOMIAL EXPANSIONS I this sectio Some Examples Otaiig the Coefficiets The Biomial Theorem I Chapter 5 you leared how to square a iomial. I this sectio you

More information

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

.04. This means $1000 is multiplied by 1.02 five times, once for each of the remaining sixmonth Questio 1: What is a ordiary auity? Let s look at a ordiary auity that is certai ad simple. By this, we mea a auity over a fixed term whose paymet period matches the iterest coversio period. Additioally,

More information

Lesson 15 ANOVA (analysis of variance)

Lesson 15 ANOVA (analysis of variance) Outlie Variability -betwee group variability -withi group variability -total variability -F-ratio Computatio -sums of squares (betwee/withi/total -degrees of freedom (betwee/withi/total -mea square (betwee/withi

More information

I. Chi-squared Distributions

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

More information

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

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

More information

Basic Elements of Arithmetic Sequences and Series

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

More information

Here are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed.

Here are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed. This documet was writte ad copyrighted by Paul Dawkis. Use of this documet ad its olie versio is govered by the Terms ad Coditios of Use located at http://tutorial.math.lamar.edu/terms.asp. The olie versio

More information

Soving Recurrence Relations

Soving Recurrence Relations Sovig Recurrece Relatios Part 1. Homogeeous liear 2d degree relatios with costat coefficiets. Cosider the recurrece relatio ( ) T () + at ( 1) + bt ( 2) = 0 This is called a homogeeous liear 2d degree

More information

Sequences and Series

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

More information

Laws of Exponents. net effect is to multiply with 2 a total of 3 + 5 = 8 times

Laws of Exponents. net effect is to multiply with 2 a total of 3 + 5 = 8 times The Mathematis 11 Competey Test Laws of Expoets (i) multipliatio of two powers: multiply by five times 3 x = ( x x ) x ( x x x x ) = 8 multiply by three times et effet is to multiply with a total of 3

More information

Solving equations. Pre-test. Warm-up

Solving equations. Pre-test. Warm-up Solvig equatios 8 Pre-test Warm-up We ca thik of a algebraic equatio as beig like a set of scales. The two sides of the equatio are equal, so the scales are balaced. If we add somethig to oe side of the

More information

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

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

More information

FOUNDATIONS OF MATHEMATICS AND PRE-CALCULUS GRADE 10

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

More information

SEQUENCES AND SERIES

SEQUENCES AND SERIES Chapter 9 SEQUENCES AND SERIES Natural umbers are the product of huma spirit. DEDEKIND 9.1 Itroductio I mathematics, the word, sequece is used i much the same way as it is i ordiary Eglish. Whe we say

More information

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

THE REGRESSION MODEL IN MATRIX FORM. For simple linear regression, meaning one predictor, the model is. for i = 1, 2, 3,, n We will cosider the liear regressio model i matrix form. For simple liear regressio, meaig oe predictor, the model is i = + x i + ε i for i =,,,, This model icludes the assumptio that the ε i s are a sample

More information

CS103X: Discrete Structures Homework 4 Solutions

CS103X: Discrete Structures Homework 4 Solutions CS103X: Discrete Structures Homewor 4 Solutios Due February 22, 2008 Exercise 1 10 poits. Silico Valley questios: a How may possible six-figure salaries i whole dollar amouts are there that cotai at least

More information

Complex Numbers. where x represents a root of Equation 1. Note that the ± sign tells us that quadratic equations will have

Complex Numbers. where x represents a root of Equation 1. Note that the ± sign tells us that quadratic equations will have Comple Numbers I spite of Calvi s discomfiture, imagiar umbers (a subset of the set of comple umbers) eist ad are ivaluable i mathematics, egieerig, ad sciece. I fact, i certai fields, such as electrical

More information

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

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

More information

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

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

More information

Chapter 5: Inner Product Spaces

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

More information

7.1 Finding Rational Solutions of Polynomial Equations

7.1 Finding Rational Solutions of Polynomial Equations 4 Locker LESSON 7. Fidig Ratioal Solutios of Polyomial Equatios Name Class Date 7. Fidig Ratioal Solutios of Polyomial Equatios Essetial Questio: How do you fid the ratioal roots of a polyomial equatio?

