Day 12, MAT115

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

(We'll talk more about that later.)

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?!