Your age as a fraction
Little kids often report their ages as "two and a half" or sometimes even "three and three quarters". These evaluations are usually based on whole months: if you were born on April 2, 1969, then on October 2, 1971 you start reporting your age as "two and a half", and, if you choose to report your age as "three and three quarters", you conventionally may begin on January 2, 1973.

However, these reports are not quite accurate. On January 2, 1973, exactly 3 years and 9 months from your birthday, you would be 1,371 days old, or 3 years plus 275 days. 275/365 = 0.7534. On January 1, you were only 3 + 274/365 days old, which is 3.7507 years, and so January 1 is the day on which you should have been allowed to start reporting your age as "three and three quarters". This slippage between days and months occurs in the other direction as well, so there may be kids wandering around declaring themselves as "three and a half" a full day before they actually reach that age.

Clearly this is one of the major problems facing our society, so I wanted to make up a table showing, for each number of days d from 1 to 365, what is the simplest fraction a/b such that when it is d days after your birthday, you are (some whole number and) a/b years. That is, I wanted a/b such that d/365 ≤ a/b < (d+1)/365.

Then, by consulting the table each day, anyone could find out what new fraction they might have qualified for, and, if they preferred the new fraction to the old, they might start reporting their age with that fraction.

There is a well-developed branch of mathematics that deals with this problem. To find simple fractions that approximate any given rational number, or lie in any range, you first expand the bounds of the range in continued fraction form. For example, suppose it has been 208 days since your birthday. Then today your age will range from (y years and) 208/365 days up to (y years and) 209/365 days.

Then we expand 208/365 and 209/365 as continued fractions:

208/365 = [0; 1, 1, 3, 12, 1, 3]
209/365 = [0; 1, 1, 2, 1, 16, 1, 2]

Then you need to find a continued fraction that lies numerically in between these two but is as short as possible. (Shortness of continued fractions corresponds directly to simplicity of the rational numbers they represent.) To do this, take the common initial segment, which is [0; 1, 1], and then apply an appropriate rule for the next place, which depends on whether the numbers in the next place differ by 1 or by more than 1, whether the first difference occurs in an even position or an odd one, mumble mumble mumble; in this case the rules say we should append 3. The result is [0; 1, 1, 3], or, in conventional notation:

Since I already had a library for calculating with continued fractions, I started extending it with functions to handle this problem, to apply all the fussy little rules for truncating the continued fraction in the right place, and so on.

Then I came to my senses, and realized there was a better way, at least for the cases I wanted to calculate. Given d, we want to find the simplest fraction a/b such that d/365 ≤ a/b < (d+1)/365. Equivalently, we want the smallest integer b such that there is some integer a with db/365 ≤ a < (d+1)b/365. But b must be in the range (2 .. 365), so we can easily calculate this just by trying every possible value of b, from 2 on up:

        use POSIX 'ceil', 'floor';

        sub approx_frac {
          my ($n, $d) = @_;
          for my $b (1 .. $d) {
            my ($lb, $ub) = ($n*$b/$d, ($n+1)*$b/$d);
            if (ceil($lb) < ceil($ub) && ceil($ub) > $ub) {
              return (int($ub), $b);
            }
          }
          return ($n, $d);
        }

        my $N = shift || 365;

        for my $i (1..($N-1)) {
          my ($a, $b) = approx_frac($i, $N);
          print "$i/$N: $a/$b\n";
        }

1/10 ≤	1/6	< 2/10	(0.1667)
2/10 ≤	1/4	< 3/10	(0.2500)
3/10 ≤	1/3	< 4/10	(0.3333)
4/10 ≤	2/5	< 5/10	(0.4000)
5/10 ≤	1/2	< 6/10	(0.5000)
6/10 ≤	2/3	< 7/10	(0.6667)
7/10 ≤	3/4	< 8/10	(0.7500)
8/10 ≤	4/5	< 9/10	(0.8000)
9/10 ≤	9/10	< 10/10	(0.9000)

The simplest fractions that are missing from this table are 1/5, which is in the [2/10, 3/10) range and is beaten out by 1/4, and 3/5, which is in the [6/10, 7/10) range and is beaten out by 2/3.

This works fine, and it is a heck of a lot simpler than all the continued fraction stuff. The more so because the continued fraction library is written in C.

For the application at hand, an alternative algorithm is to go through all fractions, starting with the simplest, placing each one into the appropriate d/365 slot, unless that slot is already filled by a simpler fraction:

        my $N = shift || 365;
        my $unfilled = $N;

        DEN:
        for my $d (2 .. $N) {
          for my $n (1 .. $d-1) {
            my $a = int($n * $N / $d);
            unless (defined $simple[$a]) {
              $simple[$a] = [$n, $d];
              last DEN if --$unfilled == 0;
            }
          }
        }

        for (1 .. $N-1) {
          print "$_/$N: $simple[$_][0]/$simple[$_][1]\n";
        }

(It might also be that you could optimize the algorithms to go faster: you can skip the body of the loop in the slot-filling algorithm whenever $n and $d have a common factor, which means you are executing the body only n log n times. But testing for common factors takes time too...)