More information

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

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

More information

CHAPTER 3 THE TIME VALUE OF MONEY

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

More information

Lesson 17 Pearson s Correlation Coefficient

Lesson 17 Pearson s Correlation Coefficient Outlie Measures of Relatioships Pearso s Correlatio Coefficiet (r) -types of data -scatter plots -measure of directio -measure of stregth Computatio -covariatio of X ad Y -uique variatio i X ad Y -measurig

More information

A Guide to the Pricing Conventions of SFE Interest Rate Products

A Guide to the Pricing Conventions of SFE Interest Rate Products A Guide to the Pricig Covetios of SFE Iterest Rate Products SFE 30 Day Iterbak Cash Rate Futures Physical 90 Day Bak Bills SFE 90 Day Bak Bill Futures SFE 90 Day Bak Bill Futures Tick Value Calculatios

More information

A probabilistic proof of a binomial identity

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

More information

The following example will help us understand The Sampling Distribution of the Mean. C1 C2 C3 C4 C5 50 miles 84 miles 38 miles 120 miles 48 miles

The following example will help us understand The Sampling Distribution of the Mean. C1 C2 C3 C4 C5 50 miles 84 miles 38 miles 120 miles 48 miles The followig eample will help us uderstad The Samplig Distributio of the Mea Review: The populatio is the etire collectio of all idividuals or objects of iterest The sample is the portio of the populatio

More information

Question 2: How is a loan amortized?

Question 2: How is a loan amortized? Questio 2: How is a loa amortized? Decreasig auities may be used i auto or home loas. I these types of loas, some amout of moey is borrowed. Fixed paymets are made to pay off the loa as well as ay accrued

More information

3. Greatest Common Divisor - Least Common Multiple

3. Greatest Common Divisor - Least Common Multiple 3 Greatest Commo Divisor - Least Commo Multiple Defiitio 31: The greatest commo divisor of two atural umbers a ad b is the largest atural umber c which divides both a ad b We deote the greatest commo gcd

More information

EGYPTIAN FRACTION EXPANSIONS FOR RATIONAL NUMBERS BETWEEN 0 AND 1 OBTAINED WITH ENGEL SERIES

EGYPTIAN FRACTION EXPANSIONS FOR RATIONAL NUMBERS BETWEEN 0 AND 1 OBTAINED WITH ENGEL SERIES EGYPTIAN FRACTION EXPANSIONS FOR RATIONAL NUMBERS BETWEEN 0 AND OBTAINED WITH ENGEL SERIES ELVIA NIDIA GONZÁLEZ AND JULIA BERGNER, PHD DEPARTMENT OF MATHEMATICS Abstract. The aciet Egyptias epressed ratioal

More information

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

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

More information

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

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

More information

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

More information

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

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

More information

Incremental calculation of weighted mean and variance

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

More information

Asymptotic Growth of Functions

Asymptotic Growth of Functions CMPS Itroductio to Aalysis of Algorithms Fall 3 Asymptotic Growth of Fuctios We itroduce several types of asymptotic otatio which are used to compare the performace ad efficiecy of algorithms As we ll

More information

INFINITE SERIES KEITH CONRAD

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

More information

Section 11.3: The Integral Test

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

More information

Infinite Sequences and Series

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...

More information

Measures of Spread and Boxplots Discrete Math, Section 9.4

Measures of Spread and Boxplots Discrete Math, Section 9.4 Measures of Spread ad Boxplots Discrete Math, Sectio 9.4 We start with a example: Example 1: Comparig Mea ad Media Compute the mea ad media of each data set: S 1 = {4, 6, 8, 10, 1, 14, 16} S = {4, 7, 9,

More information

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

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

More information

1 Correlation and Regression Analysis

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

More information

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

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

More information

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

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

More information

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

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

More information

CHAPTER 3 DIGITAL CODING OF SIGNALS

CHAPTER 3 DIGITAL CODING OF SIGNALS CHAPTER 3 DIGITAL CODING OF SIGNALS Computers are ofte used to automate the recordig of measuremets. The trasducers ad sigal coditioig circuits produce a voltage sigal that is proportioal to a quatity

