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Napier's e - o

Amy Shell-Gellasch (Beloit College)

We use e to denote the constant 2.7282… associated with continuous compound interest, natural logs and exponential functions. Euler gets the credit for choosing the letter e. And he was the first, and maybe the only one to use it so extensively, giving us so many beautiful and surprising series that converge to some expression containing e, as well as Euler’s identity: eiπ + 1 = 0. But is referring to it as Euler’s number historically accurate?

We have seen that c would have been fitting in light of Cotes’s contribution. However, that could have resulted in some messy business with the quadratic formula, not to mention relativity theory and the speed of light. Since l and j for Leibniz and Jakob Bernoulli sound promising, we will keep those on the table.

The British use of the term Napier’s constant is certainly deserving. Napier’s lasting contribution to mathematics is his invention of logarithms, a term that he coined. However, in much of the world, the only thing his name is associated with is Napier’s Bones or Rods. These ingenious aids to arithmetic from the 1600s are now wholly obsolete (but great fun to make and use in the classroom). They too provided a means (mechanical in this case) of reducing multiplication to a series of single or double digit additions [2]. So the European attribution of the constant to Napier is both realistic in light of the magnitude of his discovery, as well as more egalitarian. So for consistency’s sake, n would be the way to go. Or nb to cover all the bases. But nb and np could throw a few people for a loop.

But, truth be told, the first recorded use of the number that would later be known as e, should go to Oughtred. Continuing with the urban legend that e stands for Euler, I move that we adopt o for Oughtred for 2.71828… But then again, Newton used o to denote his vanishing fluxion in his version of the calculus. And then, there is also the small detail that o could be confused with 0. But this leads to wonderful statements such as: oiπ + 1 = 0 or 1=0.

So we have considered e, c, b, a, l, j, n, nb and o. On second thought, maybe it’s best to just leave it as it is. That is e-sier. And as I always tell my calculus students when they can’t remember how to differentiate ex, “remember, e stands for ‘easy’: to differentiate, just leave it alone.”


Euler was the first to imply that e is irrational in 1744, while Lambert published the first rigorous proof of its irrationality in 1768. The number e was shown to be transcendental by Hermite in 1873 [11].

Amy Shell-Gellasch (Beloit College), "Napier's e - o," Convergence (April 2010), DOI:10.4169/loci003209