Last week, in Fundamental Understanding of Mathematics XLIII, we continued our look at the denominator of a fraction, putting fractions on number lines and demonstrating some of the effects of our definition of division
Now, as anyone knows who has followed this series for more than a while, it does not proceed in anything resembling a straight line. In fact, readers may be forgiven if they've got the impression that I have the attention span of a woozle in a room full of kids opening presents. Which is meant to signal yet another zig in the zaggy path we're on. I rationalize chasing after the next shiny object that's caught my attention by claiming it's not the destination, it's the journey, and as long as the ground we're covering is mathematical ground, it's all good.
This week, we are going to prove something.
In the comments of an earlier diary, daulton pointed out that ancient Egyptians used the sum of unit fractions to represent any fraction. Instead of common denominators, bottom numbers, they used a common numerator or top number: 1. So, 7/8, for example, could be written 1/2 + 1/4 + 1/8. But there are other unit fractions which also add up to 7/8.
Now, the trick to getting the "correct" unit fractions to add up to make the larger fraction is this: they are the largest possible unit fractions.
An algorithm to convert any fraction into the sum of a series of largest unit fractions works something like this:
Try to subtract 1/2, then try to subtract 1/3, then try to subtract 1/4 and so on until the difference is itself a unit fraction. Simplify any intermediate results to keep the numbers from getting out of hand.
7/8 - 1/2 = 7/8 - 4/8 = 3/8
3/8 - 1/3 = 9/24 - 8/24 = 1/24
So 7/8 = 1/2 + 1/3 + 1/24
Here's another
9/11 - 1/2 = 18/22 - 11/22 = 7/22
7/22 - 1/3 = 21/66 - 22/66 = can't subtract 1/3
7/22 - 1/4 = 28/88 - 22/88 = 6/88 = 3/44
3/44 - 1/5 = 15/44*5 - 44/44*5 = can't subtract 1/5
3/44 - 1/6 = 18/44*6 - 44/44*6 = can't subtract 1/6
3/44 - 1/7 = 21/44*7 - 44/44*7 = can't subtract 1/7
3/44 - 1/8 = 24/44*8 - 44/44*8 = can't subtract 1/8
notice the pattern? I'm looking for a number, when multiplied by 3, is greater than 44. Let's try 15
3/44 - 1/15 = 45/44*15 - 44/44*15 = 1/660
So 9/11 = 1/2 + 1/4 + 1/15 + 1/660
xgy2 pointed out that, while odd, this method of writing fractions allows one to easily compare fractions to find the larger or smaller number.
Using modern fraction notation, we write: 7/8 and 9/11, and it isn't obvious from inspection which is larger. We'd have to convert to a common denominator, 77/88 and 72/88 to tell that 7/8ths is larger than 9/11ths.
Ancient Egyptians, on the other hand, could look at
1/2 + 1/3 + 1/24
and 1/2 + 1/4 + 1/15 + 1/660
and tell right away that 1/2 + 1/3 + 1/24 was the larger number. How? Well, the first unit fraction in both numbers is the same, but the second unit fraction 1/3 is larger than 1/4, so the fraction with the 1/3 is the larger one.
Now, this is all well and good, and here is the distracting shiny object: How do you prove this? Sure, it happens to work for 7/8ths and 9/11ths, and we could probably work out several other examples, but showing a lot of examples isn't a proof.
To prove this conjecture (a conjecture is something we suppose might be true, but haven't yet proved) we need to come up with a convincing argument that shows that our Egyptian Fraction Comparison method is necessarily true.
Here is one proof, and it hinges on the fact that the algorithm producing Egyptian Fractions proceeds in order, from larger unit fractions to smaller.
Suppose you have two fractions, not equivalent: one is larger than the other.
We start by subtracting 1/2 from both fractions
(I'll assume both fractions are larger than 1/2, and later show it doesn't matter whether they are or not.)
We continue to subtract from the remainder, in sequence, 1/3, 1/4, 1/5, 1/6 and so on, from both fractions' remainders, until we can no longer subtract a unit fraction from one of the two.
(notice that the segments we subtract get smaller and smaller as we go)
Eventually, we end up with this situation: a unit fraction can be subtracted from the larger number (Fraction A), but not from the smaller number (Fraction B). Let's take a closer look at that end situation:
All the unit fractions subtracted, up to this point, are identical for both Fraction A and Fraction B.
(This is true even if this is our first subtracted unit fraction. Nothing subtracted previously is identical to nothing subtracted previously.)
The smaller fraction still has a bit (which may be zero) that needs to be further divided into unit fractions. But whatever unit fractions it might consist of, they must add up to something smaller than 1/a, the unit fraction that exceeded the length of the smaller number.
If you take a look at the unlabeled diagram, you'll see why this argument also works if the original fractions are smaller than 1/2. It doesn't matter what size the first subtracted unit fraction is, only that subsequent unit fractions are smaller than the unit fractions that precede them.
So this proves the conjecture that ancient Egyptians could, by inspection, compare fractions written in their strange (to us) sum-of-unit-fractions-notation. It is considered an "informal" proof, still true for all that, because "formal" proofs are peculiar things which use symbolic logic.
Well, that's my story, and I'm sticking to it. Have fun in the comments.