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Clasa a XII-a Elemente de algebrã

Relaþii de echivalenþã. Partiþii Fie M ≠ ∅ . Numim relaþie binarã pe M orice submulþime nevidã ρ a produsului cartezian M D M. Dacã (x; y) i ρ, notãm x ρ y. Fie ρ o relaþie binarã pe mulþimea M. Spunem cã ρ este: reflexivã, dacã µ x i M, xρx ; simetricã, dacã µ x, y i M, xρy ± yρx ; tranzitivã, dacã µ x, y, z i M, xρy ºi yρz ± xρz. O relaþie binarã reflexivã, simetricã ºi tranzitivã se numeºte relaþie de echivalenþã. De multe ori, o relaþie de echivalenþã pe o mulþime M se va nota ~ ; scriem x ~ y (citim x echivalent cu y), sau scriem x : y (citim x nu este echivalent cu y). / Fie n i q* ºi x, y i m. Spunem cã x este congruent cu y modulo n ºi scriem x ≡ y (mod n) dacã n divide x y. Fie M o mulþime nevidã înzestratã cu relaþia de echivalenþã ~ . Pentru a i M, def clasa de echivalenþã a lui a este mulþimea a ={ x ∈ M | x ~ a} . ¶ Mulþimea claselor de echivalenþã se noteazã M (sau ( M / : ) ) ºi se numeºte def ¶ mulþimea factor a lui M prin relaþia ~ . M = { a | a ∈ M } . Fie ~ o relaþie de echivalenþã pe M. Clasele de echivalenþã definite de ∼ pe M, sunt disjuncte douã câte douã. Fie M o mulþime nevidã. O familie {Ci}iiI de pãrþi nevide ale lui M se numeºte partiþie a mulþimii M dacã: 1) µ i, j i I, i @ j ⇒ Ci O Cj = l; 2) µ x i M, j i i I astfel încât x i Ci . Fie {Ci}iiI partiþie a mulþimii M. M = N Ci . Pentru o relaþie de echivalenþã pe M, clasele de echivalenþã definite de aceastã relaþie formeazã o partiþie a mulþimii M. Legi de compoziþie Fie M o mulþime nevidã. O aplicaþie ϕ : M D M → M, (x, y) a ϕ(x, y), se numeºte lege de compoziþie (internã) sau operaþie (algebricã, binarã) pe mulþimea M. Elementul ϕ(x; y) i M se numeºte compusul lui x cu y prin ϕ (în aceastã ordine). De obicei, în loc de ϕ(x; y) notãm x C y sau x o y sau x T y sau x ∆ y etc. i∈I 55

Tabla lui Cayley asociatã legii de compoziþie ϕ pe mulþimea M este un tabel cu linii ºi coloane corespunzãtoare elementelor mulþimii M obþinut astfel: la intersecþia liniei ai cu coloana aj se aflã compusul lui ai cu aj prin operaþia ϕ. Ori de câte ori notãm (M, C) subînþelegem cã C este o lege de compoziþie internã pe mulþimea nevidã M.

