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I must admit that despite my concerted efforts to ignore current events so as not to further upset me, I have found myself engrossed in the news about the coronavirus pandemic. As a budding mathematician watching the spread of disease unfold on television, my eye always arrives at the numbers ticking away at the corner of the screen. I examine the number of total cases in the United States, the number of new cases, and the number of deaths — all terribly unsettling.

Each time I see this data anew, though, I struggle to fully intuit the scope of these impossibly large numbers.

As I write this piece, more than 20,000 new cases were confirmed just yesterday in the United States alone. That’s nearly four times the undergraduate population here at Princeton. It is quite an upsetting number and surprisingly large, yet at the same time, it is somewhat meaningless. I find it hard to grasp what it actually represents.

I have never seen the whole Princeton undergraduate population in the same place, for example, let alone four times that number of people. The best I can do to envision 20,000 people is to recall the first day of orientation, when all thirteen hundred-and-some incoming freshmen packed ourselves into McCarter Theatre, and imagine sixteen of those crowded theaters superimposed together. Not a helpful comparison.

In an effort to comprehend the exponential growth and the resulting large numbers associated with coronavirus, I have furnished some heuristic analogies that I hope will help display the full expanse of the disease more concretely. If we take the time to comprehend the intuition-defying numbers of confirmed and new cases of the coronavirus, these numbers, while terribly disheartening, should become more manageable.

The most rudimentary analogy I have concocted is one inspired by books about recreational mathematics (not an oxymoron!) that transfixed me in my childhood. The number of cases of coronavirus, at least before preventative measures take effect, grows at an exponential rate. Perhaps the most recognizable process exhibiting exponential growth to many of us is financial interest. The more money we put in the bank, the more money we gain through interest the following month. Similarly, the more cases of the virus that are reported, the more cases will be confirmed the following day.

Of course, in reality, a bank’s interest rates are sufficiently small to prevent our money from growing as fast as the coronavirus. However, suppose that a peculiar bank offered to double your savings after each week. In this case, a quick calculation — one I know well from my (unusual) childhood — verifies that an investment of one dollar would make you a millionaire after only twenty weeks. Similarly, one case of coronavirus would infect one million people after only twenty weeks if the virus doubled once a week; unfortunately, though, the virus doubles faster than this.

While the comparison to banking already succeeds in relating the growth of coronavirus to a process that affects us on a normal basis, even this analogy does not circumvent the difficulty of intuiting large numbers. To get at this problem, I have found two ordinary kitchen phenomena most enlightening.

Do try this at home! Place a bag of popcorn in the microwave. At first, a few kernels will pop sporadically, but as more and more kernels become pop, the popping rate increases so quickly that it becomes nearly impossible to detect individual pops within the noise produced. The number of new cases of coronavirus each day is analogous to the number of new pops within a certain time interval. At this time in the trajectory of the virus, the number of cases increases so quickly that we cannot even visualize the number of new confirmed cases per day, just as we cannot possibly count the individual pops of popcorn after a certain time in the microwave.

Similarly, try to coerce ketchup to flow out of a glass bottle. Perhaps this is more of a cartoon trope than an everyday phenomenon, but the ketchup will take a seeming eternity to first exit the bottle, then gush out so quickly as to smother your entire plate with unwanted red goo. The ketchup again flows in the same way that the virus spreads: faster as more ketchup has already been poured.

All three of these analogies give an intuitive sense of the magnitude of the large numbers appearing so frequently in the news now. As we acquire a better idea of just what these numbers mean, they become all the more viscerally upsetting; at the same time, these everyday comparisons will allow us to interpret them, rather than being daunted by them.

Ollie Thakar is a first-year from Baltimore, M.D. He can be reached at othakar@princeton.edu.

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