How The Exponential Function Flummoxes Us


First, a fable.

A long time ago, pleased with one of his courtiers, a king offered him any present he would ask for. The courtier said, quite modestly, that he was not a material man. But if the king was to insist, then he should order his servants to get a chess board.

‘A chess board?’ asked the king. ‘Whatever for?’

‘My lord,’ said the courtier, ‘please arrange for your servants to place one grain of rice on the first square of the chess board, two on the next, four on the one next to it, and so on, each time doubling the number of grains placed on the previous square. At the end of the exercise, if you could arrange for the total rice on the chess board to be transported to my house, I would be most grateful.’

The king looked doubtful. ‘Is that all you want? Are you certain I could not give you a better present?’

The courtier smiled and shook his head.

Only after the king began the process that the courtier had laid out did he realize his folly. The numbers started innocuously enough: 2, 4, 6, 8, 16, 32, 64… but soon they began to swell. By the time they reached the twentieth square, they had to place 1048576 grains on it. At the twenty-fourth square, the royal granary had to be emptied. For a sense of perspective, the total number of grains that the king would have had to give the courtier if he had followed through the task to completion would exceed the number of electrons in the known universe.

The Problem

Human beings are linear thinkers. When we look into the future or at the past, we tend to project from our present situations backwards, in a linear fashion.

Is this always a problem? No. Many of life’s situations are linear, so it works well. If you need two handfuls of grain to feed one adult, you will most likely need four handfuls to feed two, and six to feed three. If one acre of land yields two tonnes of rice, two acres of the same land will probably yield four. If one cow gave one litre of milk per day, two cows will probably give two.

Unfortunately, the real world doesn’t work on a linear path. There are many social constructs that we’ve set up that refuse to be restricted by a straight line. The most common of these is money.

Here are two quotes by Albert Bartlett, an American Professor of Physics:

The greatest shortcoming of the human race is its inability to understand the exponential function.

And on growth:

We must realize that growth is but an adolescent phase of life which stops when physical maturity is reached. If growth continues in the period of maturity it is called obesity or cancer.

Some examples from the world of money

  • It’s common to hear economists and politicians say that a country has been ‘steadily’ growing at a rate of x%. That gives us, the listeners, the impression that the growth rate is constant. In our minds, we picture a straight line. But in reality, a ‘steady’ growth of x% is not linear. It’s an exponential function. Like the grains on the chess board in our story, it requires exponentially more resources to maintain the same percentage level of growth.
  • Our lives run n the principle that more is good. If one car makes us twice as happy, we reason that buying a second car will make us four times as happy, and so on. If one house gives us a certain amount of pleasure, our second house should bring us twice that amount, and our third three times that amount and so on. But human happiness is not a linear function. As Maslow said, more is good only until the point at which our basic needs are met. From then on, more brings less and less happiness.
  • We underestimate the power of compound interest. We think small savings will not add up to much in the long run, and we think that small loans don’t matter. We think that 100 rupees saved over 12 months equals 1200 rupees, whereas in reality it equals 1257 rupees (at 8.75%). Even our debts blindside us for this very reason, because our mental calculations are all linear.

A few things we could do to conquer this bias

1. Understand that there is no such thing as continuous growth. The Earth’s natural resources are finite. Infinite growth – which our modern economics demand – cannot come from finite resources.

2. Consciously remind ourselves that more is not always better. After we’ve reached a certain stage in our economic lives, more is very often worse.

3. Take note of what Albert Einstein said: Compound interest is the eighth wonder of the world. He who understands it, earns it. He who doesn’t, pays it.

Over to you, now. How else do you think the exponential function trips us up? Do you have any more examples to add to the ones I posit above? What steps do you take to cover for this bias?

Image Courtesy: Independent


  1. The most alarming effect of exponential growth is India’s population. We started out with 33 crores in 1947 and have already crossed 124 crores in 2014. That’s an increase of 91 crores, or 175% in just 67 years. And in the next 67 years, the growth is not going to be only 175% but exponentially higher. Not a single citizen (politicians in a minority here) is aware that we are speeding towards anarchy. How will we feed the huge population 50 years hence? Where will the youth get jobs? Where will so many people live? Only if Mars is inhabited in large numbers can people on earth breathe a sigh of relief.

    Liked by 1 person

    • Hi Neelesh,

      Yes, a definitely alarming trend. Don’t you think birth-control measures are helping, though? And don’t you think increasing education levels will control population rise in the future? I tend to be an optimist in this regard – maybe unreasonably so.


      • Hi Sharath,
        I’ve thought and read about this a lot. More than the reading, I’ve observed people around me.

        For one, ALL women irrespective of the age group, have only one question to ask newly weds :”When are you giving us good news?” Meaning – when is the baby coming along?

        This, cutting across all religious, caste, state, region, language, income groups, is being echoed throughout the length and breadth of India. Married couples are under great pressure to perform (a performance that India will be better off without).

        Consider this: current population 124 crores
        Assuming 10% of the population is under 30 years and of marriageable age, 12.4 crore marriages waiting to happen within the next 5 years. If 50% of the couples give birth to a baby in that period, the next 5 years will see an addition of 6.2 crore to the population.

        Assuming again that this trend continues for 20 years, 4 x 6.2 = 24.8 crore added to the population by 2034.

        Here I am not calculating the exponential growth rate. So the above figure is far, far modest compared to actuals.

        I know that there will be deaths occurring in this period, but I just want to emphasise that the birth rate is still alarming and is showing no signs of abating.

        124 crores is already too much. India needs a population of not more than 75 crores to be on a comfort level as far as standard of living, proper educational facilities, housing, food supply, water supply and drainage, power distribution etc is concerned.

        How are we going to achieve that?


      • Hi Neelesh,

        I take your argument. Now the other side.

        We’ve always ‘underestimated’ Earth’s ability to hold the human population. For instance, in the 1920’s, when world population was around 2 billion, people were proclaiming 3 billion as the ‘breaking point’. But science kept catching up, and now, at 7 billion, the breaking point has not yet come.

        You say 124 crores is already too much. By what standard, though? Who knows how much is enough and how much is too much?

        Differences in standard of living, educational facilities, housing, food supply etc – are not due to high population, but because of inequity in DISTRIBUTION. We have enough resources today to house, clothe and feed every human being on the planet for the next few decades. That we’re not able to is a socio-economic problem.

        My stand is that even if India’s population would go down to 75 crores today, we will still have about the same percentage of people living in poverty, without access to basic facilities, while the rich get that much richer. Poverty is not a problem caused by scarcity of resources. It is a problem related to their distribution.

        Who knows? Maybe in twenty years’ time, with the population at 10 billion, we will still be looking ahead into the future for the breaking point to arrive. And in the intervening time, maybe we’ll have solved the problem of resource distribution so that we can feed, clothe and house everyone on the planet.

        In all studies that examined economic prosperity and population, it was found that economic prosperity caused population decrease. Not the other way around. To control population, therefore, solving the economic problem of resource distribution is essential.


What do you think?

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s

%d bloggers like this: