The astounding nature of exponential growth

How good were you at maths?

In 1979 I was proud to receive an A grade in O-level mathematics. I went on to use some of what I learned as I continued to study Physics and Biology to A-level.

Obviously some parts of the maths we all learned at school are useful in our day-to-day life, others not so much.

What are the chances of...?

Other than the basics of addition, subtraction, multiplication and division, one aspect of the subject that really interested me was probabilities. How often will a coin toss result in heads, how often in tails? What are the odds of pulling an ace from a deck of cards? What chance do the bookies give this horse of winning in this race?

I could see how useful an understanding of probabilities would be in life.

But I wasn't so sure about the value of understanding exponential growth, or logarithmic equations... until now.

It is 40 years since I last grappled with a logarithmic equation, when I sat down to tackle the O-level exam at age 16. Just recently I tried re-visiting logs, with surprising success.

An introduction to exponential growth

Let's begin with exponentials, which is where my recent return to the sums of my youth began.

I can clearly remember "Fat Don", our maths teacher, posing a riddle about a chess board and grains of rice. The mathematical illustration of exponential growth comes from a fable about a man, asked by the king what reward he would claim for his great deeds, suggested payment in rice.

All he wanted was grains of rice, measured out on the squares of a chess board, the number of grains doubling on each successive square.

There are 64 squares on a chess board.

If you place one grain of rice (or wheat, in the image below) on the first square, then double it, there are two grains of rice on the second square. You'll have 4 grains on the third square, and 8 grains on the fourth.

How many grains, Fat Don wanted to know, would there be on the final 64th square of the board?

There are 16, 32, 64 and 128 grains on squares 5 to 8. That's the first row done. Not too expensive for the king, is it?

The numbers on the second row grow bigger, but still affordable to the wealthy ruler of a large, prosperous kingdom: 256, 512, 1,024, 2,048, 4,096, 8,192, 16,384, and on the 16th square, 32,768 grains.

At school we were allowed to use calculators to assist us, but couldn't believe the screens of our high-tech devices weren't big enough to display the final result. It's easier to do today on a computer spreadsheet.

The astounding answer to Fat Don's riddle is that the 64th square requires 9,223,372,036,854,780,000 grains of rice. That's 9.223 quintillion! There isn't enough rice in the world for this one square alone, never mind the rest of the board.

The total number of grains on all squares is over 18 quintillion. It's a number we can't really make any sense of, or even visualise.

This illustration shows something of the power of exponential, or compounding growth.

Here's a great video which explains the same concept in a different way:

Exponential growth can be exhilarating or terrifying, depending on what is being measured.

Grow your wealth

The value of an investment, as interest paid on it compounds over the years, can be thrilling as one approaches retirement. A sum of $1,000, invested at the age of 20, compounded at 13% growth per year, over a term of 45 years will grow to an incredible hoard of wealth.

At the end of Year 1 the investment is worth $1,130, and when re-invested for the second year, will gain a further 13%. This results in a sum of $1,277 at the end of Year 2.

The maths is simple. You just take the original 1,000 and multiply it by 1.13. The answer is multiplied by 1.13 again to give the result for the end of the second period.

Continue this multiplication 45 times, and you have the resulting cash lump sum payout at age 65, from your initial investment of $1,000 at age 20.

The amazing result is $216,497.

If you'd invested just $5k at age 20, and for 45 years didn't add a single extra cent to your retirement fund, you'd retire a millionaire. $5,000 compounded at 13% over 45 years, comes out at $1,082,484.

OK, so I know what you're thinking:

"But Ian, you can't earn 13% on your capital. Also, this doesn't take in to account interest rate variables, or the devaluation of your currency due to inflation."

You're right, of course. This is just a simple example to further illustrate how compound interest, or exponential growth, works over time to create huge numbers. I've used the optimistic growth rate of 13% for a reason, as you'll soon see.

Let's look at another example, perhaps of a more relevant nature right now.

Viral growth

A virus, if unchecked, tends to spread through a population in an exponential manner.

I'm sure, if you've been watching as the current coronavirus pandemic has begun its spread across the globe, you'll have heard the term "R-naught" or R0 being used. This refers to how infectious a virus is, and is a measure of how many other people one person may infect.

An R0 of 2 means that, on average, one person will infect two others. This growth and spread from one person to two more will take place over a certain length of time, maybe several days. Which is why containment, via isolation, is so important.

So back to the chess board example. If one person infects two others, those two will infect two more each, resulting in 4 more new infections. There are now 7 infections in total, the original one, two more, then four more... Over each period of days numbers of newly infected double and re-double, just like the grains of rice on a chess board.

After 10 periods of doubling, from one initial infection, there are new 512 people infected. The 11th doubling infects 1,024 more, and the total number infected is now 2,047.

Many of the original "generations" of infected will by this time have had an "outcome". They have either recovered or died. CFR (case fatality rates) can begin to be estimated from these figures.

The 21st period of doubling produces over 1 million new cases.

How long is the doubling period?

So a key question in predicting growth is: "How long is the doubling period?"

This is what prompted the return to the mathematics of my youth.

