Day 15, MAT115

• Your Pascal homework. Let's take a look at that....
  • How can you use Pascal's triangle to do any of these problems?

    Pascal's triangle counts "in or outness". Either a friendship is included in a Facebook or it isn't. Either a friend goes in a car, or doesn't (that is, goes in a different car!). Either a bar is chosen, or it isn't.

    1. In how many different ways can you put 5 friends into 2 different vehicles for a trip to the graduation party, where 2 friends go in one car, and 3 friends go in the other?

    2. In how many different ways can you put 5 friends into 2 different vehicles for a trip to the graduation party (assuming either car could take all)?

    3. How many different ways can you choose 3 candy bars from 6 different candy bars?

  • Leibniz's hexagons:
    • Don't reduce your fractions. That makes it harder....
    • Don't use mixed numbers. Mathematicians don't like them. Carpenters and cooks like mixed numbers...
    • Use symmetry, of course: you only have to do the left or right part.
    • Notice that Leibniz's fractions are all unit fractions. What other pattern do you see in them?
      

• You have a homework due today.
• You have a homework due Thursday.

Keep up with your homework!

• The difference today is that we're going to try a new technique for doing the same thing we did last time: the unit fraction table. First let's review multiplication and our first way of doing division again:

• At the heart of Egyptian multiplication is the "Fraudini fact", or the "binary factorization":

  We started last time with some simple examples, and we should do a warmup: what's 197*54

  1	197
  2
  4
  8
  16
  32
  64	Too big!

  On the left we build the "missing part" of the product -- 54 -- and on the right we build up our answer, using the corresponding doubles of 197.

• Now review division: Let's do that product as a division problem:
  1. 197*54 (= 10638) $\iff$ 10638/54 (= 197)

  We can think of division as just using the multiplication table "backwards". So if we write the quotient (which is what we're looking for) as

  dividend/divisor = quotient

  We can think of this as a product instead:

  divisor*quotient = dividend

  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.

  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.

• Now: what happens when the division doesn't work out quite so nicely? We get "the f-word": fractions!

  • 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
    3. 8/9

  • 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?!

    Notice the numerals and fractional notation, as we see in our text on pages 13 and 29.