Tips for Mathematical Handwriting (2007)
johnkerl.org
johnkerl.org
My advice is actually that mistake journal: it was the single best thing I ever did for a math class. Figure out what mistakes are actually happening and develop a system or habit to avoid that actual mistake.
That seems reasonable. My solution was to slash my zeros, to the point that it takes conscious effort not to.
Primarily nowadays I'm writing on a tablet and need to be very precise to make it readable.
My zero is a continuous line, starting top right and angling sharply left-down after a ~360° oval. That nicely makes it writable in one fluid motion.
My null/empty set is a somewhat smaller circle and a separate slash starting and ending visibly outside the circle.
I've never heard of this before. That's a very cool idea.
One other consideration is to pay attention to your colleagues. Sometimes more specific fields have conventions, and it's probably a good idea to follow them. For example, physicists universally* write a cursive v, not the shape preferred by the article author. You should do that too, if you want to communicate with physicists.
(*At least in my experience as a US almost-PhD in physics. And yes, it does help if you have to, say, calculate the velocity of a neutrino beam....)
I do still feel though, that they are a legacy of when all these fancy old Greek-reading scientists could flex their knowledge of unfamiliar symbols and choose to make it hard for people who couldn't grasp it immediately. ("Now to simplify the equation let's introduce the zeta function into this expression!")
Lowercase Xi and Zeta were the worst. Never mind that they were the rarest and always most complex (visually) of letters to be used -- but also always horribly written by students/teachers, and made it even more incomprehensible.
I have similar feelings about the bra/ket notations in QM.
When the greek letters are used, they usually give you a clear indication that they are "something different" from the things written. It's much either to understand that alpha, beta, gamma, delta are some counterpart to a, b, c, d than if you had to use i, j, k, l (for example).
Lowercase Xi is indeed always written horribly (like a tornado), but at least in my studies, everyone wrote it equally horribly and all in the same way.
Then there are the German algebraists who developed ring theory, for whom Greek letters were not sufficient -- they reached for their Fraktur. The correspondences for certain letters -- like A and S -- can be inscrutable to neophytes.
While I get why notation shouldn't matter, I do find it to be more tiring to read new notation where I don't already have an internal pronunciation.
Also some Greek letters have such a strong conventional association with certain quantities that using Roman letters in their place can be more obfuscating, e.g. ρ = density, Δ = change, η = efficiency, θ = parameter, σ = standard deviation, μ = mean, τ = time constant, etc.
Notation is to some extent a function of a community's conventions.
0: https://physics.stackexchange.com/questions/488903/advantage... 1: https://physics.stackexchange.com/questions/533523/confusion...
Honest question: why does the horribly written matter at all? For all intents and purposes the symbol could be a smiley face or a little drawing of a tree. Indeed, in a freshman course you sometimes have at least one worksheet using trees,cars,stars, apples to make this point. As long as the smiley on page 1 and the smiley on page 3 look the same you are fine.
Same for mu, nu, kappa, rho. They could just be written as C1,C2,C3,C4, but it makes it much easier to read when you use the Greek letters, because you can be reasonably certain that for example rho has something to do with a density (while C4 tells you nothing).
This relates to internal vocalisation, a subject on HN a couple of times recently, as I don't really do that (particularly for equations) then it was never an issue for me. Though at 15 I learnt the Greek alphabet as I was into physics, that probably helped.
People who need to vocalise to read, and don't bother learning (or coming up with) a vocalisation are going to find it very difficult.
Certain variables pick up a conventional meaning, and one quickly runs out of letters.
Notation shouldn't be a problem if it's compact enough. Then the bra-ket one is foolproof and not too many physicists study functional analysis proper, so it makes sense to simply follow Dirac there.
I remember a Greek-born colleague writing the uppercase Omega as just an underlined letter O. I suppose that shape is probably the idea behind the flourishes in the typeset Omega.
The advantage of a xi is that it looks nothing like an x so coordinate transformations (xi = f(x), tau = f(t)), are easier to keep track of.
One habit I did learn in high school is writing x as two 'c' shapes in a mirror image. This is clearly an x rather than a multiplication symbol. I prefer it to the version listed in the article.
I find it hard to distinguish the author's X/χ
What's your subject?
I started crossing z's and doing a few of the other tips (some taught, some natural) in this page when it became necessary. But the pedantic in me really wanted to instill it in my kids from the get-go. I suppose experience (going through the problems of mixing up symbols) is the best teacher for the mathematical handwriting solution.
(And in both cross product and curl, if you substitute a \cdot, then that's an entirely different operation!)
This is an unbelievably ugly handwriting quirk which is sadly prevalent in some European countries.
If you do this the ghosts of dead calligraphers will come puke on your paper.
(Also, the cross product is always a less elegant alternative to the exterior product, but I guess this isn't the place for that rant.)
