Nothing, then everything
Aarav got a message on a Tuesday. It asked him to pass it on. He sent it to five friends and forgot about it.
On Thursday his cousin Tanvi sent him the same message. Tanvi lives four hundred kilometres away.
Nobody had planned that. Nobody was in charge of it. It had simply kept going while he was not looking.
On Wednesday it still felt like nothing. By Friday it was everywhere.
The idea
Some things grow in a way that keeps surprising people. They stay so small for so long that you stop watching them. Then they are past you.
The people who study this have one sentence for it.
The idea
Repeated multiplication outruns intuition.
That sentence has two hard words in it. Take them one at a time.
Multiplication repeated is a thing done to itself, over and over. The plainest kind is doubling: at every step, as much again as there already is.
Adding is the other kind. At every step the same amount goes in, whatever is sitting there already.
Pocket money is an adding. Twenty rupees a week is twenty rupees a week, in the first week and in the fortieth.
The message was a doubling: everyone who had it sent it on.
Your intuition is the answer your head gives before you have worked anything out.
It is built out of what you have met, and almost everything you meet adds. A queue gets longer by one person. A wall gets higher by one brick.
So when your head is shown something growing, it draws a straight line and follows it. That is the part of the sentence that says outruns. A doubling does not beat you by being clever. It beats you because your head went straight on and it did not.
Here is the part that catches everyone. At the start, a doubling looks like nothing at all. Beside something that is adding, it looks like less than nothing.
So the beginning of a doubling tells you nothing about its end. Monday and Tuesday will not tell you what Friday is.
What people say instead
Aarav told Ba about it at dinner. She worked it out in her head, while serving rice.
“Five on Tuesday, five more on Wednesday,” she said. “At this rate it will reach a hundred by Sunday, maybe.”
Ba was not being careless. She did what every head does. She took the rise she could see and carried it straight on.
By Sunday it was not a hundred. Aarav does not know what it was, and nor does anyone else.
The same climb, twice more
In March one desk in Aarav’s class was empty on a Monday. Nobody mentioned it.
On Tuesday there were two. On Wednesday four, and the teacher moved the front row back.
By Friday half the class was gone and the school sent everybody home.
Nothing new happened on Friday. Friday was Monday’s thing, done four more times.
That week Aarav tried something with one sheet of paper. He folded it in half. Then in half again.
By the fifth fold it was a hard little brick. At the seventh it would not bend at all.
Nothing was added to that sheet. No paper came from anywhere. It was only laid onto itself, again and again, which is what doubling is.
Spot it
Two things that grow. Ask the same question of each.
1
Money you put in a jar, the same amount every week. What happens between one week and the next?
2
A rumour in a hostel, where everyone who hears it tells two more. What happens between one week and the next?
Aarav does this with a pencil. Two things on facing halves of a page, one arrow between them, and a question written on the arrow.
In the jar, the same amount goes in every week. In the hostel, as much again as has already heard it. Three weeks of looking will not show you that difference.
So this is the question worth asking. Not how big is it now, but what happens between one step and the next.
This week, at home
Take one sheet of paper and fold it in half as many times as you can. Count the folds, not the layers. Write the number down, and then say out loud what you think would happen if you could keep going.
The same climb, three costumes
The message, the empty desks and the folded sheet are one climb in three costumes. Each one adds as much again as it already has, at every step.
Each one looks like nothing for a long while, and then it is everything at once.
None of the three was hiding. All three were in plain sight on the first day, and the first day is the day that tells you least.
Aarav’s grandfather kept an old book by a man called Aryabhata, who counted things for a living about fifteen hundred years ago. Count what you can count, he liked to say, and put a bound on the rest. Aarav could count seven folds. He could not count what came after seven, and he decided he was allowed to say so. He wrote down “thicker than the table is wide” and left it at that.
The name
People who study this call the two shapes linear and exponential. Linear comes from the word line: an adding growth, drawn out, is a straight line.
Exponential is the other one, where every step is the whole of it again. It is the one that spends a week on the floor and then leaves the page.
Nothing in the shape is hidden. It only asks to be read at the second step instead of at the end.
Where you will see this next
Next chapter: what reaches you was chosen before it reached you. Somebody or something let it through, and the ones that did not get through left no trace at all.