History of Mathematics:

Academic Content

Standards Timeline

Michelle Yee

Number, Number Sense, and Operations

Origin of Zero How 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 system

• 350 B.C. Greeks were unsure about the idea of zero– Did 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.

Number, 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 infinite number of primes.

- Author of The Elements- used to teach geometry for more than 2,000 years.

•276 B.C. ERATOSTHENES – Sieve of Eratosthenes- an algorithm for finding prime numbers.

•1777 GAUSS proved any number is a product of primes.

Number, Number Sense, and Operations

Fractions

•3000 B.C. Egyptians recognized that fractions begin with

reciprocals of whole numbers.

•Fractions were written as a sum of unit fractions. An eye was placed over the

integer to represent the reciprocal AKA unit fraction.

•Horus- Egyptian God who fought forces of darkness. His eye is the symbol of Egyptian unit fractions.

Measurement

PiThe ratio of the circumference to the diameter of a circle.

• Used by Egyptians, Babylonians, Greeks, Chinese and Hebrews

• Greeks discovered Pi was an irrational number

• 287-212 B.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.

Measurement Volume and Surface Area

287-212 B.C. ARCHIMEDES OF SYRACUSE Greek Mathematician

• Found the surface area and volume of a sphere: 4πr2 4/3πr2

• Found volume of a cone: 1/3πr2h

Measurement

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 (1733-1799)– Joseph-Louis Comte de Lagrange (1736-1813)– Pierre-Simon Laplace (1749-1827)– Gaspard Monge (1746 -1818)

• Meter comes from the Greek word metron, meaning measure.

Geometry 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 correspond

to ratios with small numbers. Example: Halving the length of a musical string gave a music note one octave higher than the first.

Geometry and Spatial Sense

Area of =(a+b)(a+b)

ab

bc

One of the many proofs for the Pythagorean theorem:

Area of

a

bc

=ab/2

Area of =c2

area

+4=(

) area

c2 + 4(ab/2) = (a+b)2

c2 + 2ab = a2 + 2ab + b2

c2 = a2 + b2

Geometry and Spatial Sense Cartesian grid

1596-1650 RENE DESCARTES French philosopher and mathematician

Descartes was sick and lying on his bed staring at a fly. The ceiling had square tiles, and Descartes noticed he could describe where the fly was in relation to the points formed from the tiles.

Patterns, Functions, and Algebra

History of AlgebraEgyptian Algebra• 1850 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.

Patterns, Functions, and Algebra

History of AlgebraBabylonian Algebra• More advanced than Egypt.

• Had a general procedure equivalent to solving quadratic equations

• Considered some problems involving more than two unknowns and a few equivalent to solving equations of higher degree.

• Problems solved were taught through examples and no reasons or explanations were given.

• Little use of symbols

• No negative or irrational numbers

Patterns, Functions, and Algebra

History of Algebra

Hindu Algebra• Introduced negative numbers to represent debts. 598-665 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 or proofs were not given.

• Included negative as well as irrational roots.

Patterns, Functions, and Algebra

History of AlgebraArabic Algebra• Improved the Hindu number symbols and

the idea of positional notation.

• Worked with irrational numbers

• Rejected negative numbers and solutions

• Solved quadratic equations and recognized two solutions

• Algebra is named after a book written by Al’ Khwarizmi.

• Algorithm and arithmetic are from modifications of Al’ Khwarizmi’s name.

Patterns, Functions, and Algebra

1707-1783 LEONARD EULER Swiss mathematician

• Contributions to all branches of mathematics including– Calculus– Geometry– Algebra– Number theory

• Inventor of graph theory.

• Developed several notations used today:– Π for pi– i for √-1– ∆y for change in y– f(x) for a function– ∑ for summation

Data Analysis and Probability

1623- 1662 BLAISE PASCALFrench mathematician

Wrote 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 combinations– If we need to know the number of combinations of n things taken r at atime, (the # of subsets of size r in a setof size n) we read entry number r of row number n from Pascal’s Triangle.

Data Analysis and Probability

1623- 1662 BLAISE PASCALProbability Theory• Originated over a dispute between Pascal and French

mathematician Pierre de Fermat about the problem of the points: Involved how to divide points between players of a game if the contestants’ scores and the score needed to win were known.

Fermat

Probability theory became an important

tool for scientists studying the physical

world.

Data Analysis and Probability

•Inventor of the box and whisker plot

•Inventor of the stem and leaf plot

1977 JOHN TUKEYAmerican Statistician

Mathematical Processes

384-322 B.C. ARISTOTLEGreek Philosopher and Mathematician

Logical thinking- conclusions must be supported by observation

• Aristotle systemized deductive logic– From general to specific

• The modern scientific method is a combination of– Deductive reasoning– Inductive reasoning

Mathematical Processes

1887-1985 George PolyaHungarian mathematician

• Author of How to Solve It (1945) describes methods of problem solving.

• Developed Polya’s Four step Problem-Solving Process1. Understand the problem

2. Devise a plan 3. Carry out the plan 4. Look back

SourcesBarrow, 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 from http://www-history.mcs.st-andrews.ac.uk/history/index.html.

Michelle Yee