Math Notation

Here are some of my controversial ideas math notation. Some of them are quite common, while others are my invention and objected to every time I bring them up.

Square root

This is how you write a square root.\sqrt{\text{This is how you write a square root.}} It has no hook at the end.\sqrt{\text{It has no hook at the end.}}


τ\tau is good. Use it.

Function exponentiation

I prefer to avoid using the notation sin1(x)\sin^{-1}(x), since it's unclear if it refers to arcsin(x)\arcsin(x) or 1sin(x)\frac{1}{\sin(x)}. Similarly, I like to write sin(x)2\sin(x)^2, which is completely inambiguous, rather than sin2(x)\sin^2(x).

Factorial of a real number

There's no reason why the notation x!x! couldn't be used for the factorial of a real number, which many people awkwardly write as Γ(x+1)\Gamma(x + 1).


The notation log(x)\log(x) should not be used since nobody agrees if it's base ee, 1010 or 22. Use ln\ln, log10\log_{10} or lg\lg respectively.

Vertical parentheses

Vertical parentheses are useful to apply a diacritic to a complex expression. For example, ddtLq˙j\frac{\mathrm d}{\mathrm dt}\frac{\partial L}{\partial \dot q_j} can be succintly written as Lq˙j˙\dot{\overgroup{\undergroup{\frac{\partial L}{\partial \dot q_j}}}}.

Extracting the limit

When dealing with limits, it often gets repetitive to write lim\lim at the beginning. For example: limnn+1n+2=limnn(1+1n)n(1+2n)=limn1+1n1+2n=1+01+0=1\lim_{n\to \infty}\frac{n+1}{n+2} = \lim_{n\to \infty}\frac{n {\left(1 + \frac{1}{n}\right)}}{n {\left(1 + \frac{2}{n}\right)}} = \lim_{n\to \infty}\frac{1 + \frac{1}{n}}{1 + \frac{2}{n}} = \frac{1+0}{1+0} = 1 In this case, we can “extract the limit”: limn(n+1n+2=n(1+1n)n(1+2n)=1+1n1+2n)=1+01+0=1\lim_{n\to \infty}{\left(\frac{n+1}{n+2} = \frac{n {\left(1 + \frac{1}{n}\right)}}{n {\left(1 + \frac{2}{n}\right)}} = \frac{1 + \frac{1}{n}}{1 + \frac{2}{n}}\right)} = \frac{1+0}{1+0} = 1 This can also apply to other things.


The notation n\exists_n means “there exist exactly nn”. For example, 1\exists_1 is equivalent to (and harder to overlook than) !\exists!, \exists_\infty means “there exist infinitely many”, and 0\exists_0 is the same as \nexists.

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