Aaron Kriegman /

notation

There is a lot of room for improvement in the notation we use for math. Here are some conventions that I use in my own work. Some of these are very simple and don’t conflict with existing notation, so they can be used without ambiguity. Others are deeper changes. I believe that many of these are a large enough improvements that it’s worth the friction of having to explain what it means. It’s especially worth it in your scratchwork, when no one has to understand your notation but you.

I use \(-\) as shorthand for \(-1\). So for example \(\frac12=2^-\). The existing uses of \(-\) are as a binary operator for subtraction and a unary operator for negation, so this does not conflict because here it’s nullary.

Similarly I write \(/\) for \(2^-\) and \(\backslash\) for \(-2^-\). So for example \(\frac{1}{\sqrt{2}}\) is \(2^\backslash\). Like with \(-\), \(/\) is currently only used as a binary operator, so using it nullarily here does not conflict.

Instead of calling the number \(/\) “a half” or “one half”, you should just call it “half”. \(\backslash\) is then “minus half”. The latin root “sesqui-” means one and half, and one and half is a pretty useful number, so it may be worth introducing a word for it, maybe “sesq”.

\(2\pi i\) is too important a number to have a three symbol name. People have tried to introduce \(\tau\) for \(2\pi\) but this has not caught on, perhaps because \(\tau\) has too many other uses. I sometimes use \(\sout{\pi}\) instead, and \(\dot\tau\) or \(\dot\pi\) for \(2\pi i\).

Instead of using \(\hbar = h/\sout\pi\), I recommend \(\dot h = h/\dot\pi\), since \(\hbar\) always appears with a factor of \(\pm i\) anyways. If we take the perspective that radians are imaginary by nature then this is well motivated.

One idea I’m toying with is quantifier annotations for sets and multivalued expressions. Instead of saying things like “Let \(\log\) be the principal branch of the logarithm, defined on the complement of the nonpositive real axis with \(-\sout\pi/2 < \operatorname{Im} \log z < \sout\pi/2\),” you can just let \(\log\) be multivalued and use it like \(\log^\forall 1 = \dot\pi\Z^\exists\). When \(\forall\) and \(\exists\) are used in the same statement, the \(\forall\) s come first.

It’s probably best to never use fractions or radicals and just use exponents instead. This is a realization that came to me in the final minutes of my Numerical Relativity exam. I had a massive algebraic bash ahead of me, and I wasn’t sure if I’d be able to finish it in time. In that moment it suddenly became clear what I had to do.

Fractions, radicals, and exponents all interact with each other in different ways, but if you only use exponents then you have one set of rules. Fractions and radicals are a special tool that you get to use when your exponent happens to be \(-\) or \(n^-\), and if you are not so lucky to be on that blessed path then you have to fall back to the general case anyways. It’s better to not have a special case to begin with.

But if you took this convention to its extreme then you’d end up writing things like \(2^{-2^-}\). I think the right compromise is to use shorthand like \(/\) in tight spaces like exponents or matrix entries, and use \(2^-\) otherwise. But maybe it’s better to get comfortable with exponent towers.

One reason why fractions and radicals are more convenient than exponents is that they also do grouping. Usually if you want to switch them out for exponents then you have to add parentheses to do the grouping instead. For example, the quadratic formula becomes

$$ (-b\pm(b^2–4ac)^/)(2a)^- $$

I’ve written before about how much I hate parentheses. To handwrite the above expression you have to either plan ahead or backtrack, and you shouldn’t have to do that. The best alternative I have found is to use overparentheses:

$$ \overgroup{-b \pm \overgroup{b^2–4ac}^/ } \overgroup{2a}^- $$

I’m using \overgroup because it’s available in Katex, but an arc shape might be better. Newton used a similar notation \(\angl{4ac}\), which might be the move tbh¹. But it looks like the Katex version \angl is somewhat limited, it doesn’t let you put exponents inside.

It is standard to write expressions in order from simple to complex. Eg we write \(-2x\) and not \(x2(-)\). However there are some good reasons to write the complex parts first. The complex parts often contextualize the simple parts. It is also often easiest to figure out the simple parts last. Often you will have an expression with three or four factors of \(-\), with each factor coming from a different part of the expression, so if you want to figure out the sign before writing down the rest of the expression then you have to hold the whole expression in your head.

We can try writing the quadratic formula from complex to simple:

$$ \overgroup{b(-) + \overgroup{b^2+ac4(-)}^/(\pm)} \overgroup{a2}^- $$

This is a little clunky. We can boldly drop the parentheses around \(-\) :

$$ \overgroup{b- + \overgroup{b^2+ac4-}^/\pm} \overgroup{a2}^- $$

This is ambiguous since \(a-+\ b\) is commonly understood to mean \(a-b\), even though it’s pretty rare to write this. So it may be best if we overrode this. Really it would be best if we flipped subtraction, so that the length of the interval from \(a\) to \(b\) was written \(a-b\). But that’s a whole other conversation that the world’s not ready for.

Underparentheses may be better for handwriting since space is more limited above your expression than below. But overparentheses make it more clear whether exponents are inside or outside.

$$ \undergroup{-b \pm \undergroup{b^2–4ac}^/} \undergroup{2a}^- $$

The big game changer though is Penrose diagrams, which I won’t go into here because they’re less widely applicable and they can’t be Katexed.