Why the nearest power of ten is the only handle on a quantity too big to picture

0:00Here is a number: twenty quadrillion. You did not picture anything, and neither did I. Past a few thousand, the digits of a quantity stop carrying any feeling for its size. One part of a large number still does, and this is about that part. So we build a ladder. One rung for each power of ten, and on each rung we put something real. At the foot, ten to the nothing — that is one, and counting still works.

0:31Northern white rhinos: two of them left alive. You can hold two in your head. That is why we start here. Climb one rung, and everything on it is ten times the rung below. Say that out loud each time: the ladder is drawn evenly, and evenness makes it easy to forget. Ten to the one: Hainan gibbons, forty-two of them. Ten to the two: kakapo, two hundred and forty-two.

1:01Ten to the three: Komodo dragons, about three thousand. Ten to the four: maned wolves, about seventeen thousand. Four rungs, from a number you could count on your fingers to one you could not count in a day. The ladder did not get taller. We took four steps. That is the trade the whole idea rests on. Ten to the five: African elephants, four lakh fifteen thousand. Ten to the six: American alligators, five million.

1:35Here a picture stops helping. Nobody sees the difference between four lakh and five million. Everybody sees the difference between two rungs. Ten to the seven has two things on it. Camels, three and a half crore. Horses, five point eight crore. Same rung, and yet the horses are two thirds again as many. So a rung fixes the decade, and the number in front of the ten refines it.

2:04Two things on one rung are not equal. They are comparable, which is more useful. Ten to the eight: water buffalo, twenty crore. Ten to the nine. And here we are. Eight point two billion people, and on the same rung, one point three billion starlings. Now be careful. Eight point two is much nearer to ten than it is to one. So the power of ten we are nearest to is the rung above.

2:35And yet we stand on rung nine. That is not a mistake, it is what the rung means: a rung is a decade, not a nearest value. Everything from one billion up to just under ten billion stands on rung nine. Hold on to that; an argument at the end turns on it. Now the thing the ladder was built for. How many people are there for every African elephant?

3:03Round both to their rungs: eight billion of us, four lakh of them. Nine minus five is four. Four rungs apart means ten to the four times as many — twenty thousand people per elephant. That subtraction of exponents is not a shortcut to the comparison. It is the comparison. And it survives the rounding: on the unrounded figures, to one digit, you get twenty thousand again. The digits we threw away were not carrying the answer.

3:36They were never carrying anything you were using. Try the other operation and watch it fail. Subtract the elephants from the people. The answer sits on rung nine — the rung the people were already on. Report it to one digit and it is the population, unchanged. You need five significant digits before the elephants show up at all. So at this size, subtracting tells you which is bigger and nothing else.

4:08Dividing tells you twenty thousand to one. Every comparison from here up is a ratio, and a ratio read off the ladder is a difference of rungs. That is the habit worth keeping. Keep climbing, past anything you could meet. Ten to the ten: chickens alive right now, thirty-three billion. Ten to the twelve: trees, three trillion. Notice what just happened — we skipped rung eleven. Not because it is missing, but because we had nothing to put on it.

4:43Ten to the fourteen: mosquitoes at a hundred and ten trillion, and Antarctic krill at five hundred trillion. Ten to the fifteen: beetles, one quadrillion. Ten to the sixteen: ants. Twenty quadrillion of them. That is the number from the first sentence, and now it has a place to stand. Ten to the twenty-one: the grains of sand on every beach and desert on Earth. Ants are on sixteen. Sand is on twenty-one.

5:17Five rungs, so the ladder says about a lakh grains for every ant. Do it properly, with the numbers in front of the tens, and you get fifty thousand. Those two answers are not even on the same rung. The rung method was out by a factor of two, landing on the rung next door. That is not a flaw. That is the promise. An order-of-magnitude answer is right within a factor of ten, and never promised more.

5:47For a quantity you could not picture, a factor of ten is an enormous amount of knowing. Two more rungs, near the top of what anyone has counted. The stars in the observable universe: two times ten to the twenty-three. The drops of water on Earth: two times ten to the twenty-five. Two rungs apart, so about a hundred drops of water for every star. And now the trap this ladder sets.

6:18A drop of water is not remotely the size of a star. The ladder measures how many, never how big, and those are different questions. Read the wrong one off it and you will conclude something absurd with complete confidence. One more: about two point four million ants for every person alive. The ladder does not care what it is counting. Build a second one, in seconds. Ten to the nothing, one second: a thrown ball is in the air for a few.

6:52Ten to the one: your blood takes about that long to go round once. Ten to the two — somewhere in the hundreds: sunlight reaching us takes about eight minutes. Ten to the three: a satellite goes round the Earth in an hour and a half. Ten to the four: a meal takes a few hours to leave your stomach. Every one of those was a different unit, and the ladder held them all.

7:19That is what one unit buys you. Here is a real argument the ladder settles. A packet of noodles claims two minutes. Nobody has had cooked noodles in two minutes; call it ten. Two minutes is a hundred and twenty seconds. Ten minutes is six hundred. One is five times the other, and yet both stand on rung two. So the packet could have said: of the order of ten to the two seconds, and been right either way.

7:50That is what an order claims: somewhere in that decade, from a hundred seconds to just under a thousand. It does not claim about a hundred, and reading it that way is the commonest mistake. The packet was not imprecise. It was precise, and wrong, which is worse. Keep climbing. A million seconds sounds enormous. It is eleven and a half days. A thousand million seconds — ten to the nine — is thirty-one point seven years.

8:24So that rung is of the order of a human lifetime. Three rungs below it, a fortnight. That is the whole distance between a fortnight and a life. Now the top. The universe is about thirteen point eight billion years old. In seconds, that is four point four times ten to the seventeen. Eight rungs above a lifetime — the entire history of everything. Count the stars at one a second and you would need six point three times ten to the fifteen years — five rungs past the age of the universe.

9:04One last thing: the ladder doing something arithmetic cannot. Somebody tells you their age is sixty-nine lakh, seventy thousand, seven hundred and ten. They forget the unit. Try seconds: that is nought point two two years. Nobody says that. Try hours: seven hundred and ninety-six years. Nobody is that. Try days: nineteen thousand years. Certainly not. Try minutes: thirteen point three years. That is a person. The longest life ever recorded is a hundred and twenty-two years. Only one reading lands under it.

9:45So the unit was minutes — not deduced, but placed: four candidates on the ladder, three fell off. That is what the ladder is for. A quantity too big to picture can still be placed, and then held against something you have seen. You will not get the digits. You were never using them. You get the rung, and the rung is the part that means something.