I’m teaching a weekend math workshop for school kids (grades 3-6) in Montclair again this spring. This week we started in on comparing fractions. I always like to work with real life problems, and my inspiration for this past Sunday’s class was a remark on teaching by the great Russian mathematician I.M. Gelfand, who once said:
You can explain fractions even to heavy drinkers. If you ask them, ‘Which is larger, 2/3 or 3/5?’ it is likely they will not know. But if you ask, ‘Which is better, two bottles of vodka for three people, or three bottles of vodka for five people?’ they will answer you immediately. They will say two for three, of course.
Well, it wasn’t quite as easy as that. Figuring that my kids might have less intuition for vodka than Gelfand might be counting on, I asked them if they’d rather have two bottles of soda for three people or three bottles for five. Sadly, I got back only random guesses and bored stares. Perhaps kids don’t drink enough soda anymore. So I decided to frame the question in terms of pizza instead.
The advantage of working with pizza is that you can divide it up into slices. Your basic large pizza has 8 slices, so two pizzas means 16 slices. If you have 3 people and two pizzas to divide up between them, then you can give each person 5 slices (3 × 5 = 15), with a single slice left over. Similarly, three pizzas means 24 slices. If you’re dividing three pizzas among five people, four of them get 5 slices (4 × 5 = 20), but then there are only 4 left for the fifth person. In other words: with two pizzas for three people, you have a bit more than five slices a person, whereas with three pizzas for five you have a bit less. So two for three is indeed better, just as Gelfand and the vodka drinkers promised. Coming back to fractions: just by counting pizza slices, we’ve shown that 2/3 is larger than 3/5. Easy-peasy, right? It worked very nicely as an activity in my class.
But there is some very famous and lovely math hiding behind our humble slice-counting. To find it, let’s take our one group of three people with two pizzas, and our other group of five people with three pizzas, and combine them — eight people, five pizzas. Thinking in terms of slices again, five pizzas means 40 slices, which is nice and even: each person gets exactly five. So the group with a little more than five slices a person exactly balanced out the other group with a little less. In terms of fractions, 3/5 < 5/8 < 2/3.
Perhaps the numbers that appear in these fractions — 2, 3, 5, 8 — look familiar to you. What you’re recognizing are some of the first numbers in the Fibonacci sequence, in which you add pairs of adjacent elements to get the next element. So we have 5 = 2 + 3, and 8 = 3 + 5. The next element after that is 5 + 8, which is 13. The next ones after that are 21, 34, 55, 89, and so on.
The pizza eater’s analog of adding the last two numbers is combining the last two groups. If we take eight people with five pizzas and five people with three pizzas and combine them, we get thirteen people with eight pizzas. Because 3/5 < 5/8 from the above, there was more pizza per person in the first group than in the second. In the combined group, the amount of pizza per person has to be somewhere in between. That average amount in the combined group is 8/13 of a pizza, which has to squeeze somewhere between the pizza per person ratios of the two smaller groups. Without any more slice counting, we’ve shown that 3/5 < 8/13 < 5/8.
And you can keep right on going. Combining our new, 13 (people)-with-8 (pizzas) group with the previous 8-with-5 group, we get a group of 21-with-13. In this group, there is more pizza per person than in the 13-with-8 group, but less than in the 8-with-5 group. Since everyone in the 21-with-13 group gets 13/21 of a pizza, 13/21 has to squeeze in between 8/13 and 5/8. In other words: 8/13 < 13/21 < 5/8. Continuing on, we get:
- 8/13 < 21/34 < 13/21,
- 21/34 < 34/55 < 13/21,
- 21/34 < 55/84 < 34/55,
and on and on.
Let’s notice a few things that happen as we iterate and the numbers get bigger:
- Each fraction is the ratio of two successive Fibonacci numbers.
- Up steps and down steps alternate. We went down from 2/3 to 3/5, then up from 3/5 to 5/8, then down from 5/8 to 8/13, then up to 13/21, down to 21/34, up to 34/55, and so on.
- As the numbers get bigger, successive ratios get squeezed into a tighter and tighter range. Each step is smaller in magnitude, and opposite in direction, to the one that preceded it.
In the same way that each Fibonacci number is the sum of the two numbers that came before it, each Fibonacci ratio has to squeeze in between the two ratios that came before it. A calculus fact known as the alternating series test guarantees that numbers that oscillate up and down forever in a tighter and tighter range like that have to be homing on something. A limit, if you will. But what is that limit here?
If we write Fi for the i-th Fibonacci number, the i-th ratio is Fi / Fi+1, and the ratio after that is Fi+1 / Fi+2, which you can rewrite as Fi+1 / (Fi+1 + Fi). Here’s a picture, with Fi = 21, Fi+1 = 34, and Fi+2 = Fi+1 + Fi = 55:
Fi / Fi+1 is the ratio of the side lengths of the small rectangle, and Fi+1 / Fi+2 is the ratio of the side lengths of the large rectangle. These ratios get closer and closer to each other as we iterate Fibonacci numbers and ratios. In the limit, they become equal, and the two rectangles become similar:
If we write b/a for the ratio in the limit, and equate the ratios of the side lengths of the small and large rectangles, we get the equation b/a = a/(a+b). To find a formula for the ratio, we can set a = 1 and solve the resulting quadratic equation for b.
The ancient Greeks thought that the picture above indicated perfect proportionality, and called the ratio (really its reciprocal) the Golden Ratio, or Golden Mean. So as you iterate the Fibonacci numbers, the ratios between them become more and more perfectly proportioned. And it all started with sharing some pizza.
So try not to despair the next time your kid insists on pizza for dinner yet again. Perhaps you can explain fractions to heavy drinkers, but look how much tasty math you can explain to pizza eaters.