ϕ a1 a2 ... a j ... an M a1 M a2 M M ai L L L ϕ( ai , a j ) M an

Fie M o mulþime nevidã ºi C o lege de compoziþie pe M. O submulþime nevidã H a lui M se numeºte parte stabilã în raport cu legea de compoziþie C dacã: µ x, y i H ⇒ x C y i H. O lege de compoziþie C se numeºte asociativã dacã: ( x ∗ y ) ∗ z = x ∗ ( y ∗ z ), ∀x , y , z ∈ M . O lege de compoziþie M D M → M, ( x ; y ) a x ∗ y se numeºte comutativã dacã x ∗ y = y ∗ x , ∀x , y ∈ M . Un element e i M se numeºte element neutru pentru legea de compoziþie C, dacã ∀x ∈ M e ∗ x = x ∗ e = x . Fie M o mulþime nevidã înzestratã cu o lege de compoziþie C cu element neutru e. Spunem cã un element x i M este simetrizabil în raport cu legea de compoziþie C, dacã existã x′ i M astfel încât x′ ∗ x = x ∗ x′ = e . Elementul x′ cu aceastã proprietate se numeºte simetricul lui x. În cazul în care legea de compoziþie este o lege de adunare (de numere, de matrice, de polinoame, de funcþii, de vectori, ...) folosim denumirea de opus în loc de simetric al unui element. Dacã legea de compoziþie este o lege de înmulþire (de numere, de matrice, de polinoame, de funcþii, ...) folosim denumirea de invers în loc de simetric al unui element. Aceeaºi denumire se foloseºte în cazul în care legea de compoziþie este o lege de compunere de funcþii. Fie n i q, n U 2. Notãm mn mulþimea claselor de echivalenþã pentru congruenþa · $ $ modulo n. Avem m n = {0; 1; 2; ...; n − 1} . Pe mn definim operaþiile numite adunarea ºi · ¶ înmulþirea claselor de resturi modulo n astfel: α + β = α + β , αβ = αβ , ∀ α , β∈ m . n Grupuri Un cuplu (G; C), format cu o mulþime nevidã G ºi cu o lege de compoziþie ∗ pe G, se numeºte grup dacã legea de compoziþie C este asociativã, are element neutru ºi orice element din M este simetrizabil. Dacã, în plus, legea C este comutativã, atunci G se numeºte grup comutativ sau abelian. Un cuplu (M, C) format cu o mulþime nevidã M ºi o lege de compoziþie C pe M, se numeºte monoid dacã legea C este asociativã ºi are elementul neutru.

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Regulile de simplificare într-un grup. Fie (G , ∗) un grup. Pentru orice a , b , c ∈ G avem: aCb = aCc ⇒ b = c ºi b ∗ a = c ∗ a ⇒ b = c Grupuri de matrice

GL2 (Z ) = { A ∈ M2 (Z ) | det A ≠ 0} înzestrat cu înmulþirea formeazã un grup numit grupul general liniar de grad 2. t 1 Submulþimile SL2 (Z ) = { A ∈ GL2 (Z ) | A = 1} , O (2) = { A ∈ GL2 (Z ) | A = A } , SO (2) = { A ∈ O (2) | det A = 1} , înzestrate cu înmulþirea matricelor formeazã grupuri de matrice, numite respectiv grupul special liniar de grad 2 peste Z, grupul ortogonal de grad 2 ºi grupul ortogonal special de grad 2. Pentru n i q* pot fi definite grupurile SLn({), SLn(Z) ºi SLn(³), numite grupul special liniar de grad n peste {, Z, respectiv ³. De asemenea, pot fi introduse grupurile O(n) ºi SO(n), numite respectiv grupul ortogonal de grad n ºi grupul ortogonal special de grad n.

Morfisme de grupuri Fie grupurile (G , o) ºi (G ′ , ∗) . Funcþia f : G → G′ se numeºte morfism de grupuri dacã: f ( x o y ) = f ( x ) ∗ f ( y ), ∀x , y ∈ G . Fie (G , o) ºi (G ′ , ∗) douã grupuri. O funcþie f : G → G′ se numeºte izomorfism de (2) f este bijectivã. grupuri dacã: (1) f ( x o y ) = f ( x ) ∗ f ( y ), ∀x , y ∈ G ; Spunem cã grupul G este izomorf cu grupul G′ ºi scriem G ; G′ , dacã existã un izomorfism f : G → G′. În caz contrar, spunem cã grupul G nu este izomorf cu grupul G′ ºi scriem G ; G ′ . Dacã G este grup, atunci un morfism (izomorfism) f : G → G se numeºte endomorfism (respectiv automorfism) al grupului G. Grupuri de permutãri Fie A o mulþime finitã cu n elemente, n i q*. O funcþie bijectivã σ : A → A se numeºte permutare a mulþimii A. Vom nota cu SA mulþimea tuturor permutãrilor mulþimii A. Pentru σ, π i SA, compunerea permutãrilor σ ºi π este funcþia σ o π : A → A , cu (σ o π)( x ) = σ(π( x )) , x i A. Funcþia σ o π este de asemenea bijectivã, deci σ o π ∈ S A . ( S A , o) este grup. Grupul permutãrilor mulþimii {1, 2, ..., n} se noteazã ( S n , o) . Subgrupuri Fie (G, ∗) un grup ºi H o parte stabilã a lui G. ( H , ∗) se numeºte subgrup al lui G dacã ( H , ∗) este grup. Fie (G, E) un grup de element neutru e ºi a i G. Spunem cã a este element de ordin finit al grupului G dacã existã m > 0 astfel încât am = e. Dacã a este element de ordin finit, atunci cel mai mic numãr m > 0 cu proprietatea am = e se numeºte ordinul lui a ºi notãm ord a = m. 57

