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An Introduction to CPI 200: Math Foundations of Informatics School of Computing & Informatics Arizona State University Dianne Hansford Questions you might have: Why is this class (math) important ? What are we going to study? Relevance to the Informatics Certificate/Concentration? How are we going to learn the topics? Why is math important? Mathematics is part of our culture Math ideas are as important as the ideas of Darwin, Marx, Voltaire Ideas shape how we perceive the world Ideas shape how we perceive our place in the world Let’s take a look at history from a math perspective Early Math: Counting 2000BC Babylonia Mesopotamia (between Tigris & Euphrates rivers) -- Iraq Writing and base 60 counting 24 hour day, 60 minutes in an hour and 60 seconds in a minute large numbers and fractions Calculation for commerce If 1 cow is worth 3 goats, then how much does 4 cows cost? Construction of tables (pre-computed squares and cubes) to aid calculations http://en.wikipedia.org/wiki/Babylonian_mathematics Babylonia con’t Pythagorian triples: a^2 + b^2 = c^2 Systems of linear equations Quadratic equations Geometric problems relating to similar figures Area and volume calculations Pi estimate Greeks 450BC: Babylonian math transferred to Greeks Thales, Pythagoras: height of pyramids, distance of ship to shore Rational numbers (a/b) cannot measure all lengths – need irrational numbers sqrt(2) Area calculation – early integration (sum over the parts) Conic section (parabola, ellipse, hyperbola) by Apollonius Trigonometry driven by astronomy Logic Euclid’s Elements – basis of geometry http://en.wikipedia.org/wiki/Greek_mathematics Greeks: Archimedes 287-212 BC – from Sicily Used math to design innovative machines Volume and surface area Archimedes screw pump Death ray See http://en.wikipedia.org/wiki/Archimedes Greeks: Aristotle 384 – 322 BC Student of Plato; teacher to Alexander the Great Wrote on many subjects! More: http://en.wikipedia.org/wiki/Aristotle Math: Contributions to logic Focused on theory over experiments rock falls faster than a feather centuries later: air resistance discovered Islamic (Arab) Math 600 – 1600 AD (Iraq, Iran, Turkey, N. Africa, Spain, India) Arithmetic (numerical calculations) and algebra Arithmetic unified math ideas: algebra, trig, geometry Al-Khwarizimi (Persian scientist) -- algorithm Key: preservation of Greek math 11th Century: brought math back to Europe Europe 16th Century Earth was assumed to be the center of the universe Copernicus and Galileo – study universe predictions of things out of human reach and beyond human control Copernicus Stars moved east to west each day – in fixed positions relative to each other Planets’ movement seemed unpredictable 1543: published sun center of universe Church: man/earth center of universe because man is God’s central creation http://en.wikipedia.org/wiki/Copernicus Galileo Father of modern science 1609: Telescope to discover Jupiter’s moons Promoted Copernicus’s heliocentric theory Punished by the Church / Inquisition Studied effects of gravity Disproved Aristotle’s finding http://en.wikipedia.org/wiki/Galileo Descartes 1596 – 1659 France Cartesian coordinate system Analytic geometry: bridged algebra and geometry Key for development of calculus Mind and mechanism ideas -> computer science http://en.wikipedia.org/wiki/Ren%C3%A9_Descartes Calculus: Newton & Leibniz Derivatives, Integrals Newton’s 3 laws of motion – basis of physics “Clockwork universe” – predictable, deterministic Awakening Math played an important role in increasing human confidence complicated movement of heavens explained by math principles sense of control Age of Enlightenment Voltaire and Rousseau power of reason and the dignity of humans overthrow of “divine right” monarchies in America (1776) and France (1789) Scientific Method Data acquisition Build model (Gather empirical data) Run model Hypothesis analyze model -- supports hypothesis? -- new data needed? -- new model needed? visualization Math is at the center of all of this. Math is the language that we use to build and test models. It also plays a role in data acquisition Empirical data = Data collected by observation or experimentation in contrast to theory. Hypothesis = a proposal intended to explain certain facts or observations; A scientific idea about how something works, before the idea has been tested. Scientists do experiments to test a hypothesis and see if the hypothesis is correct. What are we going to study? Discrete math: first-order logic, sets, graphs, relations -- how to communicate with a computer -- tools for / getting started with abstraction Numbers: number systems, floating point numbers, finite precision, scale -- understanding how numbers are represented in a computer -- ramifications, problems, and limitations Modeling: abstraction, recursion, concurrency -- How to re-pose a problem into one we know the answer for Algorithms: definition, types, and basics of complexity -- how to communicate with a computer/software Programming concepts: types of programming languages and paradigms -- Languages we use to communcate What are we going to study? Calculus concepts: differential and integral concepts, limits and continuity --Understand our model of the universe Analytic geometry basics: 2D and 3D geometry basics -- Geometry for building and manipulating models Numerical and statistical methods: linear maps, regression -- Doing stuff! Ethics: societal impact of computational mathematics Relevance to Informatics Tools for Memory: Store, Index, Retrieve Google {Earth}, XML, SQL, GIS Tools for Routine Activity: Represent, Create, Run Scripting language: on-line purchases, Rule-based language: tax advisors, Stored programs: virus scan Tools for Connectedness: Communication, Network, Interaction MySpace, YouTube, IM, Email/spam, Virtual communities, Cell Phone (iPhone) Tools for Problem Solving: Decision making, Planning Comparison shopping, Flight planners, Games Tools for Analysis: Modeling, Inference, Visualization Excel, Mathematica, Dynamic Simulation, SmartTrade Integrated Applications: Biomedical Informatics, educational informatics, Virtual worlds How are we going to learn the topics? Mathematica: http://www.wolfram.com/ ASU has license – computing sites and available for download Details later! References Eric Schlechter, Why do we study calculus, www.studyweb.com St. Andrews University History Topics: http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/History_overview.html

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posted: | 7/8/2011 |

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