Day 06, MAT115

• You have an assignment due today -- keep an eye on the assignment page.

• have a little review of the Mayan math,
• then find out how those Egyptians were doing math.

• What did we discover from 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 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? Well zero for one thing. The Mayans had one!
  • Let's do a few examples with Mayan numbers: write
    • 37 in Mayan
    • 694 in Mayan
    • 1034 in Mayan

• How does the game work?
• Can you win at the game? You should be able to do it every time!
• Each of you will be part of a team of three, taking turns playing the roles of contestent, Fraudini, and someone to check the play.
• Here's the handout for the cards, if you want to print off another copy.

(I underline "distinct" because you cannot repeat powers: otherwise you could write, for example,

Now add up those rows marked with an asterix (*), and you'll get the answer (35952).

1. 43*16
2. 21*79

• fractions -- binary decimals, oh Ra!

• First of all, division is just multiplication backwards, right?

  Let's look at the simplest example imaginable: divide 32 by 8. We can actually do it by Egyptian multiplication, since 8 divides into 32 evenly:

  1	8
  2	16
  4	32	*

  So the answer is 4 (how do we get 4?)

• Similarly we could divide 40 by 8, using the same table (again easy, since 8 divides into 40 evenly):

  1	8	*
  2	16
  4	32	*

  So the answer is 5 (how do we get 5?)

• When the denominator doesn't divide the numerator evenly, fractions make it more interesting:

  Let's look at an example: divide 35 by 8.

  In a way we turn it into a multiplication problem: what times 8 equals 35? So we know the 8, and use it to "double" -- but then to "halve", when 8 won't go evenly into 35:

  1	8
  2	16
  4	32	*
  1/2	4
  1/4	2	*
  1/8	1	*

  So the answer is 4+1/4+1/8

  (Where have we seen those fractions before? Look to the Eye of Horus!)

• the Egyptians restricted themselves to the so-called "unit fractions", which are fractions of the form 1/m: unit fraction table, which is found on the Rhind Papyrus (which dates to around 1650 BCE).

  But they didn't restrict themselves to "halving", as our next example shows. Divide 6 by 7:

  1	7
  1/2	3+1/2	*
  1/4	1+1/2+1/4	*
  1/7	1
  1/14	1/2	*
  1/28	1/4	*

  So the answer is 1/2+1/4+1/14+1/28

• Why did Egyptians do things this way? (an example division problem, using binary)

     Dominic Olivastro, "Ancient Puzzles", suggests a third reason why 
  this use of unary fractions is good. Consider the problem Ahmes poses 
  of dividing 3 loaves of bread between 5 people. We would answer "each
  person gets 3/5-ths of a loaf". If we implemented our solution, we might
  then cut 2 loaves into 3/5 | 2/5 pieces, with bread for 3 people; then
  cut one of the smaller pieces in half, giving the other two people 
  2/5 + 1/5 pieces. Mathematically acceptable, but try this with kids and
  they will insist that it is not an even division. Some have larger pieces,
  some have smaller. Ahmes would calculate 3/5 as :
     3/5 = ()3 + ()5 + ()15    [ = 1/3 + 1/5 + 1/15 ]
  Now cut one loaf into fifths, cut two more into thirds, then take one of
  the 1/3-rd pieces and cut it into 5-ths (for the 1/15-th pieces), and you
  can now distribute everyone's 3/5-ths share in a way that _looks_ equal,
  since they will have exactly the same size pieces. (And no, I don't want
  to argue about the crust.)
  

• Here's a relatively easy one: Suppose Fatima had 3 loaves to share between 4 people. How would she do it?

  (Think about what the answer means, in terms of bread.)

• A little trickier:
  1. How would you divide 5 by 7?
  2. How can we use the unit fraction table to get the same answer?

• How would you like to do story problems like this one?!