More information

HOSPITAL NURSE STAFFING SURVEY

HOSPITAL NURSE STAFFING SURVEY 2012 Ceter for Nursig Workforce St udies HOSPITAL NURSE STAFFING SURVEY Vacacy ad Turover Itroductio The Hospital Nurse Staffig Survey (HNSS) assesses the size ad effects of the ursig shortage i hospitals,

More information

Time Value of Money. First some technical stuff. HP10B II users

Time Value of Money. First some technical stuff. HP10B II users Time Value of Moey Basis for the course Power of compoud iterest $3,600 each year ito a 401(k) pla yields $2,390,000 i 40 years First some techical stuff You will use your fiacial calculator i every sigle

More information

WHEN IS THE (CO)SINE OF A RATIONAL ANGLE EQUAL TO A RATIONAL NUMBER?

WHEN IS THE (CO)SINE OF A RATIONAL ANGLE EQUAL TO A RATIONAL NUMBER? WHEN IS THE (CO)SINE OF A RATIONAL ANGLE EQUAL TO A RATIONAL NUMBER? JÖRG JAHNEL 1. My Motivatio Some Sort of a Itroductio Last term I tought Topological Groups at the Göttige Georg August Uiversity. This

More information

Running Time ( 3.1) Analysis of Algorithms. Experimental Studies ( 3.1.1) Limitations of Experiments. Pseudocode ( 3.1.2) Theoretical Analysis

Running Time ( 3.1) Analysis of Algorithms. Experimental Studies ( 3.1.1) Limitations of Experiments. Pseudocode ( 3.1.2) Theoretical Analysis Ruig Time ( 3.) Aalysis of Algorithms Iput Algorithm Output A algorithm is a step-by-step procedure for solvig a problem i a fiite amout of time. Most algorithms trasform iput objects ito output objects.

More information

LECTURE 13: Cross-validation

LECTURE 13: Cross-validation LECTURE 3: Cross-validatio Resampli methods Cross Validatio Bootstrap Bias ad variace estimatio with the Bootstrap Three-way data partitioi Itroductio to Patter Aalysis Ricardo Gutierrez-Osua Texas A&M

More information

Fast Fourier Transform

Fast Fourier Transform 18.310 lecture otes November 18, 2013 Fast Fourier Trasform Lecturer: Michel Goemas I these otes we defie the Discrete Fourier Trasform, ad give a method for computig it fast: the Fast Fourier Trasform.

More information

Convexity, Inequalities, and Norms

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

More information

Present Value Factor To bring one dollar in the future back to present, one uses the Present Value Factor (PVF): Concept 9: Present Value

Present Value Factor To bring one dollar in the future back to present, one uses the Present Value Factor (PVF): Concept 9: Present Value Cocept 9: Preset Value Is the value of a dollar received today the same as received a year from today? A dollar today is worth more tha a dollar tomorrow because of iflatio, opportuity cost, ad risk Brigig

More information

Determining the sample size

Determining the sample size Determiig the sample size Oe of the most commo questios ay statisticia gets asked is How large a sample size do I eed? Researchers are ofte surprised to fid out that the aswer depeds o a umber of factors

More information

GCE Further Mathematics (6360) Further Pure Unit 2 (MFP2) Textbook. Version: 1.4

GCE Further Mathematics (6360) Further Pure Unit 2 (MFP2) Textbook. Version: 1.4 GCE Further Mathematics (660) Further Pure Uit (MFP) Tetbook Versio: 4 MFP Tetbook A-level Further Mathematics 660 Further Pure : Cotets Chapter : Comple umbers 4 Itroductio 5 The geeral comple umber 5

More information

Solving Logarithms and Exponential Equations

Solving Logarithms and Exponential Equations Solvig Logarithms ad Epoetial Equatios Logarithmic Equatios There are two major ideas required whe solvig Logarithmic Equatios. The first is the Defiitio of a Logarithm. You may recall from a earlier topic:

More information

GCSE STATISTICS. 4) How to calculate the range: The difference between the biggest number and the smallest number.

