# Notation diversity

This morning I had a brief discussion about different notations of trigonometric functions in Europe, so I looked for an online resource dealing with these diversities in mathematical notation. I found this paper treating the issue. It links to a website called Notation census which aims to deal with the diversity by listing the variations in the notation around the world. Still, I find it quite hard to browse the site and feel that lot of things are still not included in its database. Therefore, my question is: do you know of a book or a website compiling these differences in notation in which one can find the most of the non-orthodox symbols and abbreviations used in the world?

• I would add that differences exist not only across different (geographical) locations, but across different cultures. For example, I have seen different attitudes towards notation for identity matrix among linear algebraists and others (algebraic geometeres, dynamical system-ists). Commented Jan 6, 2011 at 0:59
• The website for the Mathematical Notation Census at math-bridge.org seems to be down. It redirects to notations.hoplahup.net, and then I receive a "502 Bad Gateway" error. Commented Jun 21, 2022 at 9:43

Christine Müller has collected a variety of examples to illustrate the diversity of notation, in the Survey on Mathematical Notations. Diversity exists because of distinct reasons:

• level of expertise ($$a\div b$$ in elementary school, versus $$\frac{a}{b}$$ in high school)
• individual styles ($$A\subseteq B$$ or $$A\subseteqq B$$)
• cultural habits (anglo-saxon $$(0,1]$$ versus french $$]0,1]$$, or german 0,5 versus english 0.5; my personal example is https://hsm.stackexchange.com/questions/5644/why-did-i-learn-to-write-the-base-of-the-logarithm-differently-from-the-rest-of )
• area of application ($$i$$ in mathematics is $$j$$ in engineering, $$\neq$$ in mathematics is $$!\!\!=$$ in computer science, $$\bar{z}$$ in mathematics is $$z^\ast$$ in physics, $$A^\ast$$ in mathematics is $$A^\dagger$$ in physics – to no end of confusion ...)
• mistakes (arcsinh instead of arsinh)

This all addresses notation that is diverse but still commonly used. For non-orthodox notation I could offer Richard Feynman's trigonometric notation:

While I was doing all this trigonometry [as a teenager], I didn't like the symbols for sine, cosine, tangent, and so on. To me, "sin f" looked like s times i times n times f! So I invented another symbol, like a square root sign, that was a sigma with a long arm sticking out of it, and I put the f underneath. For the tangent it was a tau with the top of the tau extended, and for the cosine I made a kind of gamma, but it looked a little bit like the square root sign.

Now the inverse sine was the same sigma, but left-to-right reflected so that it started with the horizontal line with the value underneath, and then the sigma. That was the inverse sine, NOT $$\sin^{-1} f$$ – that was crazy! They had that in books! To me, $$\sin^{-1}$$ meant 1/sine, the reciprocal. So my symbols were better.

I didn't like f(x) – that looked to me like f times X. I also didn't like dy/dx-you have a tendency to cancel the d's, so I made a different sign, something like an & sign. For logarithms it was a big L extended to the right, with the thing you take the log of inside, and so on.

I thought my symbols were just as good, if not better, than the regular symbols – it doesn't make any difference what symbols you use but I discovered later that it does make a difference. Once when I was explaining something to another kid in high school, without thinking I started to make these symbols, and he said, "What the hell are those?" I realized then that if I'm going to talk to anybody else, I'll have to use the standard symbols, so I eventually gave up my own symbols.

• There's also the matter of whether $A \subset B$ means the same as $A \subseteq B$ or as $A \subsetneq B$. Commented Sep 24, 2022 at 14:54