Day 08, MAT115

• You have an assignment due today.
• My understanding from IT is that they believe that only IOS devices (iphones, ipads, etc.) and some android devices (e.g. tablets or phones) may not operate properly with the website (due to problems with their operating systems not supporting some web features).

  If so, this is the recommendation from IT: use only a laptop or desktop computer to access the web content. They suspect that laptops and desktops would view it fine. Is that true?

  They ask that anyone having trouble go to UC 252 (their "tech bar"), and show how your device is behaving (or misbehaving).

• No new assignment today. Please keep an eye on the assignment page. You do have an assignment due Tuesday, however.

• review the Egyptian multiplication and division (when the answer is a whole number).
• Then we'll find out how Egyptians did more complicated divisions with "unit fractions".

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

Examples: Try these:

1. 43*22 (= 946)
2. 21*79 (= 1659)

We can think of this as a product instead:

For the product we'd take one of the parts of the product (the divisor, say), and double it on the right. Making up the quotient with numbers on the left, we'd then find the dividend by adding up the corresponding numbers on the right.

Well in the division problem, we know the dividend, so we reverse the process: we find numbers on the right that sum to the dividend, and then add up the corresponding numbers on the left to give the quotient, which is what we're after.

Examples: Try these (we'll just reverse the problems above, to show how we're using the table backwards).

1. 946/43 (= 22)
2. 1659/21 (= 79)

• 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

• 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

  Notice that the Egyptians didn't use decimals -- you shouldn't either!

  Why did Egyptians do things this way? (an example division problem, 3/5)

     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.)
  

• There is another way to get these answers, using the Fraudini trick and the unit fractions table. So let's try those same examples but using the unit fraction table rather than our doubling tables.
  1. 35/8
  2. 6/7

• 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?!