# 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

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

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

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

The notation $\log(x)$ should not be used since nobody agrees if it's base $e$, $10$ or $2$. Use $\ln$, $\log_{10}$ or $\lg$ respectively.
## Vertical parentheses

Vertical parentheses are useful to apply a diacritic to a complex expression. For example, $\frac{\mathrm d}{\mathrm dt}\frac{\partial L}{\partial \dot q_j}$ can be succintly written as $\dot{\overgroup{\undergroup{\frac{\partial L}{\partial \dot q_j}}}}$.
## Extracting the limit

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

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

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