# How do I distinguish between "e" the natural log base and a variable conventionally referred to as "e"?

How does one distinguish between the natural base, usually indicated with "e" and a coefficient that, due to historical reasons, is referred to as "e" (such as in Weibull functions for dose-response)?

• You can't. If for some reason they have to appear in the same equation, you must rename one of them. Commented Jul 21 at 15:16
• +1 for a question that unfortunately needs answering sometimes. To take a physicist's example, $\mathrm{e}^{eV/(k_BT)}$, with $e$ the charge of a proton.
– J.G.
Commented Jul 21 at 15:32
• The sum of a bunch of phase shifted currents might be $i = \sum_i\, i_ie^{i\phi_i}$, which seems clear enough. Commented Jul 21 at 15:35
• @Aruralreader Which is part of the reason people often replace $i$ as in $i^2=-1$ with $\mathrm{i}$ or $j$. If they're quirky, they might even use $\iota$.
– J.G.
Commented Jul 21 at 15:58
• Context! There exist different words that are spelled exactly the same. How to tell them apart? Commented Jul 21 at 18:44

I searched the internet for "Weibull" functions for dose-response and found Christian Ritz's GitHub reference for Weibull model functions, which demonstrates a technique that seems reasonable: Since $$e$$ is the standard name of a parameter, it can't be used for the base of the natural exponential function. As such, $$\exp$$ is used instead of $$e^\cdot$$ to represent the exponential function, as in:

The four-parameter Weibull type 1 model ('weibull1') is $$f(x)=c+(d-c)\exp(-\exp(b(\log(x)-\log(e))))\text{.}$$

• That works if you don't have to reference the value of $e$ (the constant) directly. Commented Jul 21 at 15:29
• @TonyK call it $\exp1$, as long as it's not too often.
– J.G.
Commented Jul 21 at 15:30
• @J.G.: Yes, I thought of that. And then I thought $yeugh$! But tastes vary, I suppose. Commented Jul 21 at 15:40
• @TonyK $\exp \exp 0$ or $1/\exp(-1)$ might be more to your liking? j/k Commented Jul 22 at 3:50
• yuck, I saw "log(e)" and went "isn't that just one" Commented Jul 22 at 18:27

In printed mathematics, the natural base should be set in roman type: $$\mathrm e$$, while symbols representing quantities that can vary (depending on context or units) should use the usual italic font: $$e$$.

(Added in response to comment) In general, roman type should be used for all symbols in mathematics that have a fixed meaning: $$\mathrm{i,j,k}$$ for the base elements of the complex numbers and quaternions; $$\exp,\ln,\cos,$$ and so on for standard functions; $$\lim, \sup,\operatorname{argmin}, ...$$ for standard operations; the differential symbol $$\mathrm d$$; the base symbols $$\mathrm{J}$$ for the Bessel function; $$\mathrm C^n(X)$$ for the real functions on $$X$$ with continuous $$n$$th derivative; $$\mathrm{L,H,W}$$ as the base symbols for the function spaces of Lebesgue, Hardy, Sobolev, etc. Notice also that the number realms $$\Bbb{N,Z,Q,R,C,H}$$ are always set in upright, not italic, type.

• Thank you so much. I always feel like I am the only one who prefers $\int_a^b\mathrm e^{it}\,\mathrm dt$ over $\int_a^be^{it}dt$. Commented Jul 21 at 15:24
• But I would not recommend this as a way to distinguish the two meanings within a single formula. That is asking too much of the reader's eyesight! Commented Jul 21 at 15:28
• Or write $\mathrm{e}^x$ as $\exp x$.
– J.G.
Commented Jul 21 at 15:31
• I only downvoted because I agree with @TonyK—we shouldn't trust this mere font difference to distinguish between the two mathematical objects, so this isn't the best answer to the OP's specific question. (Personally I prefer the usual italic e to the roman e in math formulas, but that alone isn't a reason to downvote :) ) Commented Jul 21 at 16:26
• Would you use an upright lowercase roman for the constant relating the diameter of a circle with its perimeter, or is "$\pi$" ok? Commented Jul 22 at 10:22