What a misplaced decimal point costs

0:00Write the digits one, eight, eight on a board, and ask what they are worth. You cannot answer, and not because the question is unfair. A string of digits is not yet a quantity. It becomes one when two further things are fixed, and not before. The first is which unit it counts. A hundred and eighty eight of what? The second is where the units place sits. Is that a hundred and eighty eight, or one point eight eight?

0:31Call those two the anchors. The digits travel, and the anchors are what hold them to a size. What follows are three occasions when the digits arrived perfectly and one of the anchors did not. Nobody miscalculated in any of them. Every number was handled correctly, and every answer was wrong. The first is a city housing office, in Amsterdam, in two thousand and thirteen. Once a year it pays out housing support to tens of thousands of households.

1:06The amounts are small and there are a great many of them, so the whole run is automatic. One year the run went out with the amounts taken in cents rather than in euros. The office intended to pay one point eight million euros. What actually went out was a hundred and eighty eight million. The arithmetic was flawless. Every household was paid the number it was owed. It was simply the wrong kind of unit, and the money left the building before anybody looked at it.

1:40Not one person had to be wrong about a single digit for that to happen. So look at what treating a count of cents as a count of euros actually does. One cent is a hundredth of a euro. That is the whole relation. Take one euro and eighty cents. Written in euros it is one point eight zero. Written in cents, the very same money is one hundred and eighty.

2:08Same amount. The digits shifted two places and the unit label changed with them. That pairing is the point. The mark moves and the unit moves together, or the quantity is not the same quantity. Now hand a machine the number one hundred and eighty, and tell it those are euros. It will pay out a hundred and eighty euros, perfectly correctly, and it will be a hundred times too much.

2:37A hundred times, because a hundred cents make a euro. Nothing to do with the accuracy of anything. The second happened in the air over Canada, in nineteen eighty three. An airliner was refuelled before a long flight, and the fuel load had been worked out carefully. The aircraft needed twenty two thousand three hundred kilograms of fuel on board. It was loaded with twenty two thousand three hundred pounds.

3:09Notice what is not in that sentence. There is no decimal point in it anywhere. The number is identical. Only the unit attached to it was the wrong one. The aircraft ran out of fuel in flight, and came down on a disused airfield. Nobody was hurt, and that was the only piece of luck in the whole story. And once again, not one person had made an arithmetic mistake.

3:38How short was it? There is a rather lovely way to see this. One pound is about zero point four five three kilograms. So twenty two thousand three hundred pounds is about ten thousand one hundred kilograms. Against a requirement of twenty two thousand three hundred kilograms. Now divide what it had by what it needed, and watch what falls out. Zero point four five three. The conversion factor itself.

4:09That is not a coincidence, and it would happen every time this particular mistake is made. Because the number loaded and the number needed were the same number, the fraction left over is simply the ratio of the two units. Roughly half a tank. And the reason it was roughly half is written into the words pound and kilogram. The third is smaller, and quieter, and it has happened more than once.

4:39A dose is written as nought point nought five milligrams. It is read as nought point five. One character. The nought sitting between the point and the five. That is ten times the amount that was prescribed. Exactly ten. Not roughly ten. The digits are the same digits in the same order, and the five has moved one column. It was sitting in the hundredths, and now it sits in the tenths.

5:08This is the only one of the three where the point is the thing that moved. Which is worth stopping on, because it is not how these three sort themselves. You would expect all of them to be stories about decimal points. They are not. The dose lost the place. That one is a genuine point error. The money lost the unit. Cents against euros. The fuel lost the unit as well. Pounds against kilograms.

5:40Two of the three have no decimal point in them anywhere at all. And yet all three are one failure wearing different clothes. An anchor was dropped, and the digits sailed on without it. Which anchor went missing is what decides how badly the thing goes. Here is what makes this kind of mistake different in kind, and not merely in degree. How wrong the answer is does not depend on the number at all.

6:10It depends only on which anchor went missing. Move a point one place and the answer is out by ten, whatever the number happened to be. Two places, a hundred. Three places, a thousand. Which means there is no slip you can make with a decimal point that is worth less than a factor of ten. You cannot be five per cent wrong about where the point goes. The smallest mistake available to you is a whole rung of the ladder.

6:40And a unit error is worth whatever ratio the two units stand in. A hundred, for cents against euros. A little over two, for pounds against kilograms. It is worth saying plainly what this is not. It is not rounding. Round twenty two point three four nine to one place and you get twenty two point three. You have moved it by about a fifth of one per cent. Rounding shortens a number, and it can never cost more than half a unit of the last place you kept.

7:16These errors do not shorten anything. They pick the whole quantity up and put it somewhere else. So keep the two words apart, because they describe opposite things. One is a small, honest inaccuracy that you accept on purpose and know the size of. The other is a multiplication you did not know you had carried out. So, two checks, and both are quick enough to become habits. The first one. What unit is this in, and what unit do I want the answer in?

7:51Run it on the fuel. Twenty two thousand three hundred of what? Pounds. And the answer is wanted in kilograms. Convert, and you get ten thousand one hundred, which is not enough, and the aircraft never leaves the ground. The second one. Where is the units place? Run it on the dose. Nought point five. Put the point back one column and you have nought point nought five. Notice that neither of those is an estimate.

8:22An estimate catches a hundredfold, and it catches a tenfold, but a factor of two slides straight past it, and that factor of two nearly destroyed an aircraft. There is a third defence, and it costs nothing at all. Say the number out loud, properly. After the point, a decimal is read one digit at a time. Nought point nought five. Nought point five. Said that way, the two of them do not sound remotely alike.

8:55The extra word is the entire error, spoken. Whereas point oh five and point five, said quickly, are one careless breath apart. A string of digits is not a quantity until you fix which unit it counts and where the units place sits. Everything else in all three of those stories was done correctly, and none of it helped at all.