Grupuri de transformãri geometrice O aplicaþie T : P → P se numeºte transformare geometricã a planului P. Vom spune cã T este izometrie dacã T conservã distanþele dintre puncte: d(T(A), T(B)) = d(A, B), µ A, B i P. Notãm cu Izom(P) mulþimea tuturor izometriilor planului P. Dacã T1 ºi T2 sunt izometrii, atunci ºi T1 o T2 este o izometrie. (Izom(P ), o) este un grup, numit grupul izometriilor planului P. Fie F o figurã planã, F ⊂ P ºi T : P → P o izometrie; notãm cu T(F) = {T(P) | P i F}. Spunem cã T invariazã (global) pe F dacã T(F) = F. Notãm cu Sim(F) mulþimea tuturor izometriilor care invariazã pe F. (Sim(F), o ) este un subgrup al grupului ( Izom(P ), o) , numit grupul de simetrie al lui F. Fie n i q, n U 3 ºi Pn un poligon regulat cu n laturi din planul P. Grupul de simetrie al lui Pn se noteazã Dn = Sym(Pn) ºi se numeºte grupul diedral. Inele În cele ce urmeazã, se lucreazã numai cu inele unitare. Un triplet (R, +, E), unde R este o mulþime nevidã iar + ºi E sunt douã legi de compoziþie pe R (numite adunare ºi înmulþire), se numeºte inel dacã: (G) (R, +) este grup abelian (M) (R, ·) este monoid (D) înmulþirea este distributivã faþã de adunare: ¼ x, y, z i R , x(y + z) = xy + xz , (y + z)x = yx + zx. În inelul R, elementul neutru al legii de compoziþie · se numeºte element unitate. Spunem cã inelul R nu are divizori ai lui zero, dacã x @ 0, y @ 0 ⇒ xy @ 0 ; în caz contrar spunem cã R este inel cu divizori ai lui zero. Un inel R se numeºte comutativ dacã satisface ºi axioma: (M3) xy = yx, ¼ x, y i R. Un inel comutativ, cu cel puþin douã elemente ºi fãrã divizori ai lui zero, se numeºte domeniu de integritate (sau inel integru). Morfisme de inele Fie inelele (R, +, ·) ºi (R′, ⊕ , e ). O funcþie f : R → R′ se numeºte morfism de inele dacã, µ x, y i R: (1) f (x + y) = f (x) ⊕ f (y) ; (2) f (xEy) = f (x) e f (y); (3) f (1) = 1′, unde 1 este unitatea inelului R ºi 1′ unitatea lui R′. Un morfism de inele bijectiv se numeºte izomorfism. Vom spune cã inelul R este izomorf cu inelul R′, ºi scriem R Y R′, dacã existã cel puþin un izomorfism f : R → R′.

