An Introduction toAn Introduction toCPI 200: Math Foundations CPI 200: Math Foundations

of Informaticsof Informatics

School of Computing & InformaticsSchool of Computing & InformaticsArizona State UniversityArizona State University

Dianne HansfordDianne Hansford

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 the Informatics Relevance to the Informatics Certificate/Concentration? Certificate/Concentration?

How are we going to learn the topics?How are we going to learn the topics?

Why is math important?Why is math important?

Mathematics is part of our cultureMathematics is part of our culture

Math ideas are as important as the ideas Math ideas are as important as the ideas of Darwin, Marx, Voltaire, or Dostoyoveskyof Darwin, Marx, Voltaire, or Dostoyovesky

Ideas shape how we perceive the worldIdeas shape how we perceive the world

Ideas shape how we perceive our place in Ideas shape how we perceive our place in the worldthe world

Let’s take a look at history from a math Let’s take a look at history from a math perspectiveperspective

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 does 4 cows cost?If 1 cow is worth 3 goats, then how much does 4 cows cost?

Construction of tables (pre-computed squares and Construction of tables (pre-computed squares and cubes) to aid calculationscubes) to aid calculations

http://en.wikipedia.org/wiki/Babylonian_mathematicshttp://en.wikipedia.org/wiki/Babylonian_mathematics

Babylonia con’tBabylonia con’t

Pythagorian triples: a^2 + b^2 = c^2Pythagorian 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

Rational numbers (a/b) cannot measure all lengths – Rational numbers (a/b) cannot measure all lengths – need irrational numbers sqrt(2)need irrational numbers sqrt(2)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 http://en.wikipedia.org/wiki/Greek_mathematics

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

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/Aristotlehttp://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

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 controlchange of seasonschange of seasons

droughtdrought

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

Ideas leveraged by Brahe and KeplerIdeas leveraged by Brahe and Kepler

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

Leaning tower of Pisa storyLeaning tower of Pisa story

http://en.wikipedia.org/wiki/Galileohttp://en.wikipedia.org/wiki/Galileo

DescartesDescartes

1596 – 1659 France1596 – 1659 France

Father of Modern MathematicsFather of Modern MathematicsFather of Modern PhilosophyFather of Modern Philosophy

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

Fundamental Theorem of CalculusFundamental Theorem of Calculus integrals (areas) are the same as integrals (areas) are the same as

antiderivatives: easy way to calculate areaantiderivatives: easy way to calculate area

Derivative is a rate of changeDerivative is a rate of change Everything in our world is changing with time!Everything in our world is changing with time!

Newton 3 laws of motion – basis of Newton 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?

What are we going to study?What are we going to study?

Discrete math: first-order logic, sets, graphs, relationsDiscrete math: first-order logic, sets, graphs, relations-- how to communicate with a computer-- how to communicate with a computer-- tools for / getting started with abstraction-- tools for / getting started with abstraction

Numbers: number systems, floating point numbers, finite precision, scaleNumbers: number systems, floating point numbers, finite precision, scale-- understanding how numbers are represented in a computer-- understanding how numbers are represented in a computer-- ramifications, problems, and limitations-- ramifications, problems, and limitations

Modeling: abstraction, recursion, concurrencyModeling: abstraction, recursion, concurrency-- How to re-pose a problem into one we know the answer for-- How to re-pose a problem into one we know the answer for

Algorithms: definition, types, and basics of complexityAlgorithms: definition, types, and basics of complexity-- how to communicate with a computer/software-- how to communicate with a computer/software

Programming concepts: types of programming languages and paradigmsProgramming concepts: types of programming languages and paradigms-- Languages we use to communcate-- Languages we use to communcate

What are we going to study?What are we going to study?

Calculus concepts: differential and integral concepts, limits and Calculus concepts: differential and integral concepts, limits and continuitycontinuity--Understand our model of the universe--Understand our model of the universe

Analytic geometry basics: 2D and 3D geometry basicsAnalytic geometry basics: 2D and 3D geometry basics-- Geometry for building and manipulating models-- Geometry for building and manipulating models

Numerical and statistical methods: linear maps, regressionNumerical and statistical methods: linear maps, regression-- Doing stuff!-- Doing stuff!

Ethics: societal impact of computational mathematicsEthics: societal impact of computational mathematics

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 MySpace, YouTube, IM, Email/spam, Virtual communities, Cell Phone (iPhone) MySpace, 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?

Excel Excel

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