FRACTIONS AND THE INDEX LAWS IN ALGEBRA
|
|
- Edwin Howard
- 7 years ago
- Views:
Transcription
1 The Improving Mthemtics Eduction in Schools (TIMES) Project FRACTIONS AND THE INDEX LAWS IN ALGEBRA A guide for techers - Yers 8 9 June 2011 NUMBER AND ALGEBRA Module 32 8YEARS 9
2 Frctions nd the Index Lws in Algebr (Number nd lgebr : Module 32) For techers of Primry nd Secondry Mthemtics 510 Cover design, Lyout design nd Typesetting by Clire Ho The Improving Mthemtics Eduction in Schools (TIMES) Project ws funded by the Austrlin Government Deprtment of Eduction, Employment nd Workplce Reltions. The views expressed here re those of the uthor nd do not necessrily represent the views of the Austrlin Government Deprtment of Eduction, Employment nd Workplce Reltions. The University of Melbourne on behlf of the Interntionl Centre of Excellence for Eduction in Mthemtics (ICE EM), the eduction division of the Austrlin Mthemticl Sciences Institute (AMSI), 2010 (except where otherwise indicted). This work is licensed under the Cretive Commons Attribution- NonCommercil-NoDerivs 3.0 Unported License
3 The Improving Mthemtics Eduction in Schools (TIMES) Project FRACTIONS AND THE INDEX LAWS IN ALGEBRA A guide for techers - Yers 8 9 June 2011 NUMBER AND ALGEBRA Module 32 Peter Brown Michel Evns Dvid Hunt Jnine McIntosh Bill Pender Jcqui Rmgge YEARS 8 9
4 {4} A guide for techers FRACTIONS AND THE INDEX LAWS IN ALGEBRA ASSUMED KNOWLEDGE The rithmetic of integers nd of frctions. Bsic methods of mentl rithmetic. The HCF nd LCM in rithmetic. Algebric expressions with whole numbers nd negtives. The index lws in rithmetic with nonzero whole numbers indices. The three index lws involving products in lgebr. Liner equtions MOTIVATION The vrious uses of lgebr require systemtic skills in mnipulting lgebric expressions. This module is the third of four modules tht provide systemtic introduction to bsic lgebric skills. In the module, Algebric Expressions, we introduced lgebr using only whole numbers nd the occsionl frction for the pronumerls. Then in the module Negtives nd the Index Lws in Algebr we extended the methods of tht module to expressions involving negtives. The present module extends the methods of lgebr so tht ll negtive frctions nd decimls cn lso be substituted into lgebric expressions, nd cn pper s the solutions of lgebric equtions. Although erlier modules occsionlly used negtive frctions, this module provides the first systemtic ccount of them, nd begins by presenting the four opertions of rithmetic, nd powers, in the context of negtive frctions nd decimls. The resulting number system is clled the rtionl numbers. This system is sufficient for ll the norml clcultions of every life, becuse dding, subtrcting, multiplying nd dividing rtionl numbers (prt from division by zero), nd tking whole number powers of rtionl numbers, lwys produces nother rtionl numbers. In prticulr, scientific clcultions often involve mnipultion of equtions with decimls.
5 The Improving Mthemtics Eduction in Schools (TIMES) Project {5} The module lso begins the discussion of lgebric frctions, which routinely cuse problems for students. In this module, the discussion is quickly restricted by excluding numertors nd denomintors with more thn one term. These more complicted lgebric frctions re then covered in more detil in the fourth module, Specil Expnsions nd Algebric Frctions. Once frctions hve been introduced into lgebr, the two index lws deling with quotients cn be stted in more systemtic form, nd then integrted with lgebr. This completes the discussion of the index lws in rithmetic nd lgebr s fr s nonzero whole number indices re concerned. CONTENT THE RATIONAL NUMBERS Our rithmetic begn with the whole numbers 0, 1, 2, 3, 4, We then constructed extensions of the whole numbers in two different directions. First, in the module, Frctions we dded the positive frctions, such s 2 3 nd 9 2 = 4 1 2, which cn be written s the rtio of whole number nd non zero whole number, nd we showed how to dd, subtrct, multiply nd divide frctions. This gve system of numbers consisting of zero nd ll positive frctions. Within this system, every non zero number hs reciprocl, nd division without reminder is lwys possible (expect by zero). Only zero hs n opposite, however, nd subtrction b is only possible when b. Secondly, in the module, Integers we strted gin with the whole numbers nd dded the opposites of the whole numbers. This gve system of numbers clled the integers, consisting of ll whole numbers nd their opposites,, 3, 2, 1, 0, 1, 2, 3, nd we showed how to dd, subtrct, multiply nd divide integers. Within this system of integers, every number hs n opposite, nd subtrction is lwys possible. Only 1 nd 1 hve reciprocls, however, nd division without reminder is only possible in situtions such s 21 7 when the dividend is n integer multiple of the divisor.
6 {6} A guide for techers Arithmetic needs to be done in system without such restrictions we wnt both subtrction nd division to be possible ll the time (except for division by zero). To chieve this, we strt with the integers nd positive frctions tht we hve lredy. Then we dd the opposites of the positive frctions, so tht our number system now contin ll numbers such s 2 3 nd 9 2 = All these numbers together re clled the rtionl numbers the word rtionl is the djective from rtio. We hve ctully been using negtive frctions in these notes for some time. These remrks re intended to formlise the sitution. The usul lws of signs pply to multiplying nd dividing in this lrger system, so tht frction with negtive integers in the numertor, or in the denomintor, or in both, cn be written s either positive or negtive frction, 2 3 nd 2 3 = 2 3 nd 2 3 = 2 3 Becuse of these identities, it is usul to define rtionl number s the rtio of n integer nd non zero integer: A rtionl number is number tht cn be written s frction, where is n integer nd b is non zero integer. Thus positive nd negtive mixed numerls, terminting decimls, nd recurring decimls re rtionl numbers becuse they cn be written s frctions of integers. For exmple, = , = 1000, 1.09 = 11 In lter modules on topics such s surds, circles, trigonometry nd logrithms, we will need to introduce further numbers, clled irrtionl numbers, tht cnnot be written s frctions. Some exmples of such numbers re 2, π, sin 22, log 2 5. The module, The Rel Numbers will discuss the resulting more generl system of rithmetic. Show tht ech number below is rtionl by writing it s frction b, where nd b re integers with b 0. 5 b 3 7 c d e 4.7 f = 5 1 b 3 7 = 3 7 c 51 3 = 16 3 d = e 4.7 = f 1.3 = 4 3
7 The Improving Mthemtics Eduction in Schools (TIMES) Project {7} Constructing the rtionl numbers There re two wys to construct the rtionl numbers. The first wy is to construct the positive frctions first nd then the negtives, which is roughly wht hppened historiclly. This involves three step procedure: 1 First construct the whole numbers 0, 1, 2, 3, 4, 2 Then construct the non negtive frctions, such s 3 4 nd 9 4, s the rtio of two whole numbers, where the second is non zero. This process involves identifying frction such s 2 6, where the two whole numbers hve common fctor, with its simplest form 1 3, nd identifying frction such s 5 1, where the denomintor is 1, with the whole number 5. 3 Lstly construct the rtionl numbers by constructing the opposites of every positive frction nd whole number. The other method is to construct the integers first nd then the frctions. The three steps re now: 1 First construct the whole numbers 0, 1, 2, 3, 4,.. 2 Then construct the integers by constructing the opposites, 4, 3, 2, 1 of ll the nonzero whole numbers. 3 Lstly, construct the rtionl numbers by constructing ll the frctions such s 3 4 nd 9 4 the rtio of two integers, where the second is non zero. This process gin involves identifying equivlent frctions, nd identifying frctions with denomintor 1 s integers. These two procedures re eqully cceptble mthemticlly, nd their finl results re identicl in structure. Students lern bout rtionl numbers, however, by working with exmples of them in rithmetic nd lgebr, nd there is little to be gined by such explntions. ADDING AND SUBTRACTING NEGATIVE FRACTIONS AND DECIMALS When dding nd subtrcting frctions nd decimls tht my be positive or negtive, the rules re combintion of the rules for integers nd the rules for dding nd subtrcting positive frctions nd decimls: To subtrct negtive number, dd its opposite. Perform ddition nd subtrction from left to right. When deling with mixed numerls, it is usully esier to seprte the integer prt nd the frction.