GCSE STATISTICS. 4) How to calculate the range: The difference between the biggest number and the smallest number. GCSE STATISTICS You should kow: 1) How to draw a frequecy diagram: e.g. NUMBER TALLY FREQUENCY 1 3 5 ) How to draw a bar chart, a pictogram, ad a pie chart. 3) How to use averages: a) Mea - add up all

More information

Listing terms of a finite sequence List all of the terms of each finite sequence. a) a n n 2 for 1 n 5 1 b) a n for 1 n 4 n 2

Listing terms of a finite sequence List all of the terms of each finite sequence. a) a n n 2 for 1 n 5 1 b) a n for 1 n 4 n 2 74 (4 ) Chapter 4 Sequeces ad Series 4. SEQUENCES I this sectio Defiitio Fidig a Formula for the th Term The word sequece is a familiar word. We may speak of a sequece of evets or say that somethig is

More information

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

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

More information

Now here is the important step

Now here is the important step LINEST i Excel The Excel spreadsheet fuctio "liest" is a complete liear least squares curve fittig routie that produces ucertaity estimates for the fit values. There are two ways to access the "liest"

More information

The Stable Marriage Problem

The Stable Marriage Problem The Stable Marriage Problem William Hut Lae Departmet of Computer Sciece ad Electrical Egieerig, West Virgiia Uiversity, Morgatow, WV William.Hut@mail.wvu.edu 1 Itroductio Imagie you are a matchmaker,

More information

*The most important feature of MRP as compared with ordinary inventory control analysis is its time phasing feature.

*The most important feature of MRP as compared with ordinary inventory control analysis is its time phasing feature. Itegrated Productio ad Ivetory Cotrol System MRP ad MRP II Framework of Maufacturig System Ivetory cotrol, productio schedulig, capacity plaig ad fiacial ad busiess decisios i a productio system are iterrelated.

More information

I. Why is there a time value to money (TVM)?

I. Why is there a time value to money (TVM)? Itroductio to the Time Value of Moey Lecture Outlie I. Why is there the cocept of time value? II. Sigle cash flows over multiple periods III. Groups of cash flows IV. Warigs o doig time value calculatios

More information

Notes on exponential generating functions and structures.

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

More information

Factors of sums of powers of binomial coefficients

Factors of sums of powers of binomial coefficients ACTA ARITHMETICA LXXXVI.1 (1998) Factors of sums of powers of biomial coefficiets by Neil J. Cali (Clemso, S.C.) Dedicated to the memory of Paul Erdős 1. Itroductio. It is well ow that if ( ) a f,a = the

More information

Solutions to Selected Problems In: Pattern Classification by Duda, Hart, Stork

Solutions to Selected Problems In: Pattern Classification by Duda, Hart, Stork Solutios to Selected Problems I: Patter Classificatio by Duda, Hart, Stork Joh L. Weatherwax February 4, 008 Problem Solutios Chapter Bayesia Decisio Theory Problem radomized rules Part a: Let Rx be the

More information

Simple Annuities Present Value.

Simple Annuities Present Value. Simple Auities Preset Value. OBJECTIVES (i) To uderstad the uderlyig priciple of a preset value auity. (ii) To use a CASIO CFX-9850GB PLUS to efficietly compute values associated with preset value auities.

More information

Chapter 14 Nonparametric Statistics

Chapter 14 Nonparametric Statistics Chapter 14 Noparametric Statistics A.K.A. distributio-free statistics! Does ot deped o the populatio fittig ay particular type of distributio (e.g, ormal). Sice these methods make fewer assumptios, they

More information

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

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

More information

Building Blocks Problem Related to Harmonic Series

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

More information

SEQUENCES AND SERIES CHAPTER

SEQUENCES AND SERIES CHAPTER CHAPTER SEQUENCES AND SERIES Whe the Grat family purchased a computer for $,200 o a istallmet pla, they agreed to pay $00 each moth util the cost of the computer plus iterest had bee paid The iterest each

