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computers in education mathematics

1. 1. By: Stephanie Sirna
2. 2. Slope Intercept isthe equation of astraight line in theform y = mx + bwhere m is theslope of the lineand b is its y-intercept
3. 3. The slope of a line in theplane containing the x and yaxes is generallyrepresented by the letter m,and is defined as the changein the y coordinate dividedby the correspondingchange in the x coordinate,between two distinct pointson the line. This is describedby the following equation:
4. 4. A quadratic equation is aunivariate polynomialequation of the seconddegree. A generalquadratic equation canbe written in the formwhere x represents avariable or anunknown, and a, b, and care constant with a ≠ 0.(If a = 0, the equation is alinear equation.)
5. 5. Pythagorean TheoremA2+B2=C2To find the length of the lineopposite the hypotenuse (thelongest line) all you have to dois square the lengths of the twolongest sides and add themtogether. Their sum will be thelength of the longest linesquared. Similarly, one cansubtract the length of lines A2or B2 from C2 to fine thelength of one of the smallerlines.
6. 6. The acronym PEMDAS is useful whenlearning how to utilize the order ofoperations, a basic yet vital skillthroughout the field of mathematics.While seemingly trivial, most of thetoughest math problems boil down toPEMDAS. The “P” stands for“parenthesis.” Work in the parenthesis isalways done first. Next, the “E” is for“exponents” followed by “M” and “D” or“multiplication” and “division”respectively. Finally, the “A” and “S” standfor “addition” and “subtraction.”
7. 7. Algebra is one of the broad partsof mathematics, together with numbertheory, geometry and analysis. Algebracan essentially be considered as doingcomputations similar to thatof arithmetic with non-numericalmathematical objects. Initially, theseobjects were variables that eitherrepresented numbers that were not yetknown (unknowns) or represented anunspecified number(indeterminate or parameter), allowingone to state and prove properties that aretrue no matter which numbers aresubstituted for the indeterminate.Algebra
8. 8. Overview of Algebra
9. 9. Euclidean GeometryEuclidean geometry is a mathematical systemattributed to the Alexandrian Greekmathematician Euclid, which he described in histextbook on geometry: the Elements. Euclidsmethod consists in assuming a small set ofintuitively appealing axioms, and deducing manyother propositions(theorems) from these. Althoughmany of Euclids results had been stated by earliermathematicians, Euclid was the first to show howthese propositions could fit into a comprehensivedeductive and logical systemhe Elements beginswith plane geometry, still taught in secondaryschool as the first axiomatic system and the firstexamples of formal proof. It goes on to the solidgeometry of three dimensions. Much ofthe Elements states results of what are nowcalled algebra and number theory, couched ingeometrical language.Oxyrhynchus papyrus (P.Oxy. I 29)showing fragmentof Euclids Elements
10. 10. Trigonometry (from Greek trigōnon"triangle" + metron "measure") is a branchof mathematic that studies triangles andthe relationships between the lengths oftheir sides and the angles between thosesides. Trigonometry definesthe trigonometric functions, which describethose relationships and have applicability tocyclical phenomena, such as waves. Thefield evolved during the third century BC asa branch of geometry used extensively forastronomical studies. It is also thefoundation of the practical art of surveying.Trigonometry
11. 11. CalculusCalculus is the mathematical study ofchange,in the same way that geometry is thestudy of shape and algebra is the study ofoperations and their application to solvingequations. It has two majorbranches, differential calculus(concerningrates of change and slopes of curves),and integral calculus (concerningaccumulation of quantities and the areasunder curves); these two branches arerelated to each other by the fundamentaltheorem of calculus. Both branches make useof the fundamental notions of convergenceof infinite sequences and infinite series to awell-defined limit. Calculus has widespreaduses in science, economics, and engineeringand can solve many problems that algebraalone cannot.
12. 12. Isaac NewtonNewton completed no definitive publicationformalizing his Fluxional Calculus; rather, many of hismathematical discoveries were transmitted throughcorrespondence, smaller papers or as embeddedaspects in his other definitive compilations, such asthe Principia and Opticks. Newton would begin hismathematical training as the chosen heir of IsaacBarrow in Cambridge. His incredible aptitude wasrecognized early and he quickly learned the currenttheories. By 1664 Newton had made his first importantcontribution by advancing the binomialtheorem, which he had extended to include fractionaland negative exponents. Newton succeeded inexpanding the applicability of the binomial theorem byapplying the algebra of finite quantities in an analysisof infinite series. He showed a willingness to viewinfinite series not only as approximate devices, butalso as alternative forms of expressing a term.
