Day 6, MAT115

• Announcements:
  • Reminder: turn off your rectangles!
  • You have an assignment, due today.
  Question of the Day: How did Egyptians do their math?

• Last time we began to explore the way in which the Mayans wrote their numbers:

  We were looking 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

• Now, on to the Egyptians! (What's this got to do with Egyptians?)
  

• Let's start with a look at the history of a western understanding of Egyptian Mathematics.

• At the heart of Egyptian multiplication is the "Fraudini fact", or the "binary factorization":
  Every natural number is either a power of two, or can be expressed as a sum of distinct powers of two in a unique way.

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

  3=1+1+1 rather than 3=2+1 (which is the unique binary factorization).

• Again this is parallel to the prime factorization, which you encountered somewhere along the line in your elementary education:
  Every natural number is either prime, or can be expressed as a product of primes in a unique way.

• We'll start with a simple example: multiply 23*42.

• Let's show that we can do a multiplication in either order, by checking the product 42*23.

• Let's try a longer multiplication. Consider, for example, 321*112:

  1	321
  2
  4
  8
  16
  32
  64
  128	Too big!

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

• Examples: Try these:
  1. 43*16
  2. 21*79

• Now: multiplication's not too bad. How about Egyptian division?
  • 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?!

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