Estimating first, so a wrong answer is visible

0:00Here is a number, and the only question is whether you may ignore it. Nought point two four two two. One circuit of our planet round the Sun takes three hundred and sixty five point two four two two days. A calendar counts three hundred and sixty five. So every year something is left hanging. Under a quarter of a day, about five and three quarter hours. Small enough that the instinct is to round it away.

0:30Hold on to that instinct. We are about to do the one thing that decides whether it is right. Multiply it by a hundred. Nought point two four two two, a hundred times over, is twenty four point two two. Not hours. Days. Twenty four days adrift in a single century. More than three weeks. Over a thousand years, two hundred and forty two days. Getting on for eight months.

1:00Over ten thousand, two thousand four hundred and twenty two. More than six years out. And notice what did not happen. The number never changed. Only what it got multiplied by. So too small to bother with was never a fact about the number. It is a claim about a scale, and nobody named the scale. Put a marker on the first of January and watch where the planet really is when that date comes round.

1:31After one year it is a quarter of a day behind. Nobody could notice. After a century the calendar runs twenty four days ahead of the sky, and the date has slid nearly a month from its season. Nothing dramatic ever happens. That is the difficulty. It arrives a fifth of a day at a time. So here is a fix, and it is a good one. The leftover is close to a quarter of a day, and four quarters make one.

2:03So every fourth year, add a day. Three hundred and sixty six instead of three hundred and sixty five. Write it as a question with two answers. Is the year divisible by four? One question, two endings. That is the whole rule. Now, and this is the part worth copying, do not admire it. Check it. Check it by counting the same stretch of time two independent ways, and subtracting.

2:34Four years the way the rule counts them. Three ordinary years and a long one. One thousand four hundred and sixty one days. Four years the way the planet counts them. One thousand four hundred and sixty point nine six eight eight. The rule gives more than is needed. Over by nought point nought three one two. Tiny. And by now we know better than to trust the word tiny.

3:02So take a century. Twenty five of its years are divisible by four, so twenty five days go in. Thirty six thousand five hundred and twenty five delivered. Thirty six thousand five hundred and twenty four point two two needed. Over by nought point seven eight. That is the four year error, twenty five times over. The fix for a calendar running slow has made it run fast. Adjust again. The error is small, so the adjustment is small.

3:38Once a century, skip it. A year divisible by a hundred gets no extra day, even though four divides it. That needs a second question, and it goes on top. Divisible by a hundred? If yes, short, and we are done. If no, ask the old question. Two questions, three endings. Count the century again. Twenty four long years instead of twenty five, so one day fewer. And that flips it. Short, by nought point two two of a day.

4:11Plus nought point seven eight became minus nought point two two. The swing is exactly one, because exactly one day came out. One rule is long, the other short. The truth is between them. Before the third attempt, one habit worth stealing. A century holds twenty four long years, and we did not count them one at a time. We wrote it down first. A hundred divided by four, minus a hundred divided by a hundred.

4:45A hundred over four is the years divisible by four. A hundred over a hundred is the one taken back out. Twenty five minus one is twenty four, and seventy six ordinary years are what is left. It gets written before it gets worked out. A tally gives the answer; this gives the reason. Third attempt. We are short, so put something back. Every four hundredth year, give the day back after all.

5:17Three questions now. Divisible by four hundred, then long. If not, divisible by a hundred, then short. If not, divisible by four, then long. And otherwise, short. Three questions, four endings, and every year lands on exactly one of them. Count four hundred years. Four hundred over four, minus four hundred over a hundred, plus four hundred over four hundred. A hundred, minus four, plus one. Ninety seven long years.

5:50Both totals, and subtract. Over by nought point one two of a day. Over that same span the first rule is out by three point one two days, the second by nought point eight eight. So this is the calendar the world runs on. And it is still wrong. Take it to a thousand years and watch. Sort those thousand years by which question catches them first. Two are divisible by four hundred. Eight by a hundred but not four hundred.

6:24Two hundred and forty by four but not a hundred. And seven hundred and fifty by none of them. Two, eight, two hundred and forty, seven hundred and fifty. A thousand. Every year counted once. The long ones are the first group and the third. Two hundred and forty two. Both totals, and subtract. Short, by two tenths of a day. And that leftover was missed by nobody. It is one subtraction, and they almost certainly did it.

6:57Two tenths of a day per thousand years, and they decided it was worth living with. Only somebody who did the arithmetic gets to make that call. Push it further. Ten thousand years. Ten thousand over four, minus ten thousand over a hundred, plus ten thousand over four hundred. Two thousand five hundred, minus a hundred, plus twenty five. Two thousand four hundred and twenty five long years. Both totals, and subtract. Over, by three entire days.

7:33Now stop. Something just happened that is easy to miss. Over a thousand years the calendar was short. Over ten thousand it is long. The sign flipped. The rule repeats every four hundred years. A thousand years is two and a half cycles. It stops halfway through one. Ten thousand is twenty five whole cycles, and each runs over by nought point one two of a day. Twenty five of those is three days. Exactly what we found.

8:06So the thousand year answer was never the real error. It was the rule caught mid stride. One last habit, the same one made small. Suppose you know that seven hundred and fifty six divided by thirty six is twenty one. Then you know a family of answers, and never divide again. Seven hundred and fifty six over three point six. The bottom shrank ten times, so the answer grows ten times. Two hundred and ten.

8:40Seven point five six over nought point three six. Both shrank a hundred times, so the answer does not move at all. Still twenty one. And seventy five point six over nought point nought three six. Top down by ten, bottom down by a thousand. Two thousand one hundred. Every one of those is the same division wearing a different number of zeros. Which is how a wrong answer becomes visible. Twenty one for that last one, and you know the size is wrong.

9:16So finish with six at once. All built on eighteen. Two multiplications, two divisions, and two more with an extra zero. Put them in order, smallest first, without working out a single one. One question per expression, and always the same one. Which side of one is that second number on? Multiplying by something under one shrinks. Multiplying by something above one grows. Dividing does the opposite of both. So multiplying by nought point something lands below eighteen, and dividing by it lands above.

9:56And nought point nought nine is ten times further from one than nought point nine, so it pushes ten times harder. The largest is two hundred, the smallest one point six two. More than a hundredfold apart, and not one sum done. That is the whole habit, and it started with a quarter of a day. Size the answer first, and a wrong one has nowhere to hide.