Day 5, MAT115

• Announcements:
  • Reminder: turn off your rectangles!
  • You have an assignment, due Thursday. Remember: justify your work!
  Question of the Day: How did the Mayans write numbers?

• Last time, over in the "Old World", we introduced the Babylonians (or, rather, their mathematics), via a "detective story": I hope that you all like detective stories.... The story is called Detective in the times of Babylon.
  • Babylon, then (source)
  • Babylon, today
  What did we deduce about
  • the Babylonians and place value?
  • the purpose of the table?
  • the Babylonian's missing number(s)?

  How would the Babylonians write these numbers (using places 1, 60, 3600, 216000, etc. -- powers of 60!)?

  1. 3600
  2. 573
  3. 14001

• Let's reflect on some of the things that we discover in chapter 6 of our text: here are a few of my musings:
  1. How cumbersome are the Roman numerals! How would you write your own birth year?
  2. "biology is deeply embedded" in number systems (10 "digits" in our system; tallies of 5; 5 and 10 in Roman numerals; 10 in Babylonian; 5 and 20 for the Mayans)
  3. Babylonians based their system on -- its prime factorization gives it lots of factors (1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30).
  4. We need no new symbol for 10 in our arabic numerals: "The great innovation here is that even though this system is based on the number 10, there is no single symbol reserved for 10." We use only the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 -- and place value (hence the name of the chapter: "location, location, location").

     "Ten is marked by a position -- the tens place -- instead of a symbol."

  5. Place value suddenly makes arithmetic possible. "Just master a few facts", our author says.

     My own musing: it's interesting that we seem to be abdicating our arithmetic to machines. Many educated people can't do arithmetic anymore....

  6. The unsung hero is the digit 0 -- a symbol for nothing! (The Babylonians can't distinguish the numbers 1 from 60, without some additional context.)
  7. Binary, or base 2 math: a place-value system with just two symbols, 0 and 1. Perfect for computers....
  8. Cornell built the first telegraph line (Baltimore to Washington, D.C.): 1844. I know that the telegraph displaced the pony express, which only ran for one year, at the beginning of the Civil War (1860-1861).

• Today's new topic: the mathematics of the Mayans.... Let's have a look at what the Mayans were doing over in the new world:
  
  • Take a look at this Mayan lunar calendar:
    

    "The Maya also made their own very accurate measurement of the solar year, putting it at 365.242 days. The latest computations give us the figure of 365.242198: so the Maya were actually far nearer the true figure than the current Western calendar of 365 days (which, with leap years, gives a true average of 365.2425)." From The Universal History of Numbers, by George Ifrah.

    How can we make sense of the numbers that are appearing?

    • How would the Mayans write the number we call 1729?
    • I'll write something in Mayan, and you tell me what it is in our number system.
    • What distinguishes the Mayans from the Babylonians?
    • Let's do a few examples with Mayan numbers: write
      • 37 in Mayan
      • 694 in Mayan
      • 1034 in Mayan

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