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Grupul unitãþilor. Subinele Elementele inversabile ale unui inel R se numesc unitãþi ale lui R. Notãm cu U(R) mulþimea unitãþilor inelului R. Fie R un inel; U(R) este grup în raport cu operaþia indusã de înmulþirea lui R, numit grupul unitãþilor inelului R. Fie (R, +, ·) un inel cu elementul unitate notat 1 ºi S ⊂ R ; S se numeºte subinel al lui R dacã (S, +, ·) este inel ºi 1 i S. Exemple de inele Numerele complexe a + bi, cu a, b i m se numesc întregi ai lui Gauss (de exemplu: 2 + 3i, 1 + 2i, 4 = 4 + 0i, i = 0 + 1 · i sunt întregi ai lui Gauss). Notãm m[i] = {a + bi | a, b i m} mulþimea întregilor lui Gauss. (m[i], +, ·) este un inel integru. Fie I o mulþime nevidã ºi R un inel. Notãm RI = { f | f : I → R} mulþimea tuturor funcþiilor f : I → R . Pentru f, g i RI ºi x i I, f (x) ºi g(x) sunt elemente ale inelului R. Putem defini astfel funcþiile: f + g : I → R, ( f + g )( x ) = f ( x ) + g ( x ) , x i I ºi fg : I → R, ( fg )( x ) = f ( x ) ⋅ g ( x ) numite suma, respectiv produsul funcþiei f cu funcþia g. Fie R inel comutativ. Notãm R[X] mulþimea polinoamelor cu coeficienþii în R. (R[X], +, ·) este inel. Fie f i R[X]. Funcþia f *: R → R definitã prin f *(x) = f (x) i R, µ x i R, este numitã funcþia polinomialã asociatã polinomului f. Vom nota funcþia f tot cu f . Zerourile funcþiei polinomiale f , se numesc rãdãcini (din R) ale polinomului f. Aºadar, un element α i R este rãdãcinã (din R) a polinomului f i R[X] dacã f (α) = 0. Corpuri. Morfisme de corpuri Un inel K se numeºte corp dacã 0 @ 1 ºi orice element nenul din K este simetrizabil în raport cu înmulþirea. Dacã înmulþirea este comutativã, K se numeºte corp comutativ. O funcþie f : K → K′ de la un corp K la un corp K′ se numeºte morfism (izomorfism) de corpuri dacã este morfism (izomorfism) de la K la K′ considerate ca inele. Un izomorfism (morfism) f : R → R de la inelul (R, +, ·) în el însãºi se numeºte automorfism (respectiv endomorfism) al inelului R. Aceeaºi terminologie se foloseºte ºi pentru corpuri. Inelul (mn, +, ·) este corp dacã ºi numai dacã n este numãr prim. Aritmetica polinoamelor cu coeficienþi într-un corp comutativ Teorema împãrþirii cu rest. Fie K un corp comutativ ºi f, g i K[X], g @ 0. Existã unic determinate polinoamele q, r i K[X] astfel încât f = gq + r , unde grad r < grad g dacã r @ 0. 59

Polinoamele q ºi r din teorema împãrþirii ( f = gq + r) se numesc câtul, respectiv restul împãrþirii polinomului f prin polinomul g. Fie K corp comutativ ºi f, g i K[X]. Spunem cã f este divizibil cu g ºi notãm g | f sau f M g , dacã existã h i K[X] cu f = gEh. Fie K corp comutativ ºi f, g i K[X]. Spunem cã f este asociat în divizibilitate cu g ºi scriem f : g , dacã f | g ºi g | f. Teorema restului. Restul împãrþirii polinomului f i K[X] prin X α i K[X] este egal cu valoarea în α a polinomului f. Teorema lui Bézout. Polinomul f i K[X] se divide prin polinomul X α i K[X] dacã ºi numai dacã f (α) = 0. Fie K corp comutativ, f i K[X], a i K ºi n i q, n U 2. Spunem cã a este rãdãcinã multiplã de ordin n dacã (X a)n | f ºi ( X − a)n+1 / f . | Fie K corp comutativ ºi f = an X n + an −1 X n −1 + ... + a1 X + a0 , din K[X]. Polinomul f ′ = nan X n −1 + ( n − 1) an−1 X n − 2 + ... + a1 se numeºte derivata formalã de ordinul I a polinomului f. Derivata formalã de ordinul II a polinomului f este derivata formalã de ordinul I a polinomului f ′ ºi este notatã f ′′. Derivata formalã de ordinul k a polinomului f este derivata formalã de ordinul I a polinomului f ( k −1) . Fie K un corp comutativ ºi f i K[X] un polinom de grad f = n > 0. Spunem cã polinomul f este reductibil peste K dacã existã polinoamele g, h i K[X], de grade strict mai mici ca n, cu f = gh. În caz contrar, spunem cã f este ireductibil peste K. Orice polinom f din K[X], grad f U 1, se descompune în mod unic în produs de polinoame ireductibile peste K. Spaþii vectoriale Fie V ºi K mulþimi nevide. O aplicaþie ψ : K D V → V se numeºte lege de compoziþie externã pe V cu scalari (sau operatori) în K. Fie corpul comutativ (K, +, E). Se numeºte spaþiu vectorial peste K un grup abelian (V, +) înzestrat cu o lege de compoziþie externã cu scalari în K, (α, u) a αu, ce verificã axiomele: (S1) µ α, β i K, µ u i V, (α + β)u = αu + βu (distributivitatea înmulþirii vectorilor cu scalari faþã de adunarea scalarilor) (S2) µ α i K, µ u, v i V, α(u + v) = αu + αv (distributivitatea înmulþirii vectorilor cu scalari faþã de adunarea vectorilor) (S3) µ α, β i K, µ u i V, α(βu) = (αβ)u (asociativitatea înmulþirilor scalarilor ºi vectorilor) 60…...

