- Reminder: last time we agreed to move our first test from Friday, 2/15, to Monday, 2/18. I've changed it on our assignment page.
- You'll have your second quiz Friday. It will involve primes and prime factorization, and Babylonian math. Any questions about primes (we'll review Babylonian math today)?
- there appear to be two distinct symbols, martini glass and boomerang, representing 1 and 10 respectively.
- On the left of the single vertical line, they appear to be counting up by ones (to 20), at which point they switch to counting up by tens -- 30, 40, 50.
- On the right of the single vertical line, they appear to be counting up by nines.
- Something mysterious happened when the Babylonians got to 63 while counting up by nines.
- Something weird happened at 180 (20 times 9), too: it looks like they wrote "3". But that's 3 60s, or 180, as it should be; but there is nothing about the representation of the number itself to indicate that: you have to get the difference from context. Very dangerous.
From this, we deduce that
- the Babylonians used place value to switch to "the 60's place" (a new television sitcom?).
- this tablet is a times table for nine.
- the Babylonians didn't have the concept of zero.
How would the Babylonians write
- 3600
- 573
- 14001
- How cumbersome are the Roman numerals!
- "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)
- Babylonians based their system on
- We need no new symbol for 10 in our arabic numerals: 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").
- 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....
- 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.)
- Binary, or base 2 math: a place-value system with just two symbols, 0 and 1. Perfect for computers....
- 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).
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This system is based on only two symbols, with a position, or place value. (Dit is "dot"; Dah is "dash".)
How do you say "SOS" in Morse code?
How do you tap it out? "If the duration of a dot is taken to be one unit then that of a dash is three units. The space between the components of one character is one unit, between characters is three units and between words seven units. To indicate that a mistake has been made and for the receiver to delete the last word, send di-di-di-di-di-di-di-dit (eight dots)." (source)
Look at right, for the pattern in the digits of our number system. Isn't 0 out of order? Everyone's got a problem with zero! (Except the Mayans....)
- "We do not know the date of these mathematical
achievements but it seems certain that when the system was
devised it contained features which were more advanced than any
other in the world at the time."
- So what do we notice about their system? Observations?
- What would be the first number in the next place of their
place value system? (Assume a base 20 system.)
- Note how accurate the Mayan astronomical observations
were: day length, "365.242 days (the modern value is 365.242198
days)"; lunar month, "29.5308 days .... The modern value is
29.53059 days."
- How would the Mayans write
- 8000
- 573
- 14001
