The Number 'e'

The mathematical number ‘e’ has a celebrity status in maths, writes Steven Strogatz in The Joy of x:

“A few numbers are such celebrities that they go by single-letter stage names, something not even Madonna… can match.”

 

That line got me thinking. Why did it take me a very long to understand why ‘e’ is such a famous number? I mean, I am an engineer and all those years of maths and physics prior to that meant I had run into ‘e’ in all kinds of places. Within maths itself, the “natural” logarithm is to the base ‘e’, not 10, but that I dismissed as an arbitrary human choice. But I could also see ‘e’ in real world phenomenon, from the charging of a capacitor and inductor to the rate of radioactive decay.

 

I’d always dismissed ‘e’ popping up in all kind of real world phenomenon as a coincidence. I mean, my idea of ‘e’ was based on how schools introduce ‘e’. Via an equation that seems contrived:

How can the occurrence of such a ridiculously defined number popping up everywhere in the real world be anything but a coincidence?

 

The answer lies, as with so many things, in calculus. Which being a complex topic is taught much later. And so unfortunately, we get introduced to ‘e’ via the above equation, which creates an impression of ‘e’ being a weird, contrived number.

 

But looked at from a calculus perspective, you realize ‘e’ can be defined in a way that immediately explains why ‘e’ pops up everywhere in the real world. In calculus, ‘e’ is defined as the rate of growth of something that grows continuously:


Ignore the mathematical representation. Focus instead on the verbal definition: if something grows continuously, ‘e’ is the rate at which it would grow. And suddenly it all fits. A capacitor charges continuously. A radioactive element decays continuously.

 

Not just in those two examples. Since most change in the world is continuous/bit-by-bit change, no wonder ‘e’ shows up everywhere in the real world. Ergo, its celebrity status.

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