 # The taboo of the exponential

For Carl Sagan, “if you understand exponentials, the key to many of the secrets of the universe is in your hand”. But not everyone seems ready to unlock the secrets of the universe. Thus, in mid-November 2021, more than 18 months after the beginning of the SARS-COVID 19 pandemic, the Minister of Health stated “the circulation of the virus has accelerated for a few weeks now, with an increase of 30% to 40% per week. We are not yet in a so-called exponential phase” (quoted in Ouest France (2021)). Since an increase at a constant rate is precisely the definition of “exponential growth”, one may wonder about this statement, which reveals either a lack of numeracy on the part of our leaders, or an element of language, with the word “exponential” becoming a taboo word that should not be mentioned?

(this is the English tranlation of a post published in January 2022)

## Proportionality and linear growth

A central aspect of mathematics taught in primary and secondary schools is the notion of proportionality, observed in geometry with Thales’ theorem and found in arithmetic with the “rule of three”. As legend has it, in the 6th century B.C., Thales of Miletus wanted to determine the height of a large pyramid, and he had the idea of using the length of shadows, and the proportionality of geometric figures. If a stick of 2 meters, planted in the sand has a shadow that is projected on 3 meters, a tree of 6 meters will have a shadow of 9 meters; and conversely, if the shadow of the pyramid is 90 meters, it is that the pyramid is 60 meters high. This is the basis of proportionality, and of linear growth.

We find this concept for example in physics, when a body moves at a constant speed: by doubling the distance to be covered, we will double the travel time. This linearity is important, and heuristically, it serves as a basis for many reflections that we can have when we try to extrapolate. This idea is the basis for the calculation of loss reserves: assuming that the rate of payment remains unchanged, a principle of proportionality allows one to extrapolate future payments. In other words, when a quantity grows linearly, $y(t)=at$, for a positive $a$ parameter. In this case, $dy(t)/dt$ is constant, and it is this property that will disappear when we have an exponential growth.

## The legend of the chessboard, the folded leaves and the water lilies

According to the legend, a Persian (or Indian, depending on the source) king had promised to give one of his ministers anything he wanted, and the latter had asked his king to give him grains of wheat, respecting a precise protocol: the king (or his steward) should place one grain of wheat on the first square of a chessboard, two on the second, four on the third, doubling from square to square, until the sixty-fourth. After laughing at his minister (the request seemed relatively modest), the king asked his accountants to do the calculations. His accountants realize that all the grain in all the granaries of the world should not be enough. In one of the legends, the king gave up his throne to his minister as a reward for his wisdom. In the ninth century, Thābit ibn Qurra (ثابت بن قرة), who had wondered about the existence of infinity, had spoken of the “duplicative sequence,” while Pierre de Fermat, in the seventeenth century, spoke of the “double progression” to describe this problem. The number of grains on the $n$ square is $2^{n-1}$, so that the total sum of grains on the first $n$ squares is $2^n-1$. That is, in total 18, 446, 744, 073, 709, 551, 615 grains when $n$ is 64… or a little more than 500 billion tons of wheat (a thousand times the annual production of wheat, in the world).

Another story related to this legend of the chessboard is that of the sheet of paper that is folded in two, then folded again in two, etc. If we start with a standard sheet of paper (one tenth of a millimeter thick), the thickness doubles with each folding. In about fifty folds (if we forget for a moment the physical difficulties, which make that after about ten folds, we do not have enough strength in our thumb to fold the sheet), the thickness is $2^{50}$ tenth of millimeter, or 112,589,991 kilometers (a little less than the distance of the earth to the sun).

Finally, we can also mention the famous water lily riddle:

“A water lily in a pond doubles in size every day. It covers the entire pond in one month (30 days). On what day did it cover half the pond?

We will call it an enigma because the intuitive answer, 15 days (half the time for half the surface), is not valid: as the surface doubles every day, it will take 29 days to cover the first half of the pond, and just one day to cover the second half.

All these legends tell the same story, that of a so-called exponential growth, in the sense that $y(t)=a^t$ (not to be confused* with $y(t)=t^a$ which would correspond to a “power function”, or “polynomial” growth). We will sometimes note $y(t)=\exp(\alpha t)$ where $\alpha$ is the logarithm of $a$.

