# An Introduction to CPI Math Foundations of Informatics

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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
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
   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
-- 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
   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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 views: 17 posted: 7/8/2011 language: English pages: 21