Day 11, MAT115

• Reminder: you have an assignment due to be handed in next Wednesday.

• You also have some reading to do. By the way, the readings are fair game for the tests. I may ask you for a few things that you should have read, and we may have never discussed in class.

  These would not be minor details, by the way -- they would be major themes within the readings. So it's not an opportunity for a gotcha -- unless you never did the reading.

  You are to do the readings: they're not supposed to be optional! Better to do them early. You can also always ask questions about them, as well.

Before we return to Africa from Belize, do you have any questions about the Mayan way of representing numbers?

• Here's our "Question of the day":
  How did Egyptians do their math?

• Before we go to Egypt, we need to a with my close friend, the Great Fraudini!
  • How does the game work?
    • Fraudini asks you to imagine a whole number between 1 and 63.
    • You indicate to Fraudini whether or not your number appears on the six magic cards.
    • This permits Fraudini to read through the haze of your mind and he produces your number.

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

• At the heart of Egyptian multiplication is this fact (we might call it 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 -- meaning one and only -- binary factorization).

• This should remind you of 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 talk more about that later.)

• Let's show that we can do a multiplication in either order using the technique: let's try 23*42, and 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?!