Handout [MC,i. Home] [this semester's schedule w/links] asquared at MCC math ed stuff [Good Stuff -- free & valuable resources] Classes Napier Bones   S Sheet  

  50+ Centuries of Computation in a 21st-Century Format
























      Pre Session Warm Up Activities & Questions
1. What's a shekel?
2. For what is Napier famous?
3. Explain the "Eye of Horus."
4. How did Archimedes die?
5. Write 2510 in base 2.
6. Simplify: log(2) + log(3).
7. Introduce yourself to those around you and discuss your answers.

     

Disclaimer
      This is the "Azzolino Version" of math history based on:
   

  • research in computing devises, math history, and linguistics,
  • extensive expertise with manipulatives.

      It is left for you to determine the "truth."
                        -- Agnes (A²) Azzolino

    History    
































Consider and/or Use a Tool to Compute

  • Origin of Numbers -- Named and Unnamed
  • Talley Sticks
  • Tokens & Coins
  • Abaxes (Sand Table), Reckoning Boards, Banks & Add on a Board
  • Writing Numerals
  • Multiply "Like an Egyptian"
  • Multiply with Napier's Bones
  • Abacuses
  • Hindu-Arabic 10-Digit Computation
  • Slide Rules & Use Them to Add and Multiply


    History    





























Origins -- Named and Unnamed
Unity, Duality, Multiplicity

   Point to the body part.
    No need to name the number.
   
    unity -- only one
    The King
    The Queen
    duality -- exactly two
    pair of sox
    yoke of oxen
    multiplicity -- Collective Nouns   
    herd of antelope
    colony of ants
    swarm of ants
    army of ants
    culture of bacteria
    crowd of people
    herd of zebras
    zeal of zebras
   
    History    
































Tally Sticks

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    Diaspora   History    

































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Tokens Then Coins

      S Sheet   History  
































Abaxes (Sand Tables), Reckoning Boards, Banks

    History   S Sheet   Apices    
































Writing Numerals -- Not Just For Recording But Truely For Computing!


 
 
    S Sheet   History    
































Egyptian Multiplication     Multiply 45671 by 92.
emulti.htm

    1st 2nd 3rd 4th   Egyptian Math     History   S Sheet  
































Abacus

2 over 5: Earlier Roman, Chinese        

1 over 4: Later Japanese

    History   S Sheet  
































Hindu-Arabic 10-Digit Computation -- Remember Paper Is Now Available

    Pascal Calculator   Mult. Tables Cover    Mult. Table       comptometer     History    
































Napier's Bones

    Two-Digit     Resource Page     History   S Sheet    
































Slide Rules

    History   S Sheet  
































End Notes

    Sources     History    
































Debrief


Origin of Numbers -- Named and Unnamed
  • Numbers started out as part of names - yoke of oxen, herd of cattle, gaggle of geese.
  • Days of conch shells and taboos about counting.
Talley Sticks
  • Though conventional wisdom for years said otherwise, "primative people" did count more specifically than "many."
  • The Ishago Bone possibly of 20KYA, 20 thousand years ago, with primes 11, 13, 17, and 19 marked, and a crystal.
  • 35,000 BCE to 8,500 BCE -- Talley marks made permanent and important math records.
Tokens Then Coins
  • 10,000 BCE to 6,000 BCE -- Tokens served as movable and permanent records.
  • The use of tokens set the stage for the earliest writing.
  • 3000 BCE - Shekels, temple coins, were used in Sumeria.
  • 600 BCE -- Minted staters were used in Lydia (Turkey) and the use of coins spread quickly there after.
Abaxes (Sand Table), Reckoning Boards, Banks
  • Sand Tables, or beaches, for drawing accurate, measurable scale drawings.
  • Reckoning Boards, from the Egyptian, thorugh the Banks, merchant benches (as in bank rupt, bankrupt) of the Middle Ages.
  • With calculi (pebbles) or apices to compute.
  • With choice of lines or spaces or both or vertical or horizonal orientation with whatever base is needed -- guilder, pound, or shekel.
  • Set the stage for the Roman hand-held abacus and other abacuses.
Writing Numerals
  • ADDITIVE SYSTEM, Egyptian, Greek [8], early Roman
    - The order of the "symbols" doesn't matter though they are often ordered.
  • MULTIPLICATIVE SYSTEM, Chinese, as in modern expanded notation
    - Write from largest to smallest a string of pairs of "number of" then "value of symbols" to write the number.
  • POSITIONAL SYSTEM, decimal, Greek & Roman Numerals
    - Order or position does matter.
Egyptian Multiplication
  • Eye of Horus and a memory trigger for the power series that facilitated multiplication and division.
  • Used for division and multiplication and still used in Middle Ages as duplation and mediation.
  • FIRST! statisticians and secretive about their work.
Abacuses
  • First hand-held calculators.
  • Great for speed, portability, shakey tables and tax assesors -- think how easy a tith is to compute on a board or abacus.
Hindu-Arabic 10-Digit Computation
  • Had Popular & Commercial Uses.
  • Has Strengths and Weaknesses.
  • Still in Use.
Napier's Bones
  • MAJOR COMPUTING ACHIEVEMENT! especially for division!
  • Can you SEE the inspiration for log computation?
Slide Rules
  • Take John Napier's natural log work and add the scales of Edmund Gunter and William Oughtred's paired Gunter scales and computation may be completed ala the sand table and scale drawing.
        History    






























Contents

    History  
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© 2011, 2019, Agnes Azzolino
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