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