An Introduction toAn Introduction toCPI 200: Math Foundations CPI 200: Math Foundations
of Informaticsof Informatics
CIDSECIDSEArizona State UniversityArizona State University
Gerald FarinGerald Farin
Questions you might have:Questions you might have:
Why is this class (math) important ? Why is this class (math) important ?
What are we going to study? What are we going to study?
Relevance to Informatics Relevance to Informatics
How are we going to learn the topics?How are we going to learn the topics?
Origins of MathOrigins of Math
OrientOrient
ChineseChinese
IndianIndian
IslamIslam
EuropeEurope
Early Math: CountingEarly Math: Counting
2000BC Babylonia2000BC BabyloniaMesopotamia (between Tigris & Euphrates rivers) -- IraqMesopotamia (between Tigris & Euphrates rivers) -- IraqWriting and base 60 countingWriting and base 60 counting
24 hour day, 60 minutes in an hour and 60 seconds in a minute24 hour day, 60 minutes in an hour and 60 seconds in a minute large numbers and fractionslarge numbers and fractions
Calculation for commerceCalculation for commerce If 1 cow is worth 3 goats, then how much do 4 cows cost?If 1 cow is worth 3 goats, then how much do 4 cows cost?
Construction of tables Construction of tables of pre-computed squares and cubes of pre-computed squares and cubes to aid calculationsto aid calculations
http://en.wikipedia.org/wiki/Babylonian_mathematics
Babylonia con’tBabylonia con’t
Pythagorean triples: a^2 + b^2 = c^2Pythagorean triples: a^2 + b^2 = c^2
Systems of linear equationsSystems of linear equations
Quadratic equationsQuadratic equations
Geometric problems relating to similar Geometric problems relating to similar figures figures
Area and volume calculationsArea and volume calculations
Pi estimatePi estimate
GreeksGreeks
450BC: Babylonian math transferred to Greeks450BC: Babylonian math transferred to Greeks Thales, Pythagoras: height of pyramids, distance of ship to Thales, Pythagoras: height of pyramids, distance of ship to
shoreshore
Area calculation – early integration (sum over the parts)Area calculation – early integration (sum over the parts)Conic section (parabola, ellipse, hyperbola) by Conic section (parabola, ellipse, hyperbola) by ApolloniusApolloniusTrigonometry driven by astronomy Trigonometry driven by astronomy LogicLogicEuclid’s Elements – basis of geometryEuclid’s Elements – basis of geometryhttp://en.wikipedia.org/wiki/Greek_mathematics
Greeks: AristotleGreeks: Aristotle
384 – 322 BC 384 – 322 BC
Student of Plato; teacher to Alexander the GreatStudent of Plato; teacher to Alexander the Great
Wrote on many subjects!Wrote on many subjects!More: More: http://en.wikipedia.org/wiki/Aristotle
Math: Contributions to logicMath: Contributions to logic
Focused on theory over experimentsFocused on theory over experiments rock falls faster than a feather rock falls faster than a feather centuries later: air resistance discoveredcenturies later: air resistance discovered
Greeks: ArchimedesGreeks: Archimedes
287-212 BC – from Sicily287-212 BC – from Sicily
Used math to design innovative machinesUsed math to design innovative machines Volume and surface areaVolume and surface area Archimedes screw pumpArchimedes screw pump Death rayDeath ray See See http://en.wikipedia.org/wiki/Archimedes http://en.wikipedia.org/wiki/Archimedes
Islamic (Arab) MathIslamic (Arab) Math
600 – 1600 AD 600 – 1600 AD (Iraq, Iran, Turkey, N. Africa, Spain, India)(Iraq, Iran, Turkey, N. Africa, Spain, India)
Arithmetic (numerical calculations) and algebraArithmetic (numerical calculations) and algebra
Arithmetic unified math ideas:Arithmetic unified math ideas:algebra, trig, geometryalgebra, trig, geometry
Al-Khwarizimi (Persian scientist) -- Al-Khwarizimi (Persian scientist) -- algorithmalgorithm
Key: preservation of Greek mathKey: preservation of Greek math
1111thth Century: brought math back to Europe Century: brought math back to Europe
EuropeEurope
1616thth Century Century
Earth was assumed to be the center of the Earth was assumed to be the center of the universeuniverse
Copernicus and Galileo – study universeCopernicus and Galileo – study universe predictions of things out of human reach and predictions of things out of human reach and
beyond human controlbeyond human control
CopernicusCopernicus
Stars moved east to west each day – in Stars moved east to west each day – in fixed positions relative to each otherfixed positions relative to each other
Planets’ movement seemed unpredictablePlanets’ movement seemed unpredictable
1543: published sun center of universe1543: published sun center of universe
Church: man/earth center of universe Church: man/earth center of universe because man is God’s central creationbecause man is God’s central creation