No, often you will use nothing - simple juxtaposition.
Save your rote copying for homework time.
So my default mode was to copy absolutely everything down (using my own personal set of abbreviations) whilst not necessarily understanding everything. Occasionally I would understand everything in the lecture; occasionally I would understand nothing in the lecture; usually it would be somewhere between the two.
Later I would copy out my notes neatly, but not allowing myself to write anything into my neat copy until I understood it. If it took me a long time to finally see how a step in a proof was justified, I would write the justification into my notes. If I had to revise something to understand the new material, I would write the relevant points I had forgotten into my notes. Hence reading my notes to revise for the exam could be as fast as possible.
To the extent that I perfectly carried out the above system, I gained near to perfect marks. To the extent that I didn't carry out the system, my marks suffered.
Sitting and thinking would only allow so much to sink in, and then I'd have none of my own notes to work off later. Instead I'd furiously transcribe by hand everything the lecturer said, everything written on the blackboard/slides (very often not the same thing) and add in my own notes on top too.
Understanding would come much later, over a much longer period than an hour-long lecture.
Not everyone can.
As someone who left numerous classes early, because confusion & anxiety over the lecture caused vomiting or diarrhea, I promise you I would have given everything to just be able to "pay attention and think."
yet the given example looks like a (badly drawn) 8 to me
[0] See left variant: https://pm1.narvii.com/6331/cb50e106b141735ad714b934eb18ea55...
I do no put loops on O's, but to this day I have a habit of writing a stroke (in forward slash direction) inside zero.
Plus, as good an idea as it sounds, telling my hands to make the same letter the same way twice is hopeless.
Another thing I do is to pronounce symbols in both the Greek and the American ways. That helps them in other classes, because where I teach, the professors are about evenly divided in their pronunciation.
PS. my squiggles are basically identical to those in the article under discussion here.
Crossing the Z resolves the need to loop the 2.
1 should have an upstroke.
0 should have a loop, dot or stroke, not O.
q descender should have a stroke, not a loop.
9 should have a round bottom.
g descender should have a loop.
Latin letters should be written in script form, not block letter form, notably: ℰ, 𝒢, ℐ, 𝒥, ℒ, 𝒴, not E, G, I, J, L, Y.
I already put a loop on g. I put a little "serif" bar on the stem of the q.
I’ve added notes in red, some baselines in blue and some lines at x-height in yellow.
I want to point out a few things:
1. OP misses out the need for script and blackboard bold letters and various symbols. You want them to not be confused with letters. In particular U and set union, epsilon and set membership, and oplus and capital theta.
I think OP’s eta looks wrong. Too much like an m.
One trick to use is that letters can have ascenders and descenders. Numbers can too but I don’t use that. I also incorrectly write a rho as a letter with an ascender and no descender to help to distinguish it from a p. I also try to impart a big curve at the top so you won’t think there could be the top of a straight line hidden in there.
I draw the top of a tau just below x-height while the bar on a t is somewhat above it. This and the rest of the t being higher helps to disambiguate.
Never use an o (“oh”) or omicron and probably not upsilon either (I missed out capital upsilon because OP did too and I didn’t notice). An exception is big/little O notation but I don’t really like it.
I don’t like OP’s q. It looks too much like a double bowled g. I prefer a tick to a loop.
I tend to not have issues with 2 vs z but I do with nu vs v. I kinda dislike using nu because of it. In print I have trouble with v vs u.
I often use variant letters for some Greek letters (in particular my epsilon is what TEX calls varepsilon because it is more like handwriting and less like a print font. Similarly for theta kappa, and phi. I don’t use varpi which looks too much like an omega, cardigan which is only really to go on the end of a word in Greek, varrho because I don’t like it, and I don’t write capital omega the modern Greek way (a horizontal line with a circle above it).
Other tricky symbols are: Angle brackets vs lt/gr. Usually context suffices. Angle brackets vs parens: just don’t rely on this distinction.
Or vs v can and union vs U can usually be solved by context and spacing (you have a bigger space between a term and an operator than between factors or arguments in a term)
A final note is that I tend to write mathematics more upright than my normal handwriting. This helps distinguish inline mathematics from the rest of a sentence.
I looked through some random old notes to see if I could still read my writing and I can. The only thing I struggled with was the script C for conjugate classes because I was missing the context and struggling to guess what ccl stood for. At first I thought it was a script G.
Having written this, I realise I omitted forall, exists, nabla/del, integral sign, partial sign, therefore dots, infinity, approxequal and hbar.
A final note is that spoken names matter to o. “Twiddle” is a much better name than “tilde” because it is more naturally made into a verb. You can say “define the relation twiddle on the naturals such that a twiddles b if ...” and then you can easily use that verb when talking through a proof of eg transitivity.