I was going to paste in a bunch of tabulations, but once again I remembered that it makes more sense to just let you run the program for yourself. Here is a form that will generate the table for all the fractions 1/N .. (N-1)/N; use N=365 to generate a table of year fractions for common years, and N=366 to generate the table for leap years:

Here's a program that will take your birthday and calculate your age in fractional years. Put in your birthday in ISO standard format: 2 April, 1969 is 19690402.

[ Addendum 20070429: There is a followup to this article. ]

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Footballs?
The diary of Samuel Pepys for Tuesday, 3 January 1664/5 says:

Up, and by coach to Sir Ph. Warwicke's, the streete being full of footballs, it being a great frost, and found him and Mr. Coventry walking in St. James's Parke.

My best guess is that it's big chunks of frozen mud that you have to kick out of the way. Do any gentle readers know for sure?

The Diary of Samuel Pepys has a syndication feed you can subscribe to. You get a diary entry every day or so, with all the names and places linked to a glossary. It's fun reading.

[ Addendum 20080105: The answer. ]

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Iris is not a vegetarian
I just had a long vacation and got to spend a lot of time with Iris, which is why she's popping up so much all of a sudden. I seem to have gotten into Mimi Smartypants mode. I will return to the regularly-scheduled discussions of programming and mathematics shortly. But since I seem to have written two articles in a row ([1] [2]) on the subject of telling kids the difficult truth, I thought I'd try to finish up this one, which has been mostly done for months now. Also there was a recent episode that got me thinking about it again.

I went to visit Iris at school last week, and stayed for lunch. I was seated with Iris and three other little girls. As the food was served, one of the girls, Riley, made some joke about how the food cart contained guinea pigs instead. This sort of joke is very funny to preschoolers.

My sense of humor is very close to a preschooler's, and I would have thought that this was funny if she had said that the food cart contained clocks, or nose hairs, or a speech in defense of the Corn Laws, or the Trans-Siberian Railroad, or fish-shaped solid waste. But she said guinea pigs, and instead of laughing, I mused aloud that I had never eaten a guinea pig.

Riley informed me that "You can't eat guinea pigs! They're animals, not food."

"Sure you can," I said. "Meat is made from animals."

Riley got this big grin on her face, the one that preschoolers get when they know that the adults are teasing them, and said "Nawww!"

"Yes," I said. "Meat comes from animals."

Riley shook her head. She knew I was joking. A general discussion ensued, with Iris taking my side, and another girl, Flora, taking Riley's. In the end, I did not convince them.

"Well," I said, mostly to myself, at the end, "you girls are in for a rude awakening someday."

Now, I know that not everyone is as direct as I am. And I know that not all non-vegetarians are as concerned as I am about the ethics of eating meat. But wow. I would have thought that someone would have explained to these girls where meat came from, just as a point of interest if nothing else. Or maybe they would have made the connection between chicken-the-food and chicken-the-farm-animal. I mean, they are constantly getting all these stories set on farms. Since three-year-olds ask about a billion questions a day. they must ask around a thousand questions a day about the farms, so how is it that the subject never came up?

Iris was accidentally exposed to a movie version of Charlotte's Web on an airplane, and the plot of Charlotte's Web is that Charlotte is trying to save Wilbur from being turned into smoked ham. Left to myself I wouldn't have exposed Iris to Charlotte's Web so soon—it is too long for her, for one thing—but my point here is that the world is full of reminders of the true nature of meat, and they can be hard to avoid. So I was very surprised when it turned out that these two age-mates of Iris's were so completely unaware of it.

Anyway, Iris has known from a very early age where meat comes from. Early in her meat-eating career, probably before she was two years old, I specifically explained it to her. I wanted to make sure that she understood that meat comes from animals. Because there are serious ethical issues involved when one eats animals, and I think they must be considered. We may choose to kill and destroy thinking beings to make food, but we should at least be aware that that is what is happening. I'm not sure I think it is evil, but I want to at least be aware of the possibility.

I have never been a vegetarian, but I want to try to face the ethical results of that choice head on, and not pretend that they are not there. I did not want Iris growing up to identify meat with sterile packages in the supermarket. Meat was once alive, moving around with its own agenda, and I think it is important to understand this.

So I made an effort to bring up the subject at home, and then one day when Iris was around twenty months old we went to a Chinese restaurant that has live fish in tanks at the front of the restaurant, and you can ask them to take one of these fish into the kitchen to be cooked for your dinner. Iris has loved to eat fish since she was a tiny baby.

We ordered a striped bass, and then I took Iris to look at the fish in the tanks. I explained to her that these fish swimming in the tanks were for people to eat, and that when we ordered our fish for dinner, a waiter came out and caught one of the fish in a net, took it back to the kitchen, and they killed it and were cooking it for us.

As I said, I had made the point before, but never so directly. We had never before seen the live animals that were turned into food for us. I really did not know how Iris would respond to this. Some people have a very strong negative response when they first learn that meat comes from animals, so negative that they never eat meat again. But I thought Iris should know the truth and make her own decision about how to respond.

Iris's response was to point at one of the striped bass and say "I want to eat that one."

Then she took me to each tank in turn, and told me me which kind of fish she wanted to eat and which ones she did not want to eat. (She favored the fish-looking fish, and rejected the crabs, shrimp, and eels.)

Then when the fish arrived on our table Iris asked if it had been swimming in the tank, and I said it had. "Yum yum," said Iris, and dug in.

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