8 {8} A guide for techers The following exmples demonstrte such clcultions first in rithmeticl exmples, then in substitutions nd equtions. Simplify: b ( 23.6) ( 26.4) = b ( 23.6) ( 26.4) = = 1 8. = Evlute x y + z when x = , y = nd z = b x = 3.6, y = 5.6 nd z = x y + z = = = = Bewre tht in nottion , the negtive sign pplies to the whole number 4012, so tht mens = b x y + z = ( 3.6) ( 2.07) = = 4.07 EXERCISE 1 Simplify
9 The Improving Mthemtics Eduction in Schools (TIMES) Project {9} MULTIPLYING NEGATIVE FRACTIONS AND DECIMALS As with ddition nd subtrction, we need to combine the rules lredy discussed: First determine the sign of the product, then del with the numbers. The product of two positives or two negtives is positive. The product of positive nd negtive is negtive. In the bsence of brckets, the order of opertions is: 1 Powers, 2 Multipliction nd division from left to right, 3 Addition nd subtrction from left to right. When multiplying frctions, first convert them to improper frctions. When multiplying decimls, convert them to frctions whose denomintors re powers of 10. (Some people prefer to use the rule, Before discrding triling zeroes, the number of deciml plces in the nswer equls the totl number decimls plces in the question. ) The following exmples demonstrte these procedures first in rithmeticl exmples, then with substitutions nd equtions. Simplify: b 1.2 ( 0.2) 3 The product is negtive becuse there re five minus signs in the product = = 6 b The product is positive becuse there re four minus signs in the product. 1.2 ( 0.2) 3 = = =
10 {10} A guide for techers Simplify 3x 2 yz when: x = 1 1 2, y = nd z = b x = 1.1, y = 0.4 nd z = x 2 yz = (The 3 is not squred indices come first) = = = (Cre with mixed numerls: 63 4 mens ). = b 3x 2 yz = = = = 1.21 Solve ech eqution: x = 53 4 b x = 2 x = 53 4 x 3 = x= ( 3) In the lst line, it ws esier to clculte = thn to use improper frctions b x = 2 x 1.2 = x= = EXERCISE 2 How mny fctors must be tken in the product ( 0.8) 0.9 ( 1.0) for the product to exceed 1?
11 The Improving Mthemtics Eduction in Schools (TIMES) Project {11} DIVIDING NEGATIVE FRACTIONS AND DECIMALS We hve lredy reviewed the order of opertions. The other necessry rules re: First determine the sign of the quotient, then del with the numbers. The quotient of two positives or two negtives is positive. The quotient of positive nd negtive is negtive. When dividing by frction, multiply by its reciprocl. An improper frction must first be converted to n improper frction. To divide by deciml, write the clcultion s frction, then multiply top nd bottom by power of 10 lrge enough to mke the divisor whole number. (Some people prefer to chnge both decimls to frctions nd then use frction methods.) Simplify: b 0.9 ( 0.08) ( 0.06) = = 3 5. b 0.9 ( 0.08) ( 0.06) 2 = = (Multiply top nd bottom by ) = 20. Simplify 3 b c when: = 8 9, b = nd c = 2. b = 0.09, b = nd c = b c = b 3 b c = ( 0.1) = = = = = 9.9
12 {12} A guide for techers The reciprocl We hve seen tht the reciprocl of positive frction such s 2 5 is the frction 5 2 formed by interchnging the numertor nd denomintor, becuse = 1 Similrly, the reciprocl of negtive frction such s 2 5 is 5 2, becuse = 1 To form the reciprocl, interchnge numertor nd denomintor, nd do not chnge the sign. Reciprocls of positive nd negtive frctions re importnt in vrious lter pplictions, prticulrly perpendiculr grdients. Find, s mixed numerl when pproprite, the reciprocls of: 4 13 b c d The reciprocl of 4 13 is 13 4 = b The reciprocl of is 20 = c The reciprocl of = 31 6 is d The reciprocl of = is 77. Compound frctions There is very strightforwrd wy to simplify compound frction such s To simplify compound frction, multiply top nd bottom by the lowest common denomintor of ll the frctions. In the exmple bove, the lowest common denomintor of ll the frctions is 12, so multiplying top nd bottom by = = It is lso possible to rewrite the frction s division , but such n pproch requires more steps.
13 The Improving Mthemtics Eduction in Schools (TIMES) Project {13} Simplify b Evlute x + 2 x when x = nd = 3 5 Multiply top nd bottom by = = 1 17 b Multiply top nd bottom by x + 2 x = = = = Solving equtions with frctions An eqution with severl frctions cn be solved by the stndrd method, Move every term in x to one side, nd ll the constnts to the other. The resulting frction clcultions, however, cn be quite elborte. There is fr quicker pproch tht gets rid of ll the frctions in one single step, If n eqution involves frctions, multiply by the lowest common denomintor. Similrly, if n eqution involves decimls, multiply through by suitble power of 10. The following exmple uses first this quicker pproch, then the stndrd method, to solve two equtions. Compre the pproches nd decide for yourself. Use the quicker pproch bove, nd then the stndrd pproch, to solve these equtions. Solve: 5 6 x = b 0.024x = x = Multiply by 6 to remove ll frctions. 5x + 3 = x = 23 3 x = ( 5) or
14 {14} A guide for techers 5 6 x = x = x = 35 6 x = = b 0.024x = 0.04 Multiply by 1000 to remove ll decimls. 24x = x = x = 2.5 ( 24) or 0.024x = x = = ( 0.024) = = 2.5 EXERCISE 3 Use n efficient method to find the men of the numbers 1, 0.9, 0.8,, 2. The system of rtionl numbers We hve now introduced four number systems in turn. First, we introduced the whole numbers 0, 1, 2, 3, 4, The set of whole numbers is closed under ddition nd multipliction, which mens tht when we dd or multiply two whole numbers, we obtin nother whole number. The whole numbers re not closed under either subtrction or division, however, becuse, for exmple, 7 10 is not whole number nd 7 10 is not whole number. Next, we dded the positive frctions to the whole numbers, so tht our number system now contined ll non negtive rtionl numbers, including numbers such s 4, 2 3, nd This system is closed under division (except by zero), but is still not closed under subtrction.
