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.


This page mathnstuff.com/papers/nctmb.htm.
The older essay version is page mathnstuff.com/papers/amtnj09.htm.


      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 (AČ) Azzolino



  • Origin of Numbers -- Named and Unnamed
  • Talley Sticks
  • Tokens & Coins
  • Abaxes (Sand Table), Reckoning Boards, Banks
  • Writing Numerals
  • Egyptian Multiplication
  • Abacuses
  • Hindu-Arabic 10-Digit Computation
  • Napier's Bones
  • Slide Rules
  • End Notes
  • Debrief

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

Tally Sticks

    History   Text    



    Diaspora   History    










Tokens Then Coins

    Hands Coins History LA  

Abaxes (Sand Tables), Reckoning Boards, Banks

    Board History   LA  

Writing Numerals

    No. System   Hyro   History   LA    

Egyptian Multiplication     Multiply 45671 by 92.

    1st 2nd 3rd 4th Resource Page History   Board LA  


2 over 5: Earlier Roman, Chinese        

1 over 4: Later Japanese

    Counting Board Multiply   Abacus Multiplication   Apices   History   LA  

Hindu-Arabic 10-Digit Computation

    Pascal Calculator  Mult. Tables  Mult. Tables  History   LA  

Napier's Bones

    Two-Digit Resource History   N Bones LA  

Slide Rules

    History   Slide LA  

End Notes

    Sources History    


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.
  • 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.


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