Day 13, MAT115

On the first page of the chapter, Strogatz says "...this is the place where many students hit the mathematical wall." He attributes this in part to one of the things I like most about mathematics: there's more than one way to represent things. So, for example, he notes that 1/2 is the same as 50/100.

But, to use my terminology, he also suggests that it's a failure of students to recognize that they can just "do it again":

"What makes him insist that you can't have a quarter of a quarter? Maybe he thinks that you can take a quarter only out of a whole or from something made of four equal parts. But what he fails to realize is that everything is made of four equal parts." (p. 31: see the figure on that page)

Last time we began by considering one of the simplest examples imaginable:

divide 32 by 8.

We can actually do it by Egyptian multiplication, since 8 divides into 32 evenly:

So the answer is 4 (how do we get 4?)

In this problem you see both 8 and 32, plus the answer (4).

So the answer is 5 (how do we get 5?)

In this problem you see neither 40 nor the answer (5) -- except that they are sums of other things that you see in the problem (from those lines indicated by the asterix). And that's how we're going to do these problems in general.

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:

So the answer is 4+1/4+1/8

(Where have we seen those fractions before? Look into the Eye of Horus!)

Again, in this problem you see neither 35 nor the answer (4+1/4+1/8) -- except that they are sums of other things that you see in the problem. And notice that the answer is literally a sum of other things!

Now here's one where the Fraudini trick allows us to do this one in another way, more simply: we start by breaking up the numerator -- 35 -- using the binary decomposition. 35=32+2+1.... Separate out the fractions, and you get the same answer:

But they didn't restrict themselves to "halving", as our next example shows. Divide 6 by 7:

So the answer is 1/2+1/4+1/14+1/28

   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.)

(Think about what the answer means, in terms of bread.)

1. How would you divide 5 by 7?
2. How can we use the unit fraction table to get the same answer?