*“The greatest shortcoming of the human race is our inability to understand the exponential function.”*

Albert Allen Bartlett

These are depressing times so this month’s contribution (a bit short for time, to be honest), though topically linked to the ongoing pandemic, deserves attention from different angles in its own right.

We’re going to have a quick look at *exponential growth*.

**Disease Spread**

OK, so we’ve seen the models of coronavirus spread according to different models of social contact and infection. Essentially, if one person infects *N* people, and each of those does the same, then very soon you have more than *N ^{2}* infected and so on. That’s varying degrees of bad depending on what

*N*is. But, if

*N > 1*, it’s bad sooner or later.

But there are lots of other areas where the same maths applies. Here’s a few random ones …

**Population growth**

As Professor Bartlett himself also noted, *“Can you think of any problem in any area of human endeavor on any scale, from microscopic to global, whose long-term solution is in any demonstrable way aided, assisted, or advanced by further increases in population, locally, nationally, or globally?”* OK, yes, we can argue about if we’ve overpopulated London, the UK or the world *right now* but the exponential law tells us we have to eventually. Stephen Hawking urged us to *“get off planet Earth within 100 years”* but it’ll only be a matter of time before we fill up other planets too.

**Chess and Rice**

There’s a (clearly made-up but) lovely story about the guy who invented chess. Apparently the king, for whose amusement the game had been devised, asked the inventor to name his reward. Shunning gold and jewels, he asked for one grain of rice for the first square of the chessboard, then two for the second, four for the third, eight for the fourth, and so on. The king thought this poor reward until his advisors pointed out that the allocation for the 64^{th} square would be *2 ^{63}* and that there wasn’t that much rice in the world!

**Moore’s Law**

Astonishingly, computing power *has* managed to keep pace with the ‘doubling’ rule every two years for a few decades now. Clearly can’t for ever though.

**Futurama and Investments**

Imagine you have 10p in the bank at 2% annual interest. Go to sleep for 1000 years and you’ll have nearly £40 million when you wake up!

(This example is really important because it stresses the importance of the difference between thinking about the *future*, a *long way* into the future, and for *ever*. Rather than thinking, ‘well we’ve got away with XXX for a few decades’, we really should be asking ‘can the human race survive that for 10,000 years? Or for ever? Of course, we’ve clearly *not* doing that right now.)

These examples aren’t hard to invent: they pop up anywhere you insist on repeatedly multiplying something by a fixed quantity.

**And more**

Finally, there are ‘worse’ things! Although the *O(n!)* complexity associated with optimisation problems like the ‘*Travelling Salesman Problem’* are still classed as exponential, the factor by which they increase each step itself increases as it goes! To illustrate, how massive this is, consider walking/travelling a factorial number of metres from where I sit now, isolated at home …

*1! = 1 *1m is just the other side of the desk

*2! = 2 *2m is outside in the yard, the other side of the window

*3! = 6 *6m is out in the road

*4! = 24 *24m is the house opposite

*5! = 120 *End of the road

*6! = 720 *Rhosllanerchrugog (yes, that’s a real Welsh place!)

*7! = 5,040 *Wrexham, where I work

*8! = 40,320 *Shrewsbury, on the way to Birmingham

*9! = 362,880 *Southampton, where I was born

*10! = 3,628,800 *Istanbul

*11! = 39,916,800 *Anywhere on Earth and back (the Earth’s circumference is roughly 40,000km)

*12! = 479,001,600 *The Moon (and way beyond)

That’s impressive, isn’t it? Halfway through the list, I’m still in the local village. At the end, I’m in outer space. These numbers grow very quickly indeed and it might be better for all of us if we understood them a little more?

March 24th, 2020 at 5:04 pm

And the exponential with an imaginary argument gives you a sine wave, then with mixed real and imaginary argument gives you a decaying (or increasing) sine wave. This is great for understanding control systems which, if heavily damped, just show an exponential decay in response to a step input, but if underdamped they oscillate – all described by one function.