There’s no e in Euler

 

Recently, while admiring Melbourne’s gorgeous display of overhead wires, we had cause to mention the number e.  We also remarked that common references to e as “Euler’s number” were inaccurate. (And no, that’s not the great Swiss mathematician pictured above).  Some of our readers queried that claim. We’ll now reply, taking the opportunity to tell a small part of the story of e.

Though extremely important, e is not the most inviting of numbers. Unlike π, it cannot be explained or motivated by way of easy geometry: we cannot simply point to a circle or similar and exclaim “Look, there’s e!” 

This difficulty of e is exemplified in the VCE curriculum. The curriculum exhibits no concern for what e is, or why it is what it is. It also appears that the forthcoming national curriculum is inclined to do little more. However, though e requires some effort, it is not nearly as difficult a number as is suggested by this dereliction of duty.

The historical origins of e are clouded by the mists of time, but it seems likely that the number first arose as the result of financial considerations. It is still the easiest way to get a grasp of the number.

Imagine we come across a very generous bank, Bank Simple, offering 100% annual interest. (Yes, this is a fantasy). So, if we invested $1 then after a year our dollar would have doubled to $2. 

That is a very good offer, but then we find a second bank, Bank Compound, with a different scheme: they will give us 50% interest every six months. Would we prefer to invest with Bank Compound? Definitely! 

After six months at Bank Compound, we will have 50% on top of our original $1, amounting to $1.50; in effect, we’ve multiplied by 1.5. Then, after the next six months, we will have earned an additional 50% of that $1.50: we again multiply by 1.5, to arrive at the year’s total of $2.25.

This is illustrating the familiar and important notion of compound interest. The point is that calculating smaller interest at correspondingly smaller time intervals means that we are obtaining interest on our interest, resulting in a greater overall return on our investment.

What does this have to do with e? We’re getting there. 

Imagine we’ve found a third bank, Bank Super Compound, which returns 25% interest every three months. We would then obtain a 25% increase in our investment, compounded four times over the year. So, starting again with our faithful $1, at the end of the year we would have (1 +1/4)⁴ = $2.44.

We can keep going. We locate Bank Super Duper Compound, which calculates the interest every month, contributing an extra 1/12 on top of our investment on each occasion. The result is, at the end of the year our $1 would have grown to (1 + 1/12)¹² = $2.61. 

Finally, we come across Bank Infinity, which goes the whole hog. This last bank divides the year into a zillion nanoseconds and then calculates the appropriately tiny amount of interest at each nanosecond. The result is, at the end of the year our $1 will have returned (1 + 1/zillion)^zillion. (To be precise, Bank Infinity calculates the limit of this quantity as the number of time intervals goes beyond a zillion and tends to infinity).

What is the amount returned by Bank infinity, what is this final number? It is the number we now denote by e. It is the result of compounding interest to the theoretical limit, what is known as continuously compounded interest. 

For those who love decimals, or calculating their interest really precisely, the expansion of this special number begins

2.7182818284590452353602874713526624977572470936999…

Do those final 9s indicate that the number is actually a terminating decimal? No: we’ve simply been cheeky in choosing where to stop. As is π, our new special number is irrational.

But what does any of this have to do with Leonhard Euler? Nothing. The earliest known appearance of the number is an indirect reference in a 1618 work, probably by the English mathematician William Oughtred: it is Oughtred’s portrait that we have included above. Our new number was then used throughout the 17th Century. Around 1690, the German mathematician Gottfried Leibniz explicitly denoted this number by the letter b. That was seventeen years before Euler’s birth in 1707.

However, there is a second part to the story, which does involve Euler. Though we have indicated how Oughtred’s number naturally arises as a financial concept, this only begins to explain the central, critical role that the number plays in calculus. This was demonstrated by Sir Isaac Newton and the other great 17th Century mathematicians. 

This work on calculus was then carried to brilliant extremes by, among others, Leonhard Euler. And, along the way, Euler chose to denote Oughtred’s number by the letter e, a convention that has endured. However, there is absolutely no evidence that Euler chose the letter e to refer to himself, or for any particular reason other than that the letter was relatively unencumbered by uses in other contexts.

Now, exactly how does e claim such a central role in calculus? And how, if at all, does e capture the notion of “exponential growth”? This is yet another issue ducked by the VCE curriculum; the question is simply handballed to the universities (which then, predictably, drop the ball). We’ll provide some answers, and we’ll dispel some myths, in a future column.

Burkard Polster teaches mathematics at Monash and is the university's resident mathemagician, mathematical juggler, origami expert, bubble-master, shoelace charmer, and Count von Count impersonator. 

Marty Ross is a mathematical nomad. His hobby is smashing calculators with a hammer.

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