13. 13. Gottfried LeibnizWhile Newton began development of his fluxional calculus in 1665-1666 his findings did not become widely circulated until later. In theintervening years Leibniz also strove to create his calculus. Incomparison to Newton who came to math at an early age, Leibnizbegan his rigorous math studies with a mature intellect. He wasa polymath, and his intellectual interests and achievementsinvolved metaphysics, law, economics, politics, logic, and mathematics.In order to understand Leibniz’s reasoning in calculus his backgroundshould be kept in mind. Particularly, his metaphysics which consideredthe world as an infinite aggregate of indivisible monads, and his plansof creating a precise formal logic whereby, “a general method in whichall truths of the reason would be reduced to a kind of calculation.” In1672 Leibniz met the mathematician Huygens who convinced Leibnizto dedicate significant time to the study of mathematics. By 1673 hehad progressed to reading Pascal’s Traité des Sinus du QuarteCercleand it was during his largely autodidactic research that Leibnizsaid "a light turned on"[Like Newton, Leibniz, saw the tangent as aratio but declared it as simply the ratio between ordinatesand abscissas. He continued this reasoning to argue thatthe integral was in fact the sum of the ordinates for infinitesimalintervals in the abscissa; in effect, the sum of an infinite number ofrectangles. From these definitions the inverse relationship ordifferential became clear and Leibniz quickly realized the potential toform a whole new system of mathematics. Where Newton shied awayfrom the use of infinitesimals, Leibniz made it the cornerstone of hisnotation and calculus.
14. 14. The Calculus Controversy
15. 15. Probability theory is the branchof mathematics concerned with probability,the analysis of random phenomena. Thecentral objects of probability theoryare random variables, stochastic processes,and events: mathematical abstractions of non-deterministic events or measured quantitiesthat may either be single occurrences orevolve over time in an apparently randomfashion. If an individual coin toss or the rollof dice is considered to be a random event,then if repeated many times the sequence ofrandom events will exhibit certain patterns,which can be studied and predicted. Tworepresentative mathematical results describingsuch patterns are the law of large numbers andthe central limit theorem.Probability Theory
16. 16. Graph TheoryIn mathematics and computer science, graphtheory is the study of graphs, which aremathematical structures used to modelpairwise relations between objects. A graph inthis context is made upof vertices or nodes and linescalled edges that connect them. A graph maybe undirected, meaning that there is nodistinction between the two verticesassociated with each edge, or its edges maybe directed from one vertex to another;see graph (mathematics) for more detaileddefinitions and for other variations in thetypes of graph that are commonly considered.Graphs are one of the prime objects of studyin discrete mathematics.
17. 17. Number theory (or arithmetic) is abranch of pure mathematics devotedprimarily to the study of the integers.Number theorists study primenumbers as well as the properties ofobjects made out of integers(e.g., rational numbers) or defined asgeneralizations of the integers(e.g., algebraic integers).Number theory
18. 18. A differential equation is a mathematical equation for anunknown function of one or several variables that relates thevalues of the function itself and its derivatives of various orders.Differential equations play a prominent rolein engineering, physics, economics, and other disciplines.Differential equations arise in many areas of science andtechnology, specifically whenever a deterministic relationinvolving some continuously varying quantities (modeled byfunctions) and their rates of change in space and/or time(expressed as derivatives) is known or postulated. This isillustrated in classical mechanic, where the motion of a body isdescribed by its position and velocity as the time valuevaries. Newtons laws allow one (given the position, velocity,acceleration and various forces acting on the body) to expressthese variables dynamically as a differential equation for theunknown position of the body as a function of time. In somecases, this differential equation (called an equation of motion)may be solved explicitly.Differential Equation
19. 19. Applied mathematics is a branch of mathematics thatconcerns itself with mathematical methods that aretypically used in science, engineering, business, andindustry. Thus, "applied mathematics" isa mathematical science with specialized knowledge.The term "applied mathematics" also describesthe professional specialty in which mathematicianswork on practical problems; as a profession focusedon practical problems, applied mathematics focuseson the formulation and study of mathematicalmodels. In the past, practical applications havemotivated the development of mathematicaltheories, which then became the subject of studyin pure mathematics, where mathematics isdeveloped primarily for its own sake. Thus, theactivity of applied mathematics is vitally connectedwith research in pure mathematics.Applied mathematics
20. 20. Game theory is a study of strategic decision making.More formally, it is "the study of mathematicalmodels of conflict and cooperation between intelligentrational decision-makers."[ An alternative termsuggested "as a more descriptive name for thediscipline" is interactive decision theory. Game theory ismainly used in economics, political science, andpsychology, as well as logic and biology. The subject firstaddressed zero-sum game, such that one persons gainsexactly equal net losses of the other participant(s).Today, however, game theory applies to a wide range ofbehavioral relations, and has developed intoan umbrella term for the logical side of decision science,to include both human and non-humans, likecomputers. Classic uses include a sense of balance innumerous games, where each person has found ordeveloped a tactic that cannot successfully betterhis/her results, given the strategies of other players.Game Theory