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University of Basel

«Even mathematicians sometimes get it wrong.»

Text: Tim Schröder

Many countries were caught off guard by the rapid surge in Covid-19 cases. Mathematician Helmut Harbrecht discusses the mathematics behind exponential growth – and why we have such a hard time wrapping our heads around it.

Chess board with rice on its fields
An Indian legend illustrates exponential growth. (Photo: iStock)

UNI NOVA: Throughout the coronavirus pandemic, exponential growth has become something of a buzzword. Growth is a familiar enough concept. However, when we add the term «exponential» it gets a bit more complicated …

HELMUT HARBRECHT: Exponential growth is simply a process whereby something increases by a given factor over a fixed period of time. In the context of Covid-19, it refers to the rise in infections over a few days. Because exponential growth is something we rarely encounter in our everyday lives, most people don’t really grasp just how incredibly fast this kind of growth can be.

UNI NOVA: Can you think of an example that would make it clearer?

HARBRECHT: The classic example is the Indian legend about a wise man who invented the game of chess many hundreds of years ago. King Sher Khan was so pleased with the game that he summoned the wise man and asked him to choose his reward. He replied that for each square of the chessboard, he wanted twice as many grains of rice as on the previous one – starting with a single grain on the first square. The king laughed at the wise man’s apparent lack of ambition, but his treasurer became uneasy, realizing exactly what the request meant: For the second square, two grains of rice would be needed, followed by four on the third, eight on the fourth, and so on. For the 25th square, the fi gure is more than 1.6 billion grains of rice. By the end, it is far greater than all the rice in the world. The total amount needed is more than 1019 grains of rice – a one followed by 19 zeros.

UNI NOVA: And what does exponential growth look like in the context of the coronavirus pandemic?

HARBRECHT: That’s a little bit more complicated. Experts estimate that the generation interval is four days. Each infected person transmits the virus to around 3.5 others, assuming the population is not yet immune. So after four days, 3.5 people have been infected, each of which will infect 3.5 more within four days: that’s 3.5 times 3.5, or 12.25. Each of these will infect another 3.5 people. 12.25 times 3.5 is almost 43 new infections after 12 days. And so on and so forth, as with the chessboard. The amazing thing is that even mathematicians like myself sometimes get it wrong when it comes to exponential growth. At some point the increments become so huge that forgetting a single step means your result will be of the wrong order of magnitude altogether. The human brain is simply not built to understand it.

UNI NOVA: So non-mathematicians shouldn’t feel too bad if it gives them a headache?

HARBRECHT: Many students do in fact have some trouble with the mathematical description of exponential growth, described by the exponential function. Then there is the inverse of the exponential function, the logarithm, which is even more abstract. I always find it very interesting to see what students know when they begin university. That’s why I am a member of the examination board for the oral school-leaving examinations at Kantonsschule Olten. This gives me an opportunity to find out first-hand what the pupils have learnt. I am very happy to report that in spite of the coronavirus lockdown, pupils have a level of knowledge that is comparable to what it was before the pandemic. In any case, the main takeaway from the topic of exponential growth is that the mathematics behind it mean something can become very big very fast.

UNI NOVA: How useful is this insight in everyday life, aside from in a pandemic?

HARBRECHT: One example is compound interest, which means that your money grows at a given rate over a fixed period of time – let’s be optimistic and say two percent per year. If you invest 100 francs, after a year you have 102, after the second year 104.04, and after the third year 106.13 francs. After 35 years, the investment has doubled. Of course it’s a little more complicated than that in real life, as you have to take inflation, bank charges and other factors into account in your calculations.

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