15 The Improving Mthemtics Eduction in Schools (TIMES) Project {15} Then we introduced the integers, 3, 2, 1, 0, 1, 2, 3, consisting of ll whole numbers nd their opposites. This system is closed under subtrction, but not under division. Now tht we re working in the rtionl numbers the integers together with ll positive nd negtive frctions we finlly hve system tht is closed under ll four opertions of ddition, subtrction, multipliction nd division (except by zero). They re lso closed under the opertion of tking whole number powers. This set of rtionl numbers is therefore very stisfctory system for doing rithmetic, nd is quite sufficient for ll everydy needs. When we move to the even lrger system of rel numbers, every rel number cn be pproximted s close s we like by rtionl numbers (indeed by terminting decimls), so tht the rtionl numbers will remin extremely importnt. CANCELLING, MULTIPLYING AND DIVIDING ALGEBRAIC FRACTIONS An lgebric frction is frction involving pronumerls, x 3, 5 x, + b c, 3r s t. The pronumerls in the frction represent numbers, so we del with lgebric frctions using exctly the sme procedures tht we use in rithmetic. Substitution into lgebric frctions We hve seen in rithmetic tht the denomintor in frction cnnot be zero. This mens tht: In the expression 5 x, we cnnot substitute x = 0. In the expression + b c, we cnnot substitute c = 0. In the expression 3r s t, we cnnot substitute the sme vlue for s nd t. Aprt from this qulifiction, substitution into lgebric frctions works exctly the sme s substitution into ny lgebric expression. For exmple, if r = 5, s = 7 nd t = 4, then 3r s t = = 5 Substitution of negtive numbers requires the usul cre with signs. If r = 2, s = 7 nd t = 5, 3r s t = = 6 12 = 1 2
16 {16} A guide for techers Wht vlue of cnnot be substituted into + 3 3? b Wht vlue of mkes zero? c Evlute when = 2. When = 3, the denomintor is zero, nd the expression cnnot be evluted. b When = 3, the numertor is zero but the denomintor is not zero, so the expression is zero. c When = 2, = = 1 5 = 1 5 The rest of this module will exclude denomintors, such s s t nd 3, tht hve two or more terms. Such expressions will be considered in the lter module, Specil Expnsions nd Algebric Frctions. Cncelling lgebric frctions Becuse pronumerls re just numbers, we cn cncel both pronumerls nd numbers in n lgebric frction. For exmple, 3xy 6x = y 2 nd 2 b b =. We cnnot cncel zero, so strictly we should exclude zero vlues of the pronumerl when we cncel. This would men writing 3xy 6x = y 2 provided x 0 nd 2 b = provided 0 nd b 0. b Such qulifictions become importnt much lter in clculus. At this stge, however, it is not pproprite to bring the mtter up, unless students insist. Simplify ech lgebric frction by cncelling ny common fctors: 5 10b b 9b 12bc c 4b2 6b d 12xy2 z 12x2yz 5 10b = 1 2b b 9b 12bc = 3 4c c 4b2 6b = 2b 3 d 12xy2 z 12x2yz = y x
17 The Improving Mthemtics Eduction in Schools (TIMES) Project {17} Dividing lgebric expressions When we divide, the lws of signs re the sme s the lws for multiplying: The quotient of two negtives or two positives is positive. The quotient of positive nd negtive is negtive. The following exmples illustrte ll the possibilities: 6x2 2x = 3x 6x2 2x = 3x 6x2 2x = 3x 6x2 2x = 3x As with multipliction, there re three steps in hndling more complicted expressions: First del with the signs. Then del with the numerls. Then del with ech pronumerl in turn. 3xy 6x2 = y 2x nd (21bc 2 ) ( 7b 2 c) = 3c b The lst nswer cn lso be written s 3c b nswer 3c b, with the negtive sign out the front of the frction. 3c or even s b, but the best wy to write the Frction nottion is usully used for quotients, but the division sign cn lso be used, s in the second exmple bove. Simplify ech quotient: ( 12b) ( 3bc) b 8x2 4x c 3bc 2bc3 d 92 6b2 c ( 12b) ( 3bc) = 4 c 3bc 2bc3 = 2 2c2 b 8x2 4x = 2x d 92 6b2 = 32 2b2 Multiplying lgebric frctions We hve seen how to multiply frctions in rithmetic, = = First del with the signs. Then cncel ny common fctors from numertor nd denomintor. Then multiply the numertors, nd multiply the denomintors.
