Additive growth versus multiplicative growth

0:00Suppose somebody built a ladder to the Moon. How many rungs would it need? Before any arithmetic — guess. Thousands? Lakhs? Crores? More than that? Hold whichever one you picked; you will want it in a minute. Because the number itself is not the interesting part of this. The interesting part is that the same journey, done a different way, takes forty-six steps. Not forty-six thousand. Forty-six. And the difference between those two answers is not that one method is fast and the other slow.

0:38Start where the last idea left off: what has nobody told us? We know where the Moon is. We do not know how far apart the rungs are. And without that, there is no question here at all — the ladder could have three rungs or three trillion. So it has to be assumed, out loud. Say twenty centimetres between one rung and the next. That is a choice, not a measurement, and you may prefer a different one.

1:07Hold on to that, because we are going to come back and change it deliberately. For now: twenty centimetres a rung. The Moon is three lakh eighty-four thousand four hundred kilometres away. The rungs are twenty centimetres apart. Those two numbers cannot meet until they are in the same unit, and this is where the whole calculation is usually lost. One kilometre is one hundred thousand centimetres. So the distance, in centimetres, is thirty-eight billion, four hundred and forty million.

1:45Now the two numbers are speaking the same language. How many twenty-centimetre gaps fit into that? One division. And it comes out exactly whole, which is a small mercy. One hundred and ninety-two crore, twenty lakh rungs. Check that against the guess you were holding. That is how it is grouped here, in crores and lakhs. Read in the international grouping, the very same number is one billion, nine hundred and twenty-two million.

2:19Those are not two numbers. That is one number, grouped two ways. The digits never moved; only the commas did. So: nearly two billion rungs, resting entirely on a rung gap somebody assumed. Now the question that matters. What exactly was being held constant as that ladder went up? Every rung added the same twenty centimetres. Not twenty per cent more. Twenty centimetres. The second rung is twenty above the first, the millionth is twenty above the one below it, and the last one is twenty above that.

2:59The increase never changes, all the way to the Moon. That is what makes this kind of growth what it is, and it has a name: linear growth. Some people call it additive, which is the more useful word, because it tells you what the process is doing. It adds. And a total built by adding the same thing over and over is a very long sum. Nearly two billion terms long, in this case.

3:29Now the other way of getting there. Take a sheet of paper — about a thousandth of a centimetre thick. Fold it in half. Fold that in half. Keep going. Every fold doubles the thickness. The starting thickness is twenty thousand times smaller than one rung gap, so this begins hopelessly far behind. Fold it forty-five times and you are still short of the Moon. Fold it forty-six and you are past it.

3:59And I do mean past — forty-six folds is about seven hundred thousand kilometres, which is roughly twice the distance. It does not arrive at the Moon. It overshoots. So what was being held constant this time? Not the increase. The first fold gained a thousandth of a centimetre; the forty-sixth gained hundreds of thousands of kilometres. What stayed the same was the multiplier. Times two. Every single fold, times two.

4:33That is exponential growth, and it is also called multiplicative, which again is the more useful word. It multiplies. And a total built by multiplying the same thing over and over is a product. Forty-six factors long. Which brings the two of them next to each other. Here they are, side by side. On the left: twenty plus twenty plus twenty, and so on. How many terms? One hundred and ninety-two crore, twenty lakh.

5:08On the right: a thousandth of a centimetre, times two, times two, times two. How many factors? Forty-six. Both chains reach the Moon. One of them you could not write out in a lifetime. The other fits on a line. That is the entire topic, in one picture. The question is only ever which chain a situation builds. Put a number on the difference. Nearly two billion rungs against forty-six folds.

5:40Divide one by the other and you get about forty million. Forty million rungs for every fold. And notice what that number is not. It is not a fact about the Moon — both processes were aimed at the same Moon. It is not a fact about how big either answer is. It is a fact about the difference between adding a fixed amount and multiplying by a fixed one.

6:07The comparison is the lesson. Neither number on its own is. Which is where the word people use for this goes wrong. Exponential does not mean fast. It means multiplied by a fixed number each step — and that is compatible with being extremely slow. A quantity that doubles every hundred years is exponential, and you would never notice it. A quantity gaining a million every second is additive, and it is quick.

6:39Speed and kind are two different questions. In fact, look again at where our sheet started. A thousandth of a centimetre, against a rung of twenty. The multiplied process began twenty thousand times behind the added one. So when did it get in front? Later than you would think. Draw both on one axis and watch. After ten steps the ladder is two metres up and the paper is barely a centimetre.

7:10After fifteen the ladder is three metres and the paper is thirty-odd centimetres. At step eighteen the ladder is three hundred and sixty centimetres and the paper is two hundred and sixty-two. The ladder is still winning. At step nineteen the ladder is three hundred and eighty, and the paper is five hundred and twenty-four. That is the crossing. Eighteen steps of losing, and then it never loses again. By step thirty the ladder is six metres and the paper is ten point seven kilometres — two thousand times ahead.

7:50One loose end. That twenty centimetres was assumed. So let us change it and see what breaks. Say thirty centimetres between rungs instead. The count drops to about one point two eight billion. Exactly two thirds of what it was — which is what you would expect, since the gap went up by half. It also stops coming out whole, which is the assumption showing through. But now do the comparison again: about thirty million rungs per fold.

8:23The assumption moved the answer by a third. It moved the comparison from forty million to thirty million. Which is the point — the conclusion here is far too big to be knocked over by an assumption you could argue about. So here is the test, and it is one question long. As the process takes a step, what stays the same — the amount it gains, or the number it is multiplied by?

8:53Saving the same amount every week: the gain is constant. Additive. A candle burning down the same length each hour: also additive — the constant is negative, but it is still a constant gain. One more book on the shelf each day: additive. Every person telling two more, who each tell two more: the multiplier is constant. Multiplicative. A cell splitting in two: multiplicative. A sum growing by a tenth of itself each year: multiplicative — and slow, which is exactly why the word fast was never the right one.

9:34That last one is worth finishing, because it is the ladder and the paper again in ordinary clothes. Set a thousand growing by a tenth of itself each year against a thousand gaining five hundred a year. For twenty-eight years the fixed five hundred is ahead — fifteen thousand against about fourteen thousand four hundred. In the twenty-ninth year it is passed, and it is never in front again. Ask which quantity is being held fixed, and you already know how the story ends.