Presentation on theme: "History of Mathematics:"— Presentation transcript:
1History of Mathematics: Academic ContentStandards TimelineMichelle Yee
2Number, Number Sense, and Operations Origin of ZeroHow would math be different without zero?2000 B.C. Babylonians represented zero by leaving gaps between wedge marks on clay. The gaps were not considered a number.used base 60 numeration system350 B.C. Greeks were unsure about the idea of zeroDid not use a positional system, therefore zero was not necessary.600 A.D. Hindu-Arabic Decimal System- Zero is officially introduced.AL' KHWARIZMI – wrote math book that introduced zero to the world.1150 A.D. Hindu-Arabic Decimal System reaches Europe.
3Number, Number Sense, and Operations Prime NumbersA whole number greater than one that has exactly two factors; 1 and itself.•325 B.C. EUCLID proved there is an infinitenumber of primes.- Author of The Elements- used to teachgeometry for more than 2,000 years.•276 B.C. ERATOSTHENESSieve of Eratosthenes- an algorithm for findingprime numbers.•1777 GAUSS proved any number is a product of primes.
4Number, Number Sense, and Operations Fractions•3000 B.C. Egyptians recognized that fractions begin withreciprocals of whole numbers.•Fractions were written as a sum of unitfractions. An eye was placed over theinteger to represent the reciprocalAKA unit fraction.•Horus- Egyptian God who foughtforces of darkness. His eye is thesymbol of Egyptian unit fractions.
5The ratio of the circumference to the diameter of a circle. MeasurementPiThe ratio of the circumference to the diameter of a circle.Used by Egyptians, Babylonians, Greeks, Chinese and HebrewsGreeks discovered Pi was an irrational numberB.C. ARCHIMEDES OF SYRACUSE- Made the first theoretical calculation of Pi.used circles inscribed and circumscribed by polygons to find the value of Pi to be between 223/71 and 22/7.
6Volume and Surface Area 287-212 B.C. ARCHIMEDES OF SYRACUSE MeasurementVolume and Surface AreaB.C. ARCHIMEDES OF SYRACUSEGreek MathematicianFound the surface area and volume of a sphere:4πr /3πr2Found volume of a cone: 1/3πr2h
7Measurement Metric System Invented in France in 1970. French National Assembly told a committee from the Academy of Sciences of Paris to standardize the units of measurement.Some mathematicians on the committee:Jean Charles de Borda ( )Joseph-Louis Comte de Lagrange ( )Pierre-Simon Laplace ( )Gaspard Monge ( )Meter comes from the Greek wordmetron, meaning measure.
8Geometry and Spatial Sense 569 B.C. PYTHAGORAS OF SAMOS“Everything is number.”•Proved theorem which is named after him,Pythagorean Theorem:In a right triangle, the square of the hypotenuse isequal to the sum of the squares of the other two sides.•Discovered that notes played in music correspondto ratios with small numbers.Example: Halving the length of a musical string gave amusic note one octave higher than the first.
9Geometry and Spatial Sense One of the many proofs for the Pythagorean theorem:+4( )==(a+b)(a+b)Area ofareaareacbabc2 + 4(ab/2) = (a+b)2c2 + 2ab = a2 + 2ab + b2c2 = a2 + b2cArea ofb=ab/2aArea of=c2
10Geometry and Spatial Sense Cartesian gridRENE DESCARTESFrench philosopher and mathematicianDescartes was sick and lying on his bedstaring at a fly. The ceiling had squaretiles, and Descartes noticed he coulddescribe where the fly was in relation tothe points formed from the tiles.
11Patterns, Functions, and Algebra History of AlgebraEgyptian Algebra1850 B.C. Solved problems equivalent to a linear equation in one unknown.300 B.C. Solved problems equivalent to a system of two second degree equations in two unknowns.Did not use symbols.
12Patterns, Functions, and Algebra History of AlgebraBabylonian AlgebraMore advanced than Egypt.Had a general procedure equivalent to solving quadratic equationsConsidered some problems involving more than two unknowns and a few equivalent to solving equations of higher degree.Problems solved were taughtthrough examples and no reasonsor explanations were given.Little use of symbolsNo negative or irrational numbers
13Patterns, Functions, and Algebra History of AlgebraHindu AlgebraIntroduced negative numbers to represent debts.A.D.BRAHMAGUPTA-first known to use negative numbers.Developed correct procedures for working with irrational numbers.Used some symbolism.Steps of problems were stated but reasons orproofs were not given.Included negative as well as irrational roots.
14Patterns, Functions, and Algebra History of AlgebraArabic AlgebraImproved the Hindu number symbols andthe idea of positional notation.Worked with irrational numbersRejected negative numbers and solutionsSolved quadratic equations and recognized two solutionsAlgebra is named after a book written byAl’ Khwarizmi.Algorithm and arithmetic are from modifications ofAl’ Khwarizmi’s name.
15Patterns, Functions, and Algebra LEONARD EULERSwiss mathematicianContributions to all branches of mathematics includingCalculusGeometryAlgebraNumber theoryInventor of graph theory.Developed several notations used today:Π for pii for √-1∆y for change in yf(x) for a function∑ for summation
16Data Analysis and Probability BLAISE PASCALFrench mathematicianWrote a math book at the age of 16.Pascal’s Triangle:Triangular array of numbers studied in China and India.Pascal discovered new properties of the triangle and solved problems using it.Pascal’s Triangle has many uses.Example: Evaluating combinationsIf we need to know the number ofcombinations of n things taken r at atime, (the # of subsets of size r in a setof size n) we read entry number r of rownumber n from Pascal’s Triangle.
17Data Analysis and Probability BLAISE PASCALProbability TheoryOriginated over a dispute between Pascal and French mathematician Pierre de Fermat about the problemof the points: Involved how to divide pointsbetween players of a game if the contestants’scores and the score needed to win were known.FermatProbability theory became an important tool for scientists studying the physical world.
18Data Analysis and Probability 1977 JOHN TUKEYAmerican Statistician•Inventor of the box and whisker plot•Inventor of the stem and leaf plot
19Mathematical Processes B.C. ARISTOTLEGreek Philosopher and MathematicianLogical thinking- conclusions must be supported by observationAristotle systemized deductive logicFrom general to specificThe modern scientific method is a combination ofDeductive reasoningInductive reasoning
20Mathematical Processes George PolyaHungarian mathematicianAuthor of How to Solve It (1945) describes methods of problem solving.Developed Polya’s Four step Problem-Solving Process1. Understand the problem2. Devise a plan3. Carry out the plan4. Look back
21SourcesBarrow, J. D. (1992). Pi in the sky: Counting, thinking, and being. Oxford: Clarendon Press.Gianopoulos, A., Langone, J., & Stutz, B. (2006). Theories for everything: An illustrated history of science from the invention of numbers to string theory. Washington, DC: National Geographic.Heeren, V.E., Hornsby, J., & Miller, C.D. (2001). Mathematical ideas (9th ed.). Boston: Addison Wesley Educational Publishers.Lewinter, M. & Widulski, W. (2002). The saga of mathematics: A brief history. Upper Saddle River, NJ: Prentice Hall, Inc.Windelspecht, M. (2002). Groundbreaking scientific experiments, inventions, & discoveries of the 17th century. Westport, CT: Greenwood Press.Online SourceO’Connor, J. J. & Robertson, E.F. The mactutor history of mathematics archive retrieved October & November 2007 fromMichelle Yee