18 {18} A guide for techers Exctly the sme method pplies to multiplying lgebric frctions. For exmple, 3 x x = = 3 5x 6y 6xy 25 = x 1 x 5 = x 2 5 Simplify: 2x 3 6 x b 2b c bc 2x 3 6 x = b 2b c bc = 2 c c = 4 = 3 c2 Dividing by n lgebric frction We know the lw of signs for division, nd we know how to divide by frction. To divide by n lgebric frction, we pply these sme lws in succession 1 First determine the sign of the quotient. 2 To divide by frction, multiply by its reciprocl For exmple, 4 x 6 x = 4 x x 6 = b c 3b2 c = 32b c c 3b2 = 1 1 b = 2 3 = b Simplify ech quotient: 12 x 3 b 6x 2 x y 4x y2 c 3b c ( 6bc) d 22 c 2 c2 12 x 3 x = 12 x x 3 b 6x2 y c 3b c = x y2 = 6x2 y y 2 4y = 4 = 3xy 2 3b ( 6bc) = c 1 6bc d 22 c 2 c2 = 6x2 y y 2 4y = 22 c c2 2 = 1 c 1 2c = 1 c 1 = 1 2c2 = c
19 The Improving Mthemtics Eduction in Schools (TIMES) Project {19} The reciprocl of n lgebric frction Reciprocls of lgebric frctions re formed in exctly the sme wy s ws done in rithmetic. The reciprocl of 2x y is y 2x 2x, becuse y y 2x = 1. The reciprocl of 3 is 1 3, becuse ( 3 ) 1 3 = 1. The reciprocl of 7xy 3t is 3t 7xy, becuse 7xy 3t 3t 7xy = 1. To form the reciprocl, interchnge numertor nd denomintor, nd do not chnge the sign. Write down the reciprocl of ech lgebric frction: 2 b b 2 b c 6x2 7yz d 6x2 7yz b 2 b 2 b c 7yz d 7yz 6x 2 6x 2 THE INDEX LAWS FOR FRACTIONS The previous lgebr module Negtives nd the Index Lws in Algebr delt with the index lws for product of powers of the sme bse, for power of power, nd for power of product: To multiply powers of the sme bse, dd the indices: m n = m+n. To rise power to power, dd the indices: ( m ) n = mn. The power of product os the product of the powers: (b) n = n b n. In these identities, nd b re ny numbers, nd m nd n re ny nonzero whole numbers. We now turn to the remining two index lws, which involve quotients. The index lw for the power of quotient To see wht hppens when we tke power of frction, we simply write out the power = b = b b b b b = = 5 b 5
20 {20} A guide for techers The generl sitution cn thus be expressed s: The power of quotient is the quotient of the powers: n b = n n where nd b 0 re numbers, nd n is nonzero whole number. b The following exmples illustrte the use of the lw in rithmetic nd lgebr. Simplify: b 3 4 c 1 2 d c = b = = = = 2 d = 10 3 = 81 4 = = = When pplied to n lgebric frction, the lws previously introduced for the power of product nd for the power of power, re often required s well. For exmple, 3 2x 3 8x 3y = 3 27y 3 nd = Expnd the brckets: 2 2x 3y b 5 3 b c 2 2 b 3 d x 4x 3y = 2 b y2 b = 3 b 3 c 2 2 b = 4 3 b6 d = The index lw for the quotient of powers of the sme bse The remining index lw from rithmetic dels with dividing powers of the sme bse. The module Multiples, Fctors nd Powers developed the lw in the form, To divide powers of the sme bse, subtrct the indices. For exmple, = ( ) (7 7 7) = 7 2. Frctions, however, hd not been introduced t tht stge, so the divisor could not be
21 The Improving Mthemtics Eduction in Schools (TIMES) Project {21} higher power thn the dividend. Now tht we hve frctions, we cn express the lw in frctionl nottion nd do wy with the restriction: = = = = 7 2 = 1 = The bse cn lso be pronumerl, nd the lw cn now be written s: To divide powers of the sme bse, subtrct the indices. For exmple, 5 3 = 2 nd 4 4 = 1 nd 3 5 = 1 2. Lter, when zero nd negtive indices hve been introduced in the module The Index Lws, we will be ble to stte this lw in its usul, more concise form: m n = m n. Simplify ech expression: 3x 6 x4 b 8 b 6 6 b 8 c 34 4b8 4b4 38 d 3 b 2 6 b 6 c 6 3x 6 = 3x2 x 4 b 8 b 6 = 2 6 b c 34 4b4 = 1 8 b2 4b b d 3 b 2 6 b 6 c 6 1 = 4 3 b 4 c 6 ADDING AND SUBTRACTING ALGEBRAIC FRACTIONS When set of lgebric frctions hve common denomintor, it is strightforwrd to dd them. As lwys, we collect like terms, 3 x + x 7 x + 4 x = 5 x 4 x (which is perhps written s 5 4 x ) Simplify ech expression: 5 x x 4 x b 7 2x 11 2x 20 2x 5 x x 4 x = 1 x x, or 1 x b 7 2x 11 2x 20 2x = 24 2x = 12 x
22 {22} A guide for techers Using common denomintor When the frctions do not hve common denomintor, we need to find the lowest common denomintor, just s we did for rithmetic frctions. We shll restrict the discussion to numericl denomintors t this stge, nd leve denomintors with pronumerls to the module Specil Expnsions nd Algebric Frctions. x 4 x 6 = 3x 12 2x 12 = x 12. Simplify ech expression: x 2 x 3 + x 4 + x 12 b 3x _ 5x 8 6 x 2 x 3 + x 4 + x 12 = 6x _ 4x x 12 + x 12 b 3x _ 5x 8 6 = 9x _ 20x = 6x 12 = 11x 24 = x 2 LINKS FORWARD As mentioned in the Motivtion section, this module is the third of four modules deling with bsic mnipultion of lgebric expressions. The four modules re: Algebric Expressions Negtives nd the Index Lws in Algebr Frctions nd the Index Lws in Algebr (the present module) Specil Expnsions nd Algebric Frctions Algebric frctions hve been introduced in this module, but the lgebr of lgebric frctions cn cuse considerble difficulty if it is introduced in full strength before more fundmentl skills hve been ssimilted. Algebric frctions will therefore be considered in more detils in the fourth module Specil Expnsions nd Algebric Frctions. The index lws will be gretly expnded in scope once negtive nd frctionl indices re introduced in the module Indices. One piece of clumsiness in the present module concerned the quotient of the difference of powers, where we needed to write down three different exmples x5 = x 2 x 3 nd x4 x 4 = 1 nd x3 x 5 = 1 x 2. Once the zero index nd negtive indices re brought into lgebr, ll this cn be replced
23 The Improving Mthemtics Eduction in Schools (TIMES) Project {23} with the single lw xm xn = x m n where m nd n re integers. Squre roots nd cube roots cn lso be represented using frctionl indices for exmple, 5 is written s nd 3 5 is written s nd yet the five index lws still hold even when the index is ny rtionl number. Negtive nd frctionl indices re introduced in the module, Indices nd Logrithms. Lter in clculus, n index cn be ny rel number. ANSWERS TO EXERCISES EXERCISE 1 The lowest common denomintor is 60. Grouping the numbers in pirs before dding them ll reduces the size of the numertor. Sum = = = = EXERCISE 2 We cn ignore ll negtive products. Since ( 0.8) 0.9 ( 1.0) 1.1 is less thn 1, the first product greter thn 1 is ( 0.8) 0.9 ( 1.0) 1.1 ( 1.2) 1.3 ( 1.4). EXERCISE 3 One pproch is to notice tht when the 31 numbers re dded, the first 21 numbers cncel out. The remining 10 numbers cn then be dded by grouping them in pirs: ( 2) = = ( 1.1 2) +( ) + ( ) + ( ) + ( ) = = Hence men = = = 0.5. A better pproch is relise tht 0.5 is the middle number, nd tht the other 30 numbers fll into pirs eqully spced one to the left nd one to the right of 0.5. Hence the men is 0.5.
24 The im of the Interntionl Centre of Excellence for Eduction in Mthemtics (ICE-EM) is to strengthen eduction in the mthemticl sciences t ll levelsfrom school to dvnced reserch nd contemporry pplictions in industry nd commerce. ICE-EM is the eduction division of the Austrlin Mthemticl Sciences Institute, consortium of 27 university mthemtics deprtments, CSIRO Mthemticl nd Informtion Sciences, the Austrlin Bureu of Sttistics, the Austrlin Mthemticl Society nd the Austrlin Mthemtics Trust. The ICE-EM modules re prt of The Improving Mthemtics Eduction in Schools (TIMES) Project. The modules re orgnised under the strnd titles of the Austrlin Curriculum: Number nd Algebr Mesurement nd Geometry Sttistics nd Probbility The modules re written for techers. Ech module contins discussion of component of the mthemtics curriculum up to the end of Yer 10.
PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY
MAT 0630 INTERNET RESOURCES, REVIEW OF CONCEPTS AND COMMON MISTAKES PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY Contents 1. ACT Compss Prctice Tests 1 2. Common Mistkes 2 3. Distributive
More informationExample 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers.