More information

Cooley-Tukey. Tukey FFT Algorithms. FFT Algorithms. Cooley

Cooley-Tukey. Tukey FFT Algorithms. FFT Algorithms. Cooley Cooley Cooley-Tuey Tuey FFT Algorithms FFT Algorithms Cosider a legth- sequece x[ with a -poit DFT X[ where Represet the idices ad as +, +, Cooley Cooley-Tuey Tuey FFT Algorithms FFT Algorithms Usig these

More information

G r a d e. 5 M a t h e M a t i c s. Number

G r a d e. 5 M a t h e M a t i c s. Number G r a d e 5 M a t h e M a t i c s Number Grade 5: Number (5.N.1) edurig uderstadigs: the positio of a digit i a umber determies its value. each place value positio is 10 times greater tha the place value

More information

Theorems About Power Series

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

More information

Domain 1: Designing a SQL Server Instance and a Database Solution

Domain 1: Designing a SQL Server Instance and a Database Solution Maual SQL Server 2008 Desig, Optimize ad Maitai (70-450) 1-800-418-6789 Domai 1: Desigig a SQL Server Istace ad a Database Solutio Desigig for CPU, Memory ad Storage Capacity Requiremets Whe desigig a

More information

Repeated multiplication is represented using exponential notation, for example:

Repeated multiplication is represented using exponential notation, for example: Appedix A: The Lws of Expoets Expoets re short-hd ottio used to represet my fctors multiplied together All of the rules for mipultig expoets my be deduced from the lws of multiplictio d divisio tht you

More information

Discrete Mathematics and Probability Theory Spring 2014 Anant Sahai Note 13

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

More information

FM4 CREDIT AND BORROWING

FM4 CREDIT AND BORROWING FM4 CREDIT AND BORROWING Whe you purchase big ticket items such as cars, boats, televisios ad the like, retailers ad fiacial istitutios have various terms ad coditios that are implemeted for the cosumer

More information

Descriptive Statistics

Descriptive Statistics Descriptive Statistics We leared to describe data sets graphically. We ca also describe a data set umerically. Measures of Locatio Defiitio The sample mea is the arithmetic average of values. We deote

More information

5 Boolean Decision Trees (February 11)

5 Boolean Decision Trees (February 11) 5 Boolea Decisio Trees (February 11) 5.1 Graph Coectivity Suppose we are give a udirected graph G, represeted as a boolea adjacecy matrix = (a ij ), where a ij = 1 if ad oly if vertices i ad j are coected

More information

Elementary Theory of Russian Roulette

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

More information

1 Computing the Standard Deviation of Sample Means

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

More information

Department of Computer Science, University of Otago

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

More information

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

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

More information

Modified Line Search Method for Global Optimization

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

More information

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

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

More information

2-3 The Remainder and Factor Theorems

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

More information

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

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

More information

Working with numbers

Working with numbers 4 Workig with umbers this chapter covers... This chapter is a practical guide showig you how to carry out the types of basic calculatio that you are likely to ecouter whe workig i accoutig ad fiace. The

More information

INVESTMENT PERFORMANCE COUNCIL (IPC)

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

More information

Confidence Intervals. CI for a population mean (σ is known and n > 30 or the variable is normally distributed in the.

Confidence Intervals. CI for a population mean (σ is known and n > 30 or the variable is normally distributed in the. Cofidece Itervals A cofidece iterval is a iterval whose purpose is to estimate a parameter (a umber that could, i theory, be calculated from the populatio, if measuremets were available for the whole populatio).

More information

COMPUTER LABORATORY IMPLEMENTATION ISSUES AT A SMALL LIBERAL ARTS COLLEGE. Richard A. Weida Lycoming College Williamsport, PA 17701 weida@lycoming.

COMPUTER LABORATORY IMPLEMENTATION ISSUES AT A SMALL LIBERAL ARTS COLLEGE. Richard A. Weida Lycoming College Williamsport, PA 17701 weida@lycoming. COMPUTER LABORATORY IMPLEMENTATION ISSUES AT A SMALL LIBERAL ARTS COLLEGE Richard A. Weida Lycomig College Williamsport, PA 17701 weida@lycomig.edu Abstract: Lycomig College is a small, private, liberal

More information