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...This article is about the study of topics, such as quantity and structure. For other uses, see Mathematics (disambiguation). "Math" redirects here. For other uses, see Math (disambiguation). Euclid (holding calipers), Greek mathematician, 3rd century BC, as imagined by Raphael in this detail from The School of Athens.[1] Mathematics is the study of topics such as quantity (numbers),[2] structure,[3] space,[2] and change.[4][5][6] There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics.[7][8] Mathematicians seek out patterns[9][10] and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof. When mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity for as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano (1858–1932), David Hilbert (1862–1943), and others on axiomatic systems in the late 19th century, it has become......

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...Dates | Grading Scale [92 , 100] | 4.0 | A | | [72 , 76) | 2.3 | C+ | [88 , 92) | 3.7 | A- | | [68, 72) | 2.0 | C | [84 , 88) | 3.3 | B+ | | [63 , 68) | 1.7 | C- | [80 , 84) | 3.0 | B | | [50 , 63) | 1.0 | D | [76 , 80) | 2.7 | B- | | [0 , 50) | 0 | F | Grading Distribution Assessment | Weight | Date | Quizzes and homework | 15% | | Recitation | 5% | | Exam 1 | 22.5% | Wed Nov 5th (5:30-7:00 pm) | Exam 2 | 22.5% | Wed Dec 10th (5:30-7:00 pm) | Final Exam | 35% | | Total | 100% | | | | | | M | Explanation of Assessments | There will be in-class quizzes, in addition to two midterm tests, and a comprehensive final exam. * Most quizzes will be pre-announced at least one lecture in advance. * NO make-up quizzes will be given. However the lowest quiz will not be counted toward your final grade. * Refer to the schedule below to see the topics that will make up the material for each exam. With a valid written excuse and making immediate arrangements with the instructor, a missed exam might be replaced with the grade of the final exam and/or the average grade of all tests (including final) and/or quizzes. * Attendance: It is the university policy that attendance is compulsory. A student missing more than 15% (7 UTR classes or 5 MW classes) of the total allocated course hours will receive a WF. | N | Student Academic Integrity Code Statement | Student must adhere to the Academic Integrity code......