2 Rtionl Numbers Integers such s 5 were importnt when solving the eqution x+5 = 0. In similr wy, frctions re importnt for solving equtions like 2x = 1. Wht bout equtions like 2x + 1 = 0? Equtions of this
More informationSPECIAL PRODUCTS AND FACTORIZATION
MODULE - Specil Products nd Fctoriztion 4 SPECIAL PRODUCTS AND FACTORIZATION In n erlier lesson you hve lernt multipliction of lgebric epressions, prticulrly polynomils. In the study of lgebr, we come
More informationAlgebra Review. How well do you remember your algebra?
Algebr Review How well do you remember your lgebr? 1 The Order of Opertions Wht do we men when we write + 4? If we multiply we get 6 nd dding 4 gives 10. But, if we dd + 4 = 7 first, then multiply by then
More informationOperations with Polynomials
38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: Write polynomils in stndrd form nd identify the leding coefficients nd degrees of polynomils Add nd subtrct polynomils Multiply
More informationPolynomial Functions. Polynomial functions in one variable can be written in expanded form as ( )
Polynomil Functions Polynomil functions in one vrible cn be written in expnded form s n n 1 n 2 2 f x = x + x + x + + x + x+ n n 1 n 2 2 1 0 Exmples of polynomils in expnded form re nd 3 8 7 4 = 5 4 +
More informationFactoring Polynomials
Fctoring Polynomils Some definitions (not necessrily ll for secondry school mthemtics): A polynomil is the sum of one or more terms, in which ech term consists of product of constnt nd one or more vribles
More informationMath 135 Circles and Completing the Square Examples
Mth 135 Circles nd Completing the Squre Exmples A perfect squre is number such tht = b 2 for some rel number b. Some exmples of perfect squres re 4 = 2 2, 16 = 4 2, 169 = 13 2. We wish to hve method for
More informationBinary Representation of Numbers Autar Kaw
Binry Representtion of Numbers Autr Kw After reding this chpter, you should be ble to: 1. convert bse- rel number to its binry representtion,. convert binry number to n equivlent bse- number. In everydy
More informationFUNCTIONS AND EQUATIONS. xεs. The simplest way to represent a set is by listing its members. We use the notation
FUNCTIONS AND EQUATIONS. SETS AND SUBSETS.. Definition of set. A set is ny collection of objects which re clled its elements. If x is n element of the set S, we sy tht x belongs to S nd write If y does
More informationVectors 2. 1. Recap of vectors
Vectors 2. Recp of vectors Vectors re directed line segments - they cn be represented in component form or by direction nd mgnitude. We cn use trigonometry nd Pythgors theorem to switch between the forms
More informationMathematics. Vectors. hsn.uk.net. Higher. Contents. Vectors 128 HSN23100
hsn.uk.net Higher Mthemtics UNIT 3 OUTCOME 1 Vectors Contents Vectors 18 1 Vectors nd Sclrs 18 Components 18 3 Mgnitude 130 4 Equl Vectors 131 5 Addition nd Subtrction of Vectors 13 6 Multipliction by
More informationGraphs on Logarithmic and Semilogarithmic Paper
0CH_PHClter_TMSETE_ 3//00 :3 PM Pge Grphs on Logrithmic nd Semilogrithmic Pper OBJECTIVES When ou hve completed this chpter, ou should be ble to: Mke grphs on logrithmic nd semilogrithmic pper. Grph empiricl
More informationBabylonian Method of Computing the Square Root: Justifications Based on Fuzzy Techniques and on Computational Complexity
Bbylonin Method of Computing the Squre Root: Justifictions Bsed on Fuzzy Techniques nd on Computtionl Complexity Olg Koshelev Deprtment of Mthemtics Eduction University of Texs t El Pso 500 W. University
More information4.11 Inner Product Spaces
314 CHAPTER 4 Vector Spces 9. A mtrix of the form 0 0 b c 0 d 0 0 e 0 f g 0 h 0 cnnot be invertible. 10. A mtrix of the form bc d e f ghi such tht e bd = 0 cnnot be invertible. 4.11 Inner Product Spces
More informationSection 5-4 Trigonometric Functions
5- Trigonometric Functions Section 5- Trigonometric Functions Definition of the Trigonometric Functions Clcultor Evlution of Trigonometric Functions Definition of the Trigonometric Functions Alternte Form
More information9.3. The Scalar Product. Introduction. Prerequisites. Learning Outcomes
The Sclr Product 9.3 Introduction There re two kinds of multipliction involving vectors. The first is known s the sclr product or dot product. This is so-clled becuse when the sclr product of two vectors
More informationOr more simply put, when adding or subtracting quantities, their uncertainties add.
Propgtion of Uncertint through Mthemticl Opertions Since the untit of interest in n eperiment is rrel otined mesuring tht untit directl, we must understnd how error propgtes when mthemticl opertions re
More informationLINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES
LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES DAVID WEBB CONTENTS Liner trnsformtions 2 The representing mtrix of liner trnsformtion 3 3 An ppliction: reflections in the plne 6 4 The lgebr of
More information5.6 POSITIVE INTEGRAL EXPONENTS
54 (5 ) Chpter 5 Polynoils nd Eponents 5.6 POSITIVE INTEGRAL EXPONENTS In this section The product rule for positive integrl eponents ws presented in Section 5., nd the quotient rule ws presented in Section
More informationMATH 150 HOMEWORK 4 SOLUTIONS
MATH 150 HOMEWORK 4 SOLUTIONS Section 1.8 Show tht the product of two of the numbers 65 1000 8 2001 + 3 177, 79 1212 9 2399 + 2 2001, nd 24 4493 5 8192 + 7 1777 is nonnegtive. Is your proof constructive
More informationAppendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered:
Appendi D: Completing the Squre nd the Qudrtic Formul Fctoring qudrtic epressions such s: + 6 + 8 ws one of the topics introduced in Appendi C. Fctoring qudrtic epressions is useful skill tht cn help you
More information6.2 Volumes of Revolution: The Disk Method
mth ppliction: volumes of revolution, prt ii Volumes of Revolution: The Disk Method One of the simplest pplictions of integrtion (Theorem ) nd the ccumultion process is to determine so-clled volumes of
More informationNQF Level: 2 US No: 7480
NQF Level: 2 US No: 7480 Assessment Guide Primry Agriculture Rtionl nd irrtionl numers nd numer systems Assessor:.......................................... Workplce / Compny:.................................
