Why e, the Transcendental Math Constant, Is Just the Best

The solution to our puzzle about Euler’s number explains why e pops up in situations that involve optimality.

November 24, 2021

Last month, we presented three puzzles that seemed ordinary enough but contained a numerical twist. Hidden below the surface was the mysterious transcendental number e. Most familiar as the base of natural logarithms, Euler’s number e is a universal constant with an infinite decimal expansion that begins with 2.7 1828 1828 45 90 45… (spaces added to highlight the quasi-pattern in the first 15 digits after the decimal point). But why, in our puzzles, does it seemingly appear out of nowhere?

Before we attempt to answer this question, we need to learn a little more about e’s properties and aliases. Like its transcendental cousin π, e can be represented in countless ways — as the sum of infinite series, an infinite product, a limit of infinite sequences, an amazingly regular continued fraction, and so on.

I still remember my first introduction to e. We were studying common logarithms in school, and I marveled at their ability to turn complicated multiplication problems into simple addition just by representing all numbers as fractional powers of 10. How, I wondered, were fractional and irrational powers calculated? It is, of course, easy to calculate integer powers such as 102 and 103, and in a pinch you could even calculate 102.5 by finding the square root of 105. But how did they figure out, as the log table asserted, that 20 was 101.30103? How could a complete table of logarithms of all numbers be constructed from scratch? I just couldn’t imagine how that could be done.

Later I learned about the magic formula that enables this feat. It gives a hint of where the “natural” in “natural logarithms” came from:

ex = 1 + x1!+x22!+x33!+x44!+x55!+⋯.

For negative powers, alternate terms are negative as expected:

e-x = 1 – x1!+x22!−x33!+x44!−x55!+⋯.

These powerful formulas enable the calculation of any power of the mysterious e for any real number, integer or fraction from negative infinity to infinity, to any desired precision. They allow the construction of a complete table of natural logarithms and, from that, common logarithms, from scratch.

The special case of this formula for x = 1 gives this famous representation of e:

e = 1 + 11!+12!+13!+14!+15!+⋯.

In addition, e has many amazing properties, some of which we’ll uncover in the solutions to our problems.

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