## Drawing an exponential

Exponentials are a somewhat surprising shape, depending on the time scale (or time step) we consider, as in Figure 1, $y(t)=2^t$ , corresponding to the number of grains in the chessboard legend. Figure 1: exponential growth, with different time scales

The figure on the right corresponds to what happens with the chessboard game: when we reach large values of $t$, $y(t)$ becomes “infinitely large”. Indeed, noting that $2^{10}=1024$, we can say that $y(t)=2^t\approx 1000^{t/10}$, in other words, in 10 steps (the tenth square on the board), we have 1,000 grains, 1,000,000 grains on the twentieth, 1,000,000,000 grains on the thirtieth, etc. One could almost say that the writing of the number (in the classical base 10, with the 10 usual digits, 0,1,2,3,4,5,6,7,8,9), the size of the number grows in a linear way. On the figure on the right, we have the impression that the number of grains “explodes” from 90, but it is just that $y(100)\approx 1000\times y(90)$, and indeed, the value in 90 is a thousand times smaller than the one in 100.

But the most interesting representation, the most classical, is probably the one in the center of Figure 1, we have the illusion that there are three phases in the geometrical growth

• a very flat part, at the beginning (say between 0 and 4)
• a reasonable increase (between 4 and 8)
• a sharp increase between (8 and 10)

This is the idea that has been circulating in recent months on social networks, shown in Figure 2 (or Figure 3) Figure 2: on the right, the mathematician facing an exponential growth of a phenomenon, and on the left, the politician facing the same curve (source: Jens von Bergmann, inspired by the style of the webcomic xkcd by Randall Munroe). Figure 3:  exponential growth, with a log scale on the right (source: author, inspired by René Magritte).

Exponential gives the illusion of a break in the trend, but formally, growth is always “exponential”. This can be explained simply by noting that the exponential is the solution of a differential equation of the form $dy(t)/dt=ay(t)$, that is, the slope, $dy(t)/dt$, is proportional to $y(t)$: the growth is therefore indeed small at the beginning (because $y(t)$ is small) and much more consequent afterwards (because $y(t)$ is larger).

## Exponentials and financial discounting

The big difference between linear models ($y(t+1)-y(t)$ constant, i.e. $y(t)=at$ and multiplicative models ($y(t+1)/y(t)$ constant, or $y(t)=a^t$), between the straight lines and the exponentials, is that in one case the absolute variation is constant, and in the other case it is the relative variation. This gives well-known paradoxes. For example, a decrease of 20% followed by an increase of 20% does not return to the starting point. This is a so-called remarkable identity, since $(1-r)(1+r)=1-r^2\neq 1$. The same logic is found in compound interest, since $(1+r)(1+r)-1=2r+r^2\neq 2r$ (the return is “a little” greater than twice $r$). Strangely enough, it is the accumulation of all these small quantities, negligible at first order, that seems to lose all reason by dint of their accumulation.

For when we compound interest, we obtain that by placing a capital $C$ on a financial product yielding a rate every year, the wealth at the end of $t$ years grows exponentially fast with time, since $y(t)=C(1+r)^t$, as Fibonacci had described, in the Liber Abaci, published in 1202, as Charpentier (2016) recalled.

This exponential growth had been underlined in 1772 by Richard Price, who tried to imagine what would have been the return on the investment of a simple penny in a (virtual) bank at the birth of Jesus Christ, “one penny, put out at our Saviour’s birth at 5 per cent, compound interest, would, before this time, have increased to a greater sum, than would have been contained in a hundred and fifty millions of earths, all solid gold.

It is not certain that 150 million times the volume of the earth in gold is a good indication, but let’s remember that if $r$ is worth 5%, $(1+r)^{100}$ is worth a little more than 100, $(1+r)^{500}$ is worth a little more than 40 billions, and $(1+r)^{1772}$ is a 28-digit number.

