(Not) Understanding Large Numbers

Among the many quirks of the human brain, one is our inability to conceive of large number growth. I want to share a few interesting examples I’ve run across that are fun to consider. Notice how, when you read them, your first reaction is that they seem impossible. That reaction is normal, and in today’s world, it’s a problem.

Understanding how exponential curves work is helpful for two reasons. The first is that many of the business growth stories (see Amazon, Facebook, etc) and public health stories (COVID spread) of today and tomorrow operate on these strange curves. An Industrial Age company needn’t consider the possibility of a competitor’s rapid exponential growth because all competitors were hamstrung by the same constraints of capital, labor, equipment, etc. So its helpful to understand how these curves work in the real world of today because many of those constraints are either greatly reduced or completely eliminated in the digital era.

Secondly, our inability to conceive of these numbers cause us to be surprised and unprepared in the face of health and environmental disasters. I saw COVID spread/fatality estimates in March of 2020 that struck me as entirely implausible at the time. When only a small handful of Americans had died at that point, the concept that the death toll could rise to – or even exceed – a WW II-sized fatality count in the 300,000 range in a short amount of time was something I had a difficult time accepting (at the time). But it wasn’t just a difficulty of “accepting” or “believing” in those projections that was a mental hurdle for me; I also was (in retrospect) having difficulty in appreciating the power of exponential growth.

First, some quick terms (keeping it simple, because I’m simple): both arithmetic growth and geometric growth can be referred as exponential growth, because their rate of change is based upon a growth exponent from a previous point. Arithmetic growth is 2, 4, 6, 8, 10 and so on. You can see each number increasing by the same rate: 2.  Geometric growth is 2, 4, 8, 16, 32 and so forth. You can see that the number doubles each time. There are other, steeper curves – like logarithmic growth, which was more the COVID story – which make the curve even steeper and more difficult to comprehend.

Often, to understand, we need an illustration, which are what these three examples are. The first is a story I remember reading in The Second Machine Age some years ago. It is actually an ancient story that the authors had read in a book by Ray Kurzweil.

The game of chess originated in present-day India during the sixth century CE, the time of the Gupta Empire. As the story goes, it was invented by a very clever man who traveled to Pataliputra, the capital city, and presented his brainchild to the emperor. The ruler was so impressed by the difficult, beautiful game that he invited the inventor to name his reward.

The inventor praised the emperor’s generosity and said, “All I desire is some rice to feed my family.” Since the emperor’s largess was spurred by the invention of chess, the inventor suggested they use the chessboard to determine the amount of rice he would be given. “Place one single grain of rice on the first square of the board, two on the second, four on the third, and so on,” the inventor proposed, so that each square receives twice as many grains as the previous”

“Make it so” the emperor replied, impressed by the inventor’s apparent modesty.

Sixty-three instances of doubling yields a fantastically big number, even when starting with a single unit. If his request were fully honored, the inventor would wind up with more than eighteen quíntillion grains of rice. A pile of rice this big would  dwarf Mount Everest; it’s more rice than has been produced in the history of the world. Of course, the emperor could not honor such a request. In some versions of the story, once he realizes that he’s been tricked, he has the inventor beheaded.

The next two examples of exponential growth – one involving a penny, the other a piece of paper – come from Michael Lewis’s new book The Premonition, which I’m currently reading. In this book about the COVID spread, he is writing about a trio of public health professionals who were trying to anticipate the next global health pandemic back in 2006 by looking back at the 1918 Spanish Flu outbreak and – in particular – the case of Philadelphia, which had a particularly high fatality rate. From Lewis’s book (I bolded some text for emphasis):

Why was it still possible, in 2006, to say something original and important about the events of 1918? Why had it taken nearly a century to see a simple truth about the single most deadly pandemic in human history? Only after three amateur historians studied the various interventions, and the various death tolls in individual American cities, did the importance of timing became obvious. Carter wondered why this had been so hard to see. A big part of the answer, he decided, was in the nature of pandemics. They were exponential processes.

If you took a penny and doubled it even day for thirty days, you’d  have more than five million dollars: people couldn’t imagine disease spread any better than they could imagine a penny growing like that. “I think it’s because of the way our brains are wired” said Carter. “Take a piece of paper and fold it in half, then fold it in half again, for a total of 50 times folding it in half. If a piece of paper is 0.004 inches thick to begin with, by the time you fold it 50 times, it is more than 70 million miles thick” Again, it feels impossible. The same mental glitch that leads people to not realize the power of compound interest blinds them to the importance of intervening before a pathogen explodes”.

One of our fundamental leadership challenges is to envision how markets, or changes to a natural system such as a pandemic or climate change, are likely to unfold so that we can make the right anticipatory moves and provide our organizations with some basic degree of clarity.

A great way to help explain the concept of exponential growth is exactly what the authors above did: break it down into a relatable example. Most of us don’t appreciate complicated numbers when they are expressed as equations, but we do when they are expressed as stories.

We can’t be linear thinkers in an increasingly exponential world.

Good luck!