More informationUnit 6: Exponents and Radicals
Eponents nd Rdicls -: The Rel Numer Sstem Unit : Eponents nd Rdicls Pure Mth 0 Notes Nturl Numers (N): - counting numers. {,,,,, } Whole Numers (W): - counting numers with 0. {0,,,,,, } Integers (I): -
More informationChapter 2 The Number System (Integers and Rational Numbers)
Chpter 2 The Number System (Integers nd Rtionl Numbers) In this second chpter, students extend nd formlize their understnding of the number system, including negtive rtionl numbers. Students first develop
More informationExperiment 6: Friction
Experiment 6: Friction In previous lbs we studied Newton s lws in n idel setting, tht is, one where friction nd ir resistnce were ignored. However, from our everydy experience with motion, we know tht
More informationCHAPTER 11 Numerical Differentiation and Integration
CHAPTER 11 Numericl Differentition nd Integrtion Differentition nd integrtion re bsic mthemticl opertions with wide rnge of pplictions in mny res of science. It is therefore importnt to hve good methods
More informationIntegration by Substitution
Integrtion by Substitution Dr. Philippe B. Lvl Kennesw Stte University August, 8 Abstrct This hndout contins mteril on very importnt integrtion method clled integrtion by substitution. Substitution is
More informationRepeated 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 informationwww.mathsbox.org.uk e.g. f(x) = x domain x 0 (cannot find the square root of negative values)
www.mthsbo.org.uk CORE SUMMARY NOTES Functions A function is rule which genertes ectl ONE OUTPUT for EVERY INPUT. To be defined full the function hs RULE tells ou how to clculte the output from the input
More informationEcon 4721 Money and Banking Problem Set 2 Answer Key
Econ 472 Money nd Bnking Problem Set 2 Answer Key Problem (35 points) Consider n overlpping genertions model in which consumers live for two periods. The number of people born in ech genertion grows in
More informationBasic Analysis of Autarky and Free Trade Models
Bsic Anlysis of Autrky nd Free Trde Models AUTARKY Autrky condition in prticulr commodity mrket refers to sitution in which country does not engge in ny trde in tht commodity with other countries. Consequently
More informationIntegration. 148 Chapter 7 Integration
48 Chpter 7 Integrtion 7 Integrtion t ech, by supposing tht during ech tenth of second the object is going t constnt speed Since the object initilly hs speed, we gin suppose it mintins this speed, but
More informationReasoning to Solve Equations and Inequalities
Lesson4 Resoning to Solve Equtions nd Inequlities In erlier work in this unit, you modeled situtions with severl vriles nd equtions. For exmple, suppose you were given usiness plns for concert showing
More informationALGEBRAIC FRACTIONS,AND EQUATIONS AND INEQUALITIES INVOLVING FRACTIONS
CHAPTER ALGEBRAIC FRACTIONS,AND EQUATIONS AND INEQUALITIES INVOLVING FRACTIONS Although people tody re mking greter use of deciml frctions s they work with clcultors, computers, nd the metric system, common
More information1.2 The Integers and Rational Numbers
.2. THE INTEGERS AND RATIONAL NUMBERS.2 The Integers n Rtionl Numers The elements of the set of integers: consist of three types of numers: Z {..., 5, 4, 3, 2,, 0,, 2, 3, 4, 5,...} I. The (positive) nturl
More information5.2. LINE INTEGRALS 265. Let us quickly review the kind of integrals we have studied so far before we introduce a new one.
5.2. LINE INTEGRALS 265 5.2 Line Integrls 5.2.1 Introduction Let us quickly review the kind of integrls we hve studied so fr before we introduce new one. 1. Definite integrl. Given continuous rel-vlued
More informationRIGHT TRIANGLES AND THE PYTHAGOREAN TRIPLETS
RIGHT TRIANGLES AND THE PYTHAGOREAN TRIPLETS Known for over 500 yers is the fct tht the sum of the squres of the legs of right tringle equls the squre of the hypotenuse. Tht is +b c. A simple proof is
More informationHomework 3 Solutions
CS 341: Foundtions of Computer Science II Prof. Mrvin Nkym Homework 3 Solutions 1. Give NFAs with the specified numer of sttes recognizing ech of the following lnguges. In ll cses, the lphet is Σ = {,1}.
More informationA.7.1 Trigonometric interpretation of dot product... 324. A.7.2 Geometric interpretation of dot product... 324
A P P E N D I X A Vectors CONTENTS A.1 Scling vector................................................ 321 A.2 Unit or Direction vectors...................................... 321 A.3 Vector ddition.................................................
More informationUse Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions.
Lerning Objectives Loci nd Conics Lesson 3: The Ellipse Level: Preclculus Time required: 120 minutes In this lesson, students will generlize their knowledge of the circle to the ellipse. The prmetric nd
More informationModule Summary Sheets. C3, Methods for Advanced Mathematics (Version B reference to new book) Topic 2: Natural Logarithms and Exponentials
MEI Mthemtics in Ection nd Instry Topic : Proof MEI Structured Mthemtics Mole Summry Sheets C, Methods for Anced Mthemtics (Version B reference to new book) Topic : Nturl Logrithms nd Eponentils Topic
More informationEQUATIONS OF LINES AND PLANES
EQUATIONS OF LINES AND PLANES MATH 195, SECTION 59 (VIPUL NAIK) Corresponding mteril in the ook: Section 12.5. Wht students should definitely get: Prmetric eqution of line given in point-direction nd twopoint
More informationTreatment Spring Late Summer Fall 0.10 5.56 3.85 0.61 6.97 3.01 1.91 3.01 2.13 2.99 5.33 2.50 1.06 3.53 6.10 Mean = 1.33 Mean = 4.88 Mean = 3.
The nlysis of vrince (ANOVA) Although the t-test is one of the most commonly used sttisticl hypothesis tests, it hs limittions. The mjor limittion is tht the t-test cn be used to compre the mens of only
More informationPentominoes. Pentominoes. Bruce Baguley Cascade Math Systems, LLC. The pentominoes are a simple-looking set of objects through which some powerful
Pentominoes Bruce Bguley Cscde Mth Systems, LLC Astrct. Pentominoes nd their reltives the polyominoes, polycues, nd polyhypercues will e used to explore nd pply vrious importnt mthemticl concepts. In this
More information9 CONTINUOUS DISTRIBUTIONS
9 CONTINUOUS DISTIBUTIONS A rndom vrible whose vlue my fll nywhere in rnge of vlues is continuous rndom vrible nd will be ssocited with some continuous distribution. Continuous distributions re to discrete
More informationRegular Sets and Expressions
Regulr Sets nd Expressions Finite utomt re importnt in science, mthemtics, nd engineering. Engineers like them ecuse they re super models for circuits (And, since the dvent of VLSI systems sometimes finite
More informationRotating DC Motors Part II
Rotting Motors rt II II.1 Motor Equivlent Circuit The next step in our consiertion of motors is to evelop n equivlent circuit which cn be use to better unerstn motor opertion. The rmtures in rel motors
More informationLecture 3 Gaussian Probability Distribution
Lecture 3 Gussin Probbility Distribution Introduction l Gussin probbility distribution is perhps the most used distribution in ll of science. u lso clled bell shped curve or norml distribution l Unlike
More informationMultiplication and Division - Left to Right. Addition and Subtraction - Left to Right.