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...houses for sale. http://www.trulia.com/ 3. Find a website that shows one how to do fraction to decimal conversions. http://www.webmath.com/fract2dec.html 4. Using the internet, find a website that one can use to find the national average cost of food for an individual, as well as for a family of 4 for a given month. http://www.cnpp.usda.gov/sites/default/files/usda_food_plans_cost_of_food/CostofFoodJan2012.pdf 5. Find a website for your local city government. http://www.usa.gov/Agencies/Local.shtml 6. Find the website for your favorite sports team (state what that team is as well by the link). http://blackhawks.nhl.com/ (Chicago Blackhawks) 7. Many of us do not realize how often we use math in our daily lives. Many of us believe that math is learned in classes, and often forgotten, as we do not practice it in the real world. Truth is, we actually use math every day, all of the time. Math is used everywhere, in each of our lives. Math does not always need to be thought of as rocket science. Math is such a large part of our lives, we do not even notice we are computing problems in our lives! For example, if one were interested in baking, one must understand that math is involved. One may ask, “How is math involved with cooking?” Fractions are needed to bake an item. A real world problem for baking could be as such: Heena is baking a cake that requires two and one-half cups of flour. Heena poured four and one-sixth cups of flour into a bowl. How much flour should......

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...solving an equation with integer solutions. If you can see how to split up the original equation into its factor pair, this is the quickest and allows you to solve the problem in one step. Week 9 capstone part 1 Has the content in this course allowed you to think of math as a useful tool? If so, how? What concepts investigated in this course can apply to your personal and professional life? In the course, I have learned about polynomials, rational expressions, radical equations, and quadratic equations. Quadratic equations seem to have the most real life applications -- in things such as ticket sales, bike repairs, and modeling. Rational expressions are also important, if I know how long it takes me to clean my sons room, and know how long it takes him to clean his own room. I can use rational expressions to determine how long it will take the two of us working together to clean his room. The Math lab site was useful in some ways, since it allowed me to check my answers to the problems immediately. However, especially in math 117, it was too sensitive to formatting of the equations and answers. I sometimes put an answer into the math lab that I knew was right, but it marked it wrong because of the math lab expecting slightly different formatting Week 9 capstone part 2 I really didn't use center for math excellence because i found that MML was more convenient for me. I think that MML reassures you that you’re doing the problem correctly. MML is extra support because......

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...cityu.edu.hk. Texts: Single Variable Calculus, by James Stewart, 6E. In this semester, we will cover the majority of Chap 1-4, 7, 12. Upon completion of this course, you should be able to understand limit, derivatives, and its applications in mathematical modeling and inﬁnite series. 1 2 1. Functions and Models In this chapter, we will brieﬂy recall functions and its properties covered by high school. 1.1. Basic concepts of functions. Text Sec1.1: 5, 7, 39, 57, 67. Deﬁnition 1.1. A function f is a rule that assigns to each element x in a set D exactly one element, called f (x), in a set E. Usually, we write a function f : x → f (x) where (1) x ∈ D, i.e. x belongs to a set D , called the Domain; (2) f (x) ∈ E, i.e. f (x) belongs to a set E, called the Range; (3) x is independent variable, (4) f (x) is dependent variable. 3 For a function f , its graph is the set of points {(x, f (x)) : x ∈ D} in xy-plane. One can also use a table to represent a function. Example 1.1. Sketch the graph of following two piecewise deﬁned functions. (1) f (x) = |x|. i.e. Absolute value of x. (2) f (x) = [x]. i.e. largest integer not greater than x. The graph of a function is a curve. But the question is: which curves are graphs of functions? Proposition 1.2 (Vertical Line Test). A curve in the xyplane is the graph of a function if and only if no vertical line intersects the curve more than once. VLT is equivalent to following statements: for any given input x, the output f (x)......

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... Identify and discuss the one in which you are weakest AND the one in which you are strongest. For my strongest ability I would say that is “Making decisions”. According to the “The Case for Quantitative Literacy handout”, “making decisions is the ability to use mathematics, make decisions and solve problems in everyday life”. In my everyday life I have to keep the balance in my check book, pay bills, take care of kids, run my house, cook, clean etc. With cooking I am using math, measuring how much food to make for four people (I still haven’t mastered that one). With bills I am using math, how much each company gets, to how much money I have to spare (which these days is not much). In my everyday life I do use some form of a math. It might not be how I was taught, but I have learned to adapt to my surroundings and do math how I know it be used, the basic ways, none of that fancy stuff. For my weakest ability I would say I fall into “Confidence with Mathematics”. Math has never been one of my favorite subjects to learn. It is like my brain knows I have to learn it, but it puts up a wall and doesn’t allow the information to stay in there. The handout “The Case for Quantitative Literacy” states I should be at ease with applying quantitative methods, and comfortable with quantitative ideas. To be honest this class scares the crap out of me, and I am worried I won’t do well in this class. The handout also says confidence is the opposite of “Math Anxiety”, well I can......