http://en.wikipedia.org/wiki/Copernicushttp://en.wikipedia.org/wiki/Copernicus
GalileoGalileo
Father of modern scienceFather of modern science1609: Telescope to discover Jupiter’s 1609: Telescope to discover Jupiter’s moonsmoonsPromoted Copernicus’s heliocentric theoryPromoted Copernicus’s heliocentric theoryPunished by the Church / InquisitionPunished by the Church / InquisitionStudied effects of gravity Studied effects of gravity Disproved Aristotle’s finding Disproved Aristotle’s finding
http://en.wikipedia.org/wiki/Galileohttp://en.wikipedia.org/wiki/Galileo
DescartesDescartes
1596 – 1659 France1596 – 1659 France
Cartesian coordinate systemCartesian coordinate system
Analytic geometry: bridged algebra and Analytic geometry: bridged algebra and geometrygeometry Key for development of calculusKey for development of calculus
Mind and mechanism ideas -> computer scienceMind and mechanism ideas -> computer sciencehttp://en.wikipedia.org/wiki/Ren%C3%A9_Descartes http://en.wikipedia.org/wiki/Ren%C3%A9_Descartes
Calculus: Newton & LeibnizCalculus: Newton & Leibniz
Derivatives, IntegralsDerivatives, Integrals
Newton’s 3 laws of motion – basis of Newton’s 3 laws of motion – basis of physicsphysics
““Clockwork universe” – predictable, Clockwork universe” – predictable, deterministicdeterministic
AwakeningAwakening
Math played an important role in Math played an important role in increasing human confidenceincreasing human confidence complicated movement of heavens explained complicated movement of heavens explained
by math principlesby math principles sense of controlsense of control
Age of EnlightenmentAge of Enlightenment Voltaire and RousseauVoltaire and Rousseau
power of reason and the dignity of humanspower of reason and the dignity of humansoverthrow of “divine right” monarchies in America overthrow of “divine right” monarchies in America (1776) and France (1789)(1776) and France (1789)
Scientific MethodScientific Method
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.
Hypothesis
Data acquisition(Gather empirical data) Build model
Run model
visualization
analyze model -- supports hypothesis? -- new data needed? -- new model needed?
TopicsTopicsComputational basics: Computational basics: -- floating point numbers, finite precision-- floating point numbers, finite precision
Algorithms: Algorithms: -- definition, types, and basics of complexity-- definition, types, and basics of complexity
CalculusCalculus: : -- differential and integral concepts, limits, continuity-- differential and integral concepts, limits, continuity
Linear algebraLinear algebra: : -- 2D and 3D geometry basics-- 2D and 3D geometry basics
Statistical methodsStatistical methods
VisualizationVisualization
Relevance to InformaticsRelevance to InformaticsTools for Memory:Tools for Memory:
Store, Index, RetrieveStore, Index, Retrieve Google {Earth}, XML, SQL, GIS Google {Earth}, XML, SQL, GIS
Tools for Routine Activity:Tools for Routine Activity: Represent, Create, RunRepresent, Create, Run Scripting language: on-line purchases,Scripting language: on-line purchases,
Rule-based language: tax advisors,Rule-based language: tax advisors,Stored programs: virus scanStored programs: virus scan
Tools for Connectedness:Tools for Connectedness: Communication, Network, InteractionCommunication, Network, Interaction Facebook, YouTube, IM, Email/spam, Virtual communities, Cell Phone (iPhone) Facebook, YouTube, IM, Email/spam, Virtual communities, Cell Phone (iPhone)
Tools for Problem Solving:Tools for Problem Solving: Decision making, Planning Decision making, Planning Comparison shopping, Flight planners, Games Comparison shopping, Flight planners, Games
Tools for Analysis:Tools for Analysis: Modeling, Inference, VisualizationModeling, Inference, Visualization Excel, Mathematica, Dynamic Simulation, SmartTradeExcel, Mathematica, Dynamic Simulation, SmartTrade
Integrated Applications:Integrated Applications: Biomedical Informatics, educational informatics, Virtual worldsBiomedical Informatics, educational informatics, Virtual worlds
How are we going to learn the How are we going to learn the topics?topics?
Mathematica: http://www.wolfram.com/Mathematica: http://www.wolfram.com/ ASU has license – computing sites and ASU has license – computing sites and
available for downloadavailable for download Details later!Details later!
ReferencesReferences
Eric Schlechter, Why do we study Eric Schlechter, Why do we study calculus, calculus, www.studyweb.comwww.studyweb.com
St. Andrews University History Topics: St. Andrews University History Topics: http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/History_overview.htmlhttp://www-groups.dcs.st-and.ac.uk/~history/HistTopics/History_overview.html