I'll not go too deep into what I had to re-learn, but try to keep it simple here. Let's assume numbers of infected double every 7 days. Another way to express this is in terms of a daily increase in numbers. The required formula for this produces an answer of an increase each day by a factor of 1.104090.

In other words, a daily increase in numbers of 10.4% will result in a doubling every 7 days.

A multiplier of 1.122462 (daily growth at 12.2%) results in doubling every 6 days.

1.148698 (14.7% daily growth) results in numbers doubling every 5 days.

1.189207 (18.9%) will result in 4 day doubling.

Graphing the growth

So I've been graphing the growth in cases of coronavirus for a couple of weeks now, and am shocked by the results.

I've taken case numbers (not including China) on a daily basis from this source and graphed them on a spreadsheet, with calculations of the daily growth rate.

I've then taken these figures, and projected them forward to the end of March.

The results are truly frightening.

I am creating a new graph each day, based on the end-of-day figures from the previous day, and project forward using three measures:

  1.  The previous day's multiplier (this is calculated very simply, by taking the current day's increase in numbers, and dividing it by the previous day's increase)
  2.  An average of the multipliers from the previous 10 days
  3.  The lowest multiplier since 21st Feb

You can find my graphs here:

The lowest daily multiplier (at time of writing) occurred at the end of 10th March, and is used in today's graph (for end-of-day 11th March). That figure is 1.13507003300919.

This means that by using the lowest daily increase from the past 3 weeks, we can expect numbers to grow on a daily basis by 13.5% more than the previous day. Hence my use of a 13% multiplier in the optimistic pension pot projection. The figures are similar.

Using this lowest multiplier to project forward from today (Thursday 12th March) suggests that worldwide number of infections by the end of the month, less than 3 weeks away, is over half a million.

Even a small change in the multiplier create a huge difference in the outcome further down the line.

The other two multipliers I use on the graph both suggest figures of over a million by 31st March.

Whichever turns out to be the most accurate, either result will place catastrophic burdens on health systems around the world.

I tried running the projections forward into April, but the numbers grow so quickly that the abilities to test that number of people will in all likelihood be beyond the resources of any country.

At that level, reported figures will not be able to keep up with actual numbers, and any kind of accurate modelling would then have to be based on estimates rather than reported figures.

Two things to remember:

  1. I'm using reported numbers, which rely on testing, and may be much lower than real numbers
  2. I'm also using the smallest recent multiplier for my lowest projections, and the numbers are still huge.

I'm sure that behind the scenes official bodies are using similar predictive models, but their scientists will have much better data than I have access to, and probably have better mathematical qualifications than my 40-year-old maths O-level (Grade A).

And with access to such information, they make decisions such as the US Government's unprecedented move, just last night, to block all travel from Europe to USA.

At the same time, the often repeated advice is not to "panic", and that most citizens are at very low risk.

Maybe "most of us" are at low risk, but I'm sure we all have loved ones who are in high risk categories.

I believe that watching the actions of governments around the world, rather than listening to their words, provides a clearer picture of what is going on behind the scenes.

All of us will be touched by the consequences of this outbreak.

Something else I find frustrating is that the nature of exponential growth is rarely mentioned in the news. Often reporters will talk about a "7-fold increase in numbers", when figures go from 1,000 to 7,000.

But this would suggest a linear progression to most people. That's how our brains function, and it is what we can understand and visualise most easily.

As a species we're not very good at thinking in exponential numbers, as they quickly grow beyond our comprehension, and beyond belief too.

But the facts suggest that we're in for BIG trouble ahead.

You can see the daily update of my graphs around 10am UTC (UK time) each morning here:

Let me know what you think in the comments below. Are you prepared for what is to come...?

Further reading / watching:

Much of my recent thinking has been heavily influenced by daily video updates on the development of the coronavirus crisis from Chris Martenson at Peak Prosperity. I've been watching them since he first caught my attention on 25th January:

I discovered Chris Martenson through Mike Maloney at, who I have followed for many years. Mike has a fantastic video series called "The Hidden Secrets of Money". It's a great education:
Hidden Secrets of Money (YouTube playlist)

Also very informative for a broader perspective is Chris Martenson's "Crash Course":
The Crash Course (YouTube playlist)

Thanks also to Ken Standfield, and his graphs, which inspired me to begin some research of my own:

Leave a Reply 4 comments

Nicky - March 13, 2020 Reply

Wow. This is mind blowing…not only because O-Level Maths has finally proven useful 🙃 but those numbers are off the scale. Thanks for the insight Ian, and I suppose some of the credit must go to Fat Don too…

    Ian Usher - March 13, 2020 Reply

    Haha. I never thought that 40 years later Fat Don would still be getting credit.
    Just be glad I didn’t run the numbers out to the end of April… talk about off the scale !!

Deborah - March 13, 2020 Reply

Wow, Ian, this is amazing… and truly terrifying. You explained it so well, and you’re right, people don’t realize how quickly the number can get so huge. Thanks for spelling it out in layman’s terms.

    Ian Usher - March 13, 2020 Reply

    Hi Deborah,
    I’m glad you found the explanation clear.
    It really is so hard to comprehend how these sorts of numbers can be correct.
    I think it is so important for people to have some understanding of this as we move deeper into this crisis.

Leave a Reply: