Day 07, MAT115

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Announcements:
- You have an assignment due next time, and a new assignment today. Please keep an eye on the assignment page.
Here's our "Question of the day":

How did Egyptians do their multiplication?
Today we're going to
- review the Fraudini trick,
- then find out how it applied to Egyptians multiplication (and division, as well!).
We began with a "Fun and Game" last time: a visit from the Great Fraudini!
- Have you tried it to win fame and fortune, cash and candy?
- How does the game work?
- Can you win at the game? You should be able to do it every time!
- Here's the handout for the cards, if you want to print off another copy.
We discovered that the history of a western understanding of Egyptian Mathematics is fairly recent, and hinged on the deciphering of the Rosetta Stone.
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 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.
Let's show that we can do a multiplication in either order using the technique: let's try 23*42, and 42*23.
How do these examples relate to the Fraudini trick?
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?). Rather than building the other half of a product on the left side of the table, we're building the number we're dividing (32 in this case) on the right side of the table. Then we add up the starred entries on the left hand side of the table. It's the opposite of multiplication.
  32 was pretty easy to "build", because it appeared on the right hand side directly. It's usually a little harder, as we'll see below.
- 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?)
- Let's do one more: 966 divided by 42.
- Now you try this one: divide 117 by 9 like an Egypian.
- 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?!

Website maintained by Andy Long. Comments appreciated.

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

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

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	*