Order of Opertions r of Opertions Alger P lese Prenthesis - Do ll grouped opertions first. E cuse Eponents - Second M D er Multipliction nd Division - Left to Right. A unt S hniqu Addition nd Sutrction
More informationAA1H Calculus Notes Math1115, Honours 1 1998. John Hutchinson
AA1H Clculus Notes Mth1115, Honours 1 1998 John Hutchinson Author ddress: Deprtment of Mthemtics, School of Mthemticl Sciences, Austrlin Ntionl University E-mil ddress: John.Hutchinson@nu.edu.u Contents
More informationP.3 Polynomials and Factoring. P.3 an 1. Polynomial STUDY TIP. Example 1 Writing Polynomials in Standard Form. What you should learn
33337_0P03.qp 2/27/06 24 9:3 AM Chpter P Pge 24 Prerequisites P.3 Polynomils nd Fctoring Wht you should lern Polynomils An lgeric epression is collection of vriles nd rel numers. The most common type of
More informationSection 7-4 Translation of Axes
62 7 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY Section 7-4 Trnsltion of Aes Trnsltion of Aes Stndrd Equtions of Trnslted Conics Grphing Equtions of the Form A 2 C 2 D E F 0 Finding Equtions of Conics In the
More informationUnderstanding Basic Analog Ideal Op Amps
Appliction Report SLAA068A - April 2000 Understnding Bsic Anlog Idel Op Amps Ron Mncini Mixed Signl Products ABSTRACT This ppliction report develops the equtions for the idel opertionl mplifier (op mp).
More informationand thus, they are similar. If k = 3 then the Jordan form of both matrices is
Homework ssignment 11 Section 7. pp. 249-25 Exercise 1. Let N 1 nd N 2 be nilpotent mtrices over the field F. Prove tht N 1 nd N 2 re similr if nd only if they hve the sme miniml polynomil. Solution: If
More informationMODULE 3. 0, y = 0 for all y
Topics: Inner products MOULE 3 The inner product of two vectors: The inner product of two vectors x, y V, denoted by x, y is (in generl) complex vlued function which hs the following four properties: i)
More information2012 Mathematics. Higher. Finalised Marking Instructions
0 Mthemts Higher Finlised Mrking Instructions Scottish Quliftions Authority 0 The informtion in this publtion my be reproduced to support SQA quliftions only on non-commercil bsis. If it is to be used
More informationNovel Methods of Generating Self-Invertible Matrix for Hill Cipher Algorithm
Bibhudendr chry, Girij Snkr Rth, Srt Kumr Ptr, nd Sroj Kumr Pnigrhy Novel Methods of Generting Self-Invertible Mtrix for Hill Cipher lgorithm Bibhudendr chry Deprtment of Electronics & Communiction Engineering
More informationNumeracy across the Curriculum in Key Stages 3 and 4. Helpful advice and suggested resources from the Leicestershire Secondary Mathematics Team
Numercy cross the Curriculum in Key Stges 3 nd 4 Helpful dvice nd suggested resources from the Leicestershire Secondry Mthemtics Tem 1 Contents pge The development of whole school policy 3 A definition
More informationThinking out of the Box... Problem It s a richer problem than we ever imagined
From the Mthemtics Techer, Vol. 95, No. 8, pges 568-574 Wlter Dodge (not pictured) nd Steve Viktor Thinking out of the Bo... Problem It s richer problem thn we ever imgined The bo problem hs been stndrd
More information0.1 Basic Set Theory and Interval Notation
0.1 Bsic Set Theory nd Intervl Nottion 3 0.1 Bsic Set Theory nd Intervl Nottion 0.1.1 Some Bsic Set Theory Notions Like ll good Mth ooks, we egin with definition. Definition 0.1. A set is well-defined
More informationReview guide for the final exam in Math 233
Review guide for the finl exm in Mth 33 1 Bsic mteril. This review includes the reminder of the mteril for mth 33. The finl exm will be cumultive exm with mny of the problems coming from the mteril covered
More informationSmall Business Networking
Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd processes. Introducing technology
More informationMA 15800 Lesson 16 Notes Summer 2016 Properties of Logarithms. Remember: A logarithm is an exponent! It behaves like an exponent!
MA 5800 Lesson 6 otes Summer 06 Rememer: A logrithm is n eponent! It ehves like n eponent! In the lst lesson, we discussed four properties of logrithms. ) log 0 ) log ) log log 4) This lesson covers more
More informationPHY 140A: Solid State Physics. Solution to Homework #2
PHY 140A: Solid Stte Physics Solution to Homework # TA: Xun Ji 1 October 14, 006 1 Emil: jixun@physics.ucl.edu Problem #1 Prove tht the reciprocl lttice for the reciprocl lttice is the originl lttice.
More informationWarm-up for Differential Calculus
Summer Assignment Wrm-up for Differentil Clculus Who should complete this pcket? Students who hve completed Functions or Honors Functions nd will be tking Differentil Clculus in the fll of 015. Due Dte:
More informationAll pay auctions with certain and uncertain prizes a comment
CENTER FOR RESEARC IN ECONOMICS AND MANAGEMENT CREAM Publiction No. 1-2015 All py uctions with certin nd uncertin prizes comment Christin Riis All py uctions with certin nd uncertin prizes comment Christin
More information1. Find the zeros Find roots. Set function = 0, factor or use quadratic equation if quadratic, graph to find zeros on calculator
AP Clculus Finl Review Sheet When you see the words. This is wht you think of doing. Find the zeros Find roots. Set function =, fctor or use qudrtic eqution if qudrtic, grph to find zeros on clcultor.
More information2 DIODE CLIPPING and CLAMPING CIRCUITS
2 DIODE CLIPPING nd CLAMPING CIRCUITS 2.1 Ojectives Understnding the operting principle of diode clipping circuit Understnding the operting principle of clmping circuit Understnding the wveform chnge of
More informationLECTURE #05. Learning Objective. To describe the geometry in and around a unit cell in terms of directions and planes.
LECTURE #05 Chpter 3: Lttice Positions, Directions nd Plnes Lerning Objective To describe the geometr in nd round unit cell in terms of directions nd plnes. 1 Relevnt Reding for this Lecture... Pges 64-83.
More informationExponential and Logarithmic Functions
Nme Chpter Eponentil nd Logrithmic Functions Section. Eponentil Functions nd Their Grphs Objective: In this lesson ou lerned how to recognize, evlute, nd grph eponentil functions. Importnt Vocbulr Define
More informationaddition, there are double entries for the symbols used to signify different parameters. These parameters are explained in this appendix.
APPENDIX A: The ellipse August 15, 1997 Becuse of its importnce in both pproximting the erth s shpe nd describing stellite orbits, n informl discussion of the ellipse is presented in this ppendix. The
More informationDlNBVRGH + Sickness Absence Monitoring Report. Executive of the Council. Purpose of report
DlNBVRGH + + THE CITY OF EDINBURGH COUNCIL Sickness Absence Monitoring Report Executive of the Council 8fh My 4 I.I...3 Purpose of report This report quntifies the mount of working time lost s result of
More informationBasically, logarithmic transformations ask, a number, to what power equals another number?