These numbers, which can make you dizzy (especially when talking about financial sums), are reminiscent of the mechanism set up by Charles Ponzi, a little before 1920 (and which inspired many investors, such as Bernard Madoff): he proposed to offer a 50% return in three months! Let’s suppose that he manages to find 10 investors ready to invest $100 when he launches the product. Three months later, he has to pay them$150, which is the same as finding 15 investors willing to invest $100. Six months later, since he will have to pay them$225, he will have to find 23 investors ready to invest \$100, etc. After 5 years, he needs a little more than 330,000 investors. At the end of 10 years, the whole world must have invested in the investment offered by Charles Ponzi! In other words, the number of investors must grow as fast as the capital, i.e. exponentially fast! This need to double the number of investors almost every six months corresponds to a pyramid scheme (even if a pyramid would correspond more to a linear growth). This doubling of the size of the investor population is reminiscent of early demographic models.

## Malthus and demographic models

At the end of the 18th century, Malthus proposed a model of exponential population growth, which has left its mark on people’s memories, even though John Graunt, to whom we owe the first mortality tables, and then William Petty had considered the question a century earlier, and taken up by Leonard Euler, in 1748, who wondered whether it was possible to have had exponential population growth since Adam and Eve, or rather starting from six human beings who had begun to repopulate the earth after the Flood. Voltaire was critical of this model, pointing out that “one does not propagate in geometric progression. All the calculations that one made on this alleged multiplication are absurd chimeras”.

What is (mathematically) interesting is that demographers quickly proposed an interesting alternative to exponential growth, under the impetus of Pierre-François Verhulst, around 1840, which can be summarized as follows, considering a dynamic system, described by a differential equation:

if $dy(t)/dt$ is constant, we have a linear growth, characterized by a relation of proportionality that we mentioned in the introduction ($y(t+h)-y(t)$ will be proportional to $h$)
if $dy(t)/dt$ is proportional to $y(t)$, we have an exponential growth (or more generally if $dy(t)/dt=a+by(t)$)
if $dy(t)/dt$ is quadratic in , i.e. if $dy(t)/dt=a+by(t)+cy(t)^2$, we can obtain the logistic model proposed by Pierre-François Verhulst that we will find in demography models, and more generally in population dynamics, and in pandemic modeling.

The growth following a logistic model gives a growth resembling an exponential, at the beginning, before reaching an inflection point, before slowing down and stabilizing. In fact, the logistic model of Verhulst (formally $dy(t)/dt=ay(t)[b-y(t)]$) was proposed by Daniel Bernoulli around 1750 to model the dynamics of a smallpox epidemic. So we come back to the starting point, with a link between exponentials and pandemics…

## The exponential explosion in a pandemic

Charpentier & Barry (2020) reminded us that in classical pandemic models, the number of infected people increases exponentially fast, at the beginning, the factor of increase being linked to the number of potential contacts of a contagious individual. If it is difficult to stop an exponential growth, $y(t)=a^t$, Ghys (2020) reminded us that it is possible to slow it down very strongly by forcing social distancing, to have $y(t)=b^{t}$, with $b, where $y(t)$ is the number of infected (and contagious) persons. And as Stevens (2020) said, “That is math, not prophecy. The spread can be slowed, public health professionals say, if people practice ‘social distancing’ by avoiding public spaces and generally limiting their movement. And, almost paradoxically, a small change can have major consequences, precisely because of this “exponential” amplification. This is the optimistic vision proposed by Young (2021) when he wrote “the infectious nature of a virus means that a tiny bad decision can cause exponential harm, but also that a tiny wise decision can do exponential good“.

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* In fact, exponential growth also scares computer scientists: an algorithm that solves linearly with time (or with the number of calculations to be performed, such as finding the largest element among $t$), quadratically (such as ordering a list of $t$ elements), or more generally polynomially (such as multiplying $t$ matrices) is considered to be “simple. A problem that is solved in a number of steps that grows exponentially (or even longer) will be much harder (see the notion of “NP-hard” (NP standing for “non-deterministic polynomial”), which includes problems like finding the shortest path through $t$ cities). And if exponential growth is (very) fast, “exponential decay” is also very fast. Also, in probability, if $\mathbb{P}(X>t)=a^{-t}$ corresponding to an exponential decay, one will speak of “fine tails” contrary to a polynomial decay, $\mathbb{P}(X>t)=t^{-a}$, which corresponds to Pareto type tails, and thus “(potentially) extreme risks”.

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