There is a widely clipped and shared video of astrophysicist Neil deGrasse Tyson asking the question: There is algae growing on a lake, you see that the area covered by the algae doubles every day. You go away for a month, come back and see that half of the area of the lake is covered by algae. The question is: How long do you have to wait for the entire lake to be covered by algae? Take a guess before you read on!
Let’s use the question as motivation to understand what exponential growth is and how it differs from its more common counterpart, linear growth. Those who cannot wait, can check the answer here.
What is exponential growth?
Exponential growth is when a quantity increases by a factor over regular intervals. In other words, the quantity is multiplied by a number on a regular basis. In the case of our example, the area covered by algae is multiplied by 2 every day.
Let’s say that you find this blog post fascinating and share it with 10 people. Of course, they will each find it equally fascinating and share it with 10 of their friends. If this pattern keeps going, the number of people reading the post is being multiplied by 10 at each step. So, after 9 steps, the post will have reached 1 billion people. I am allowed to dream, aren’t I?
A practical example of exponential growth: Compound interest
Cody has just become a grandpa to a little girl, Colette. He decides to open a savings account for her with $1000. The bank will add 10%, compounding annually. This means that, in the first year, the bank will add $100 (10% of 1000) to Colette’s savings. The account’s balance is now $1100. In the second year, it will add $110, or 10% of $1100. The balance will go to $1210. To increase a number by 10%, we multiply it by 1.1. This is why compound interest is an example of exponential growth.
Simon, has also become grandpa to a little girl, Sienna. Simon mistrusts banks and likes to keep things simple. He puts $1000 in an envelope and writes “Sienna” on its back. On New Year’s eve, he adds $100 to the envelope. He adds the same amount at the end of every new year. Here, we are adding $100 each year. This is an example of linear growth.
Here’s a comparison of the two girls’ savings until they turn 18:
As we can see in the above table, when the two girls turn 18, Colette will collect nearly twice as much as Sienna. Looking at the values in grey and concentrating on Colette’s money, we can see that it takes 7 years for the savings to grow, more or less, by the first thousand dollars. The second $1000 comes after 5 years. The next thousand only needs 3 more years. The fourth extra $1000 only needs 2 years.
It is a feature of exponential growth that the rate of growth itself accelerates over time. That is to say that the time it takes for a fixed amount of growth to occur gets shorter over time. Is it any wonder that compounding has been called “the eighth wonder of the world”?
The answer to the original question
Back to our original question. Most people would think that the algae, having taken a month to cover half of the lake, needs another month to cover the remaining half. This is what deGrasse Tyson calls “linear thinking”. In actual fact, since the area covered by algae doubles every day, only one day is needed for the algae to go from covering half of the lake to covering the entirety of it!
Did you guess correctly? If so, congratulations! Otherwise, I hope that you now have a better understanding of exponential growth.
This was part 3 of a series on counter-intuitive mathematics. If you enjoyed it, check out:
Part 1: How many cricket lineups? This post explains permutations.
Part 2: How to win a car on a game show? This post explores a famous probability problem.
Topic suggestions
“Exponential growth” is a term that we hear in daily life. Are there other such terms that you hear and that you would like me to try to explain in a future post? If so, let me know in the comment.
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