Wht i logrithm? To nwer thi, firt try to nwer the following: wht i x in thi eqution? 9 = 3 x wht i x in thi eqution? 8 = 2 x Biclly, logrithmic trnformtion k, number, to wht power equl nother number? In
More information2005-06 Second Term MAT2060B 1. Supplementary Notes 3 Interchange of Differentiation and Integration
Source: http://www.mth.cuhk.edu.hk/~mt26/mt26b/notes/notes3.pdf 25-6 Second Term MAT26B 1 Supplementry Notes 3 Interchnge of Differentition nd Integrtion The theme of this course is bout vrious limiting
More informationSolving BAMO Problems
Solving BAMO Problems Tom Dvis tomrdvis@erthlink.net http://www.geometer.org/mthcircles Februry 20, 2000 Abstrct Strtegies for solving problems in the BAMO contest (the By Are Mthemticl Olympid). Only
More informationHow To Network A Smll Business
Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd processes. Introducing technology
More informationModule 2. Analysis of Statically Indeterminate Structures by the Matrix Force Method. Version 2 CE IIT, Kharagpur
Module Anlysis of Stticlly Indeterminte Structures by the Mtrix Force Method Version CE IIT, Khrgpur esson 9 The Force Method of Anlysis: Bems (Continued) Version CE IIT, Khrgpur Instructionl Objectives
More informationRadius of the Earth - Radii Used in Geodesy James R. Clynch February 2006
dius of the Erth - dii Used in Geodesy Jmes. Clynch Februry 006 I. Erth dii Uses There is only one rdius of sphere. The erth is pproximtely sphere nd therefore, for some cses, this pproximtion is dequte.
More informationAREA OF A SURFACE OF REVOLUTION
AREA OF A SURFACE OF REVOLUTION h cut r πr h A surfce of revolution is formed when curve is rotted bout line. Such surfce is the lterl boundr of solid of revolution of the tpe discussed in Sections 7.
More informationPure C4. Revision Notes
Pure C4 Revision Notes Mrch 0 Contents Core 4 Alger Prtil frctions Coordinte Geometry 5 Prmetric equtions 5 Conversion from prmetric to Crtesin form 6 Are under curve given prmetriclly 7 Sequences nd
More informationSmall Business Networking
Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd processes. Introducing technology
More information15.6. The mean value and the root-mean-square value of a function. Introduction. Prerequisites. Learning Outcomes. Learning Style
The men vlue nd the root-men-squre vlue of function 5.6 Introduction Currents nd voltges often vry with time nd engineers my wish to know the verge vlue of such current or voltge over some prticulr time
More informationg(y(a), y(b)) = o, B a y(a)+b b y(b)=c, Boundary Value Problems Lecture Notes to Accompany
Lecture Notes to Accompny Scientific Computing An Introductory Survey Second Edition by Michel T Heth Boundry Vlue Problems Side conditions prescribing solution or derivtive vlues t specified points required
More informationExample A rectangular box without lid is to be made from a square cardboard of sides 18 cm by cutting equal squares from each corner and then folding
1 Exmple A rectngulr box without lid is to be mde from squre crdbord of sides 18 cm by cutting equl squres from ech corner nd then folding up the sides. 1 Exmple A rectngulr box without lid is to be mde
More informationHelicopter Theme and Variations
Helicopter Theme nd Vritions Or, Some Experimentl Designs Employing Pper Helicopters Some possible explntory vribles re: Who drops the helicopter The length of the rotor bldes The height from which the
More informationWhy is the NSW prison population falling?
NSW Bureu of Crime Sttistics nd Reserch Bureu Brief Issue pper no. 80 September 2012 Why is the NSW prison popultion flling? Jcqueline Fitzgerld & Simon Corben 1 Aim: After stedily incresing for more thn
More informationPhysics 43 Homework Set 9 Chapter 40 Key
Physics 43 Homework Set 9 Chpter 4 Key. The wve function for n electron tht is confined to x nm is. Find the normliztion constnt. b. Wht is the probbility of finding the electron in. nm-wide region t x
More informationIncreasing Q of Waveguide Pulse-Compression Cavities
Circuit nd Electromgnetic System Design Notes Note 61 3 July 009 Incresing Q of Wveguide Pulse-Compression Cvities Crl E. Bum University of New Mexico Deprtment of Electricl nd Computer Engineering Albuquerque
More informationAn Undergraduate Curriculum Evaluation with the Analytic Hierarchy Process
An Undergrdute Curriculum Evlution with the Anlytic Hierrchy Process Les Frir Jessic O. Mtson Jck E. Mtson Deprtment of Industril Engineering P.O. Box 870288 University of Albm Tuscloos, AL. 35487 Abstrct
More informationBayesian Updating with Continuous Priors Class 13, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom
Byesin Updting with Continuous Priors Clss 3, 8.05, Spring 04 Jeremy Orloff nd Jonthn Bloom Lerning Gols. Understnd prmeterized fmily of distriutions s representing continuous rnge of hypotheses for the
More informationSINCLAIR COMMUNITY COLLEGE DAYTON, OHIO DEPARTMENT SYLLABUS FOR COURSE IN MAT 1470 - COLLEGE ALGEBRA (4 SEMESTER HOURS)
SINCLAIR COMMUNITY COLLEGE DAYTON, OHIO DEPARTMENT SYLLABUS FOR COURSE IN MAT 470 - COLLEGE ALGEBRA (4 SEMESTER HOURS). COURSE DESCRIPTION: Polynomil, rdicl, rtionl, exponentil, nd logrithmic functions
More informationCOMPLEX FRACTIONS. section. Simplifying Complex Fractions
58 (6-6) Chpter 6 Rtionl Epressions undles tht they cn ttch while working together for 0 hours. 00 600 6 FIGURE FOR EXERCISE 9 95. Selling. George sells one gzine suscription every 0 inutes, wheres Theres
More informationSmall Business Networking
Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd business. Introducing technology
More information3 The Utility Maximization Problem
3 The Utility Mxiiztion Proble We hve now discussed how to describe preferences in ters of utility functions nd how to forulte siple budget sets. The rtionl choice ssuption, tht consuers pick the best
More informationThe Velocity Factor of an Insulated Two-Wire Transmission Line
The Velocity Fctor of n Insulted Two-Wire Trnsmission Line Problem Kirk T. McDonld Joseph Henry Lbortories, Princeton University, Princeton, NJ 08544 Mrch 7, 008 Estimte the velocity fctor F = v/c nd the
More informationWeek 7 - Perfect Competition and Monopoly
Week 7 - Perfect Competition nd Monopoly Our im here is to compre the industry-wide response to chnges in demnd nd costs by monopolized industry nd by perfectly competitive one. We distinguish between
More informationMATHEMATICS FOR ENGINEERING BASIC ALGEBRA
MATHEMATICS FOR ENGINEERING BASIC ALGEBRA TUTORIAL - INDICES, LOGARITHMS AND FUNCTION This is the oe of series of bsic tutorils i mthemtics imed t begiers or yoe wtig to refresh themselves o fudmetls.
More information