xigoi

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.}}

Tau

τ\tau is good. Use it.

Function exponentiation

I prefer to avoid using the notation sin⁡−1(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 sin⁡2(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).

Logarithm

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

Vertical parentheses

Vertical parentheses are useful to apply a diacritic to a complex expression. For example, ddt∂L∂q˙j\frac{\mathrm d}{\mathrm dt}\frac{\partial L}{\partial \dot q_j} can be succintly written as ∂L∂q˙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: lim⁡n→∞n+1n+2=lim⁡n→∞n(1+1n)n(1+2n)=lim⁡n→∞1+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”: lim⁡n→∞(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.

Existence

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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