Thinking Exponentially Isn’t As Hard As You Think

Thinking Exponentially Isn’t As Hard As You Think

Humans, generally speaking, struggle with exponential thinking.

This isn’t a new idea. People tend to underestimate the long-term growth of their investments, the speed at which a disease can spread throughout a population, or the rate at which technology is improving.

Real quick:

  • Linear: Consistent incremental change (red line)
  • Exponential: Consistent rate of growth (blue and green lines)
exponential growth vs linear growth
Photo credit: Wikipedia.

The thing is, exponential growth is a fairly easy concept to grasp, and mathematically isn’t that difficult. Anyone that’s ever calculated a tip gets the idea of multiplying a number by a certain percentage and adding it back to the original figure, arriving at a new larger number.

It’s just that things get tricky when you have to re-calculate the tip over and over and over.

Here are a few thought experiments to demonstrate the point:

Can you guess what $100 growing by 10% per year be after 20 years?

Answer: Click to reveal.

What about $100 growing by 20% per year after 20 years?

Answer: Click to reveal.

Finally, what about $100 growing by 20% after 30 years?

Answer: Click to reveal.

This is where we struggle with exponential prediction and understanding. 20% doesn’t feel that much bigger than 10%. And 30 years doesn’t feel that much longer than 20 years. But these differences have a multiplier effect on outcomes that’s difficult to know, making it almost impossible to estimate future outcomes.

The fact is, when it comes to exponential growth over multiple re-calculations, we just get lost in the math and have a poor intuitive sense of what the variables actually mean.

Controlling for time

So let’s think about exponential growth differently. Instead of controlling for time and trying to predict outcomes, let’s control for outcomes and think about time.

What sounds more impactful to you:

  • The population grows at a rate of 2% per year.
  • The population will double in size every 35 years.

When you learn something grows by 2%, you just sort of go ‘Um ok…that sounds small…‘. But it’s tough to know what to make of that. It doesn’t mean anything.

But you have a much better intuitive sense for doubling. If you have $500, knowing when it will become $1000 is more impactful than knowing it is growing by a certain percentage. It’s just an easier concept for your brain to grasp.

So here’s a different way of thinking exponentially: Given any growth rate, how long before something doubles in size? When will it be four times its size? Or one thousand times? A million?

Luckily, there’s a handy way to do all of these calculations in your head quite rapidly.

Granted, these may just be interesting theoretical exercises rather than practical tools, but I think they’re valuable for teaching you to have a better appreciation for how numbers can grow (or shrink) on an exponential level.

The Rule of 72

The Rule of 72 is a simple formula you can use to figure out the doubling time for any given rate of exponential growth. It’s popular in financial circles for quickly estimating the time at which an invested dollar will turn to two. And it goes like this:

Time to double = 72 divided by the % rate of growth.

For example, let’s say you have a population of 1,000 people that grows at a rate of 5% every year. How long before that population becomes 2,000?

Take 72. Divide it by 5. Answer…14.4 years.

Of course the Rule of 72 isn’t 100% accurate–but given how simple the math is, it’s pretty damn close. In the example above, the true doubling time is actually 14.2 years. (Click here if you’re interested in the logarithmic math behind the Rule of 72).

Feel free to play around with different growth rates below to see how it affects doubling time.

Estimating doubling time: Divide 72 by the percentage of growth.

If you invested $1 in the markets, and it grew by % per year, it would become $2 in:

True double time:

Error:

Just for fun…2x Visualized

The surface area of the second circle is exactly twice the size as the first. Not that mind-blowing, I know, but you’ll see why I’m showing you this later.

1x

2x

Calculating 4x time in your head

So now that you can rapidly calculate the doubling time of something. Figuring out how long until it becomes 4x size its original value is a breeze.

Think about how many times you would need to double something to make it 4x bigger. The sequence of doubling the number ‘1’ goes like this: 2…4.

Just twice!

Therefore to figure out how long something will take to quadruple in size, just double its doubling time. Easy.

Estimating time to 4x: Divide 72 by the percentage of growth, double that number.

If you invested $1 in the markets, and it grew by %, it would become $4 in:

True 4x time:

Error:

Just for fun…4x Visualized

The third circle is 4x as large as the first.

1x

2x

4x

Calculating 1000x time in your head

Keeping with the same approach to calculate 4x, if you continued to double ‘1’, it would take 10 ‘doubling times’ until you reached 1024.

Therefore the time to 1000x something is roughly 10 times its doubling rate.

Technically you’d need to double it 9.965784 times to get to 1000 exactly, but since the goal is to become better at quick approximation, we’ll just call it an even 10.0.

Play around with different growth rates and compare them to the actual time to 1000x below.

Estimating time to 1000x: Divide 72 by the percentage of growth, multiply that by 10.

If you invested $1 in the markets, and it grew by % per year, it would become $1000 in:

True 1000x time:

Error:

Just for fun…1000x Visualized

The fourth circle is 1000x as large as the first.

1x

2x

4x

1000x

Calculating 1,000,000x time in your head

Continuing on with this logic, if something gets 1000x bigger in Y years, then in another Y years, it would be another 1000x bigger. Make sense?

Therefore all you need to do to calculate time to 1,000,000x is double the time to 1,000x.

Estimating time to 1,000,0000x: Divide 72 by the percentage of growth, multiply by 10, double that.

If you invested $1 in the markets, and it grew by % per year, it would become $1 million in:

True 1,000,000x time:

Error:

Just for fun…1,000,000x Visualized

Hopefully this visual gives you some context for how big a million actually is compared to a thousand. The fifth circle (you’ll have to scroll down for a bit before it comes into view) is 1,000,000x bigger than the first circle.

1x

2x

4x

1000x

1,000,000x

Test your undertanding

Problem A

You get 1% better every day, as is the fashion. When do you become 100% better?

Answer: Click to reveal.

Problem B

It’s 2024 and Earth’s population has grown to exactly 8 billion people. A new virus (totally non-deadly) has infected 4,000 people and spreads at a rate of 6% per day. How long before everyone on Earth gets the virus?

Hint: Click to reveal.

Answer: Click to reveal.

Problem C

Going back to Problem A, if you improve by 1% every day, about how much better are you after a year?

Hint: Click to reveal.

Answer: Click to reveal.

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