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... | | |Location | | |on-line | | | | | |Times | | |on-line, or in Maier Hall , Math Lab, Peninsula College | | | | | |Start Date | | |Sept. 21, 2015 End Date Dec. 9, 2015 | | | | | |Course Credits ...

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...brought about when Mathematics, especially problem solving processes, Mathematics interest and Mathematics aptitude. Problem solving, which the National Council for Teacher in Mathematics (NCTM) 1980’s widely heralded statement in its agenda for action and problem solving has been the theme of the council. Knowledge and skills of Mathematics problem solving is believed to help school goers solve problems in their day to day of existence. The ancient claimed “Mathematics is the queen of Knowledge,” hence it is only right to say that Mathematics enhanced students understanding of the important principles in math, that is, as a cooperative and never ending process. Mathematics also made them exert more effort in improving their achievement and inspire them in relating Mathematics and applications for the solution of human problems. There are many ways of solving problems. The one presented by Polya – a Mathematician of the 18th Century. According to Polya, to solve problems in Mathematics, a student must follow certain steps of processes. Polya’s steps for solving Mathematics problems are; (a) Understand the Problem, (b) Devise a Plan, (c) Carry out the Plan, and (d) Look Back (McCoy 1994). Due to this vision, the Department of Education continues their research on how to improve the quality of the Education. Also new methods, strategies and tools in teaching Mathematics and other subject areas have been implemented. This is also to ensure that learning is acquired by the......

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...Math was always the class that could never quite keep my attention in school. I was a daydreamer and a poor student and applying myself to it was pretty much out of the question. When I would pay some attention I would still forget the steps it had taken me to find the solution. So, when the next time came around I was lost. This probably came about because as a kid I wasn’t real fond of structure. I was more into abstract thought and didn’t think that life required much more than that at the time. I was not interested in things I had to write down and figure out step by step on a piece of paper. I figured I could be Tom Sawyer until about the age of seventy two. My thoughts didn’t need a rhyme or reason and didn’t need laws to keep them within any certain limits. The furthest I ever made it in school was Algebra II and I barely passed that. The reason wasn’t that I couldn’t understand math. It was more that I didn’t apply myself to the concepts of it, or the practice and study it took to get there. I was always more interested in other concepts. Concepts that were gathered by free thinkers, philosophers, idealists. Now I knew that a lot of those figures I read about tried their hand in the sciences, physics, and mathematics in their day, but I was more interested in their philosophical views on everyday life. It was not until I started reading on the subject of quantum physics and standard physics that I became interested in math. The fact that the laws of standard......

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...v XACC/280Week-Seven CheckPoint Ratio, Vertical, and Horizontal Analyses Financial statement analysis is the process of examining relationships among financialstatement elements and making comparisons with relevant information. There are a variety of toolsused to evaluate the significance of financial statement data. Three of the commonly used tools are the ratio analysis, horizontal analysis, andvertical analysis. Ratio analysis is a method of analyzing data to determine the overall financial strength of a business. These ratios are most useful when compared to other ratios such as the comparable ratios of similar businesses or the historical trend of a single business over several business cycles.Horizontal analysis is a type of fundamental analysis in which certain financial data is used toasses a company's performance over a period of time. Horizontal analysis can be assessed on a singlecompany over a period of time, comparing the same items or ratios, or it can be performed on multiplecompanies in the same industry to assess a company's performance relative to competitors.Vertical analysis is a method of analyzing financial statements in which each item in thestatement is represented as a percentage of a single larger item. Vertical analysis makes comparisons between two or more companies in the same industry easier. It also allows a company to weigh......

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