Keeping Count

1
Keeping Count

• Writing Whole Numbers
• K. McGivney
• MAT400

2
Introduction

• Civilizations that developed writing also had
  mathematical knowledge.
• The early history of math is traced by
  identifying records that indicate the number
  systems used within a society. Were the numbers
  used for accounting? for solving problems of
  commerce? for academic problems?
• Other evidence of math is based on
  accomplishments that (we believe) require
  mathematical sophistication for example, the
  Great Pyramids of Gizeh.

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Main topics

• Counting and numbers
• Some ancient systems
• Egyptian, Babalonyian, Mayan, Greek, Chinese
• Roman numerals
• Hindu-Arabic numbers

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Counting

• Some evidence from the early and diverse records
  of human societies
• Tally marks found on bones in Zaire (around 6000
  BCE)
• Quipu knots from Incas in Peru (1400 CE)
  non-writing recording device

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Egyptian number system

• (as early as 3000 BCE)
• Hieroglyphic system.
• Used papyrus as paper
• Base 10 grouping system
• Additive not positional.
• Rhind papyrus (1650 BCE)
• How would you write 3,244? 21,237?

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Egyptian number system

• Egyptian (as early as 3000 BCE)
• How would you write 3,244?
• How would you write 21,237?

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Babylonian number system

• Babylonian (as early as 2000 BCE) originated in
  Mesopotamia (now part of Iraq)
• Cuneiform on clay tablets, used two symbols
• Sexagesimal system. Base 10 for the digits up
  to 59, and base 60 for large numbers. (Today
  trig, clock). Multiply successive groups of
  symbols by increasing powers of 60 -- similar
  to our system we multiply successive digits by
  increasing powers of 10.
• Place value system with no 0 that is, it used
  the position of the symbols to determine the
  value of a symbol combination.

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Babylonian number system
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Representing Numbers 60 and Beyond

• Numbers between 60 and 3599 were represented by
  two groups of symbols second is placed to the
  left of the first and separated by a space or
  comma.
• The value of the entire quantity is found by
  adding the values of the symbols within each
  group and then multiplying the value of the left
  group by 60 and adding in to the value of the
  group on the right.
• For numbers 3600 and beyond, use more
  combinations of the two basic shapes and place
  the groups further to the left. Each group was
  multiplied by successive powers of 60.

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Examples of Babylonian Number System

• Convert the following numbers to Base 10
  numbers                 

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Problems with the Babylonian System

• Spacing between symbol groups.

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Mayan number systems

• Around 300 BCE Central America
• Similar to the Babylonian system, but without the
  spacing problems.
• Two basic symbols Dots and lines for 1s and
  5s
• Written vertically
• Sort of base 20 (vigesimal) with a strange use of
  18
• Place value system with a 0
• Examples

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Greek and Roman Systems

• More primitive than the Babylonians.
• Mayan culture was not known to the Europeans
  until several centuries later so the system had
  no influence on the development of number systems
  in Western culture.

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Greek (alphabetic) number system (450BCE)

• One of two Greek systems
• Ciphered numeral system
• Greek letters stand for numbers
• Non-positional (additive) decimal system

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Chinese-Japanese number system

• Multiplicative grouping system
• Symbols for digits and symbols for value
• Essentially like the way we write number names
  four hundred and three, or one thousand twenty
  five
• See Wikipedia entries for more

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Roman numerals

• As late as 500 CE
• I 1, V 5, X 10, L 50, C 100, D 500,
  M 1000
• Simple grouping system (later) with subtractive
  principle
• No symbol for 0
• Used in eighteenth century academic papers, and
  still used today in limited form

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Roman numerals

• Write the following as modern numerals
• MDCCCXXVIII
• CDXCV
• Without translating to modern numerals, find
  XV11 XX ___

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Hindu-Arabic numerals

• Developed in India around 600 CE
• Transmitted by Islamic expansion into India
  around 700 CE
• Invented 0
• Spread to the West initially by Latin
  translations of Arabic texts as early as 1100 CE
• Western trade with the Middle East at the end of
  Europes Dark Ages helped spread the system
• Fibonaccis Liber Abaci (1202 CE) begins with a
  page explaining these numerals

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Summary

• The concept of number developed independently
  in every culture, much like language.
• The various systems are similar in many ways.
  Some are positional, some have zero, and some are
  still used today.
• Commerce and communication led to widespread use
  of the Hindu-Arabic system based on its elegance
  and computational ease.

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References

• Berlinghoff and Gouvea
• MacTutor Math History Archive
• Jamie Hubbards Mayan Numerals web page (8/31/04)
  at http//mathcentral.uregina.ca/RR/database/RR.09
  .00/hubbard1/MayanNumerals.html
• Victor J. Katz, A History of Mathematics,
  Pearson/ Addison Wesley, 2004
• Howard Eves, An Introduction to the History of
  Mathematics, Saunders College Publishing, 1991.
• Wikipedia entry on Number Names (8/31/04) at
  http//en.wikipedia.org/wiki/Number_names
• http//www.michielb.nl/maya/math.html

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Approximate Timeline for the Development of
Numbers

• 3000 BCE Egyptian numerals
• 2000 BCE Babylonian (Iran/Iraq)
• 1000 BCE Chinese-Japanese
• 600 BCE500 CE Roman Empire
• 300 BCE Mayan (Central America)
• 600 CE Hindu-Arabic numerals
• 500 CE1100 CE Dark Ages in Europe
• 1100 CE Arabic texts translated
• 1202 CE Fibonacci publishes Liber Abaci

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Test questions (Note These should not be part
of your PowerPoint presentation.)

• Which of the following numbers is the largest?
  Which is the smallest? Which is illegal?
• XV
• XL
• IC
• LI
• IL

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Test questions

• Which book begins with a page explaining
  Hindu-Arabic numerals?
• The Elements, by Euclid, around 300BCE
• Liber Abaci, by Leonardo Pisano, around 1200 CE.
• Principia Mathematica, by Isaac Newton, around
  1700 CE.

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Test questions

• Give two numbers (in our modern notation) that
  when translated to Babylonian system might be
  confused with one another.

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Test questions

• Give an example of how each of the following
  number systems is still used today.
• Babylonian
• Roman
• Hindu-Arabic

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Test questions

• How many different symbols must be memorized to
  write all of the numbers less than 1000 in each
  of the following systems?
• Hindu-Arabic
• Babylonian cuneiform
• Egyptian hieroglyphics