Tips for Mathematical Handwriting (2007)(johnkerl.org)
johnkerl.org
Tips for Mathematical Handwriting (2007)
https://johnkerl.org/doc/ortho/ortho.html
78 comments
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.
I've never heard of this before. That's a very cool idea.
I've never heard of this before. That's a very cool idea.
>my solution to the letter O is just to never, ever use them in math
That seems reasonable. My solution was to slash my zeros, to the point that it takes conscious effort not to.
That seems reasonable. My solution was to slash my zeros, to the point that it takes conscious effort not to.
How do you visually differentiate nulls from zeroes? That's what stopped me from slashing my zeroes, I've considered dotting them, but it's not relevant for me anymore. I slash my 7s though, and form my 1 like that character, I don't miss off the up-tick.
Primarily nowadays I'm writing on a tablet and need to be very precise to make it readable.
Primarily nowadays I'm writing on a tablet and need to be very precise to make it readable.
>How do you visually differentiate nulls from zeroes?
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.
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.
The response from a(nother) mathematician: don't worry too much about this specific advice; it's one guy's convention, not dogma. But do have very definite conventions of your own to which you adhere carefully and faithfully, whether they're this author's or anyone else's, or your own custom blend. As the author mentions, too many undergraduates don't pay attention to distinctions among similar-but-different symbols, but I've seen even professional mathematicians who don't distinguish properly among, for example, w, ϖ, and ω. (I even refereed a paper once whose author didn't distinguish between o and 0 as subscripts!) Don't be one of them!
I agree, I managed to make it through a PhD in physics and I never had a problem. Neither do I recall any of my class mates having a problem with hand written notation.
[deleted]
This, exactly.
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....)
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'm of mixed feelings about all the Greek letters used in physics, math, etc. Eventually (after years of hard learning how to understand the underlying equations physically) I got over the hurdle of them always instantly making any equation more complicated and hard to understand -- and instead seeing them as just constants that could be basically ignored for most of the procedure (in many settings).
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.
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.
I take your point but there are downsides to restricting oneself to only Roman letters. In a few fields I've been a student of, there's a bias against introducing Greek letters. The downside is one does run out of letters. Eventually one needs Greek letters if one wants to not communicate in an amateurish manner -- either that, or one is forced to use multi-letter variables like "Eff", "Density", which are somewhat unwieldy, especially in complex expressions.
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.
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.
> but also always horribly written by students/teachers, and made it even more incomprehensible.
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).
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).
Some (maybe most) people find it easier to remember and comprehend things they can vocalize, so using an inscrutable character, even consistently, can cause problems.
Greek letters have simpler vocalisations than "c4" or whatever, though?!
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.
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.
I think it's mostly an artifact of mathematicians using a single letter for variables. (Which one could also argue against.)
Certain variables pick up a conventional meaning, and one quickly runs out of letters.
Certain variables pick up a conventional meaning, and one quickly runs out of letters.
It's pretty common in physics to use subscripts to label variables, like m-sub-e for mass of an electron, say.
It's just that at some point you run out of letters, having some 35 glyphs more just comes in handy. Very much so if you have indices for different spaces.
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.
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.
To write the lowercase xi and zeta, I imagine that I am looping around the horizontal bars in the corresponding uppercase letters, and they tend to come out approximately right.
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.
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.
I found it disproportionately mindblowing realizing omega and omicron are literally "big o" and "little o."
Zeta takes some practice to make legible. Michael Covington's sheet on how to write Greek letters, while oriented towards linguists and language learners, is very clear and aesthetically appealing:
http://www.covingtoninnovations.com/pens/#GREEK
I had one professor in university who called lowercase xi “squiggle” and wouldn’t make much effort to draw it properly. He would talk about, for example, “d squiggle d t,” and so on.
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.
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.
I don't get the issue. It's just notation. You could "feel" the same way about the integral sign or fractions or about underused letters in your own alphabet.
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.
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.
For my first encounter with lowercase xi, I was sure the professor was taking advantage of his tenure and trying to pass off a scribble as a variable, and that was with my having made a point to familiarize myself with the Greek alphabet the previous semester. Somehow I don't remember him bothering to pronounce it!
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.
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.
Just notation? I don't see that it's a trivial matter, any more than good writing is trivial. I'm reminded of a recent HackerNews thread on Why is Maxwell's theory so hard to understand?
https://news.ycombinator.com/item?id=22812969
https://news.ycombinator.com/item?id=22812969
Drawing bottom up: "'S' (with a slightly flattened bottom), then a 'c', then a little hook"; sometimes find myself even muttering 's,c,hook' as I write one even today.
While I'm with you on the Greek, bra/ket notation is one of the best anywhere in physics. Here [0] are some examples of why it generally looks simpler, and here's [1] a derivation which would be a nightmare to write without it.
0: https://physics.stackexchange.com/questions/488903/advantage... 1: https://physics.stackexchange.com/questions/533523/confusion...
0: https://physics.stackexchange.com/questions/488903/advantage... 1: https://physics.stackexchange.com/questions/533523/confusion...
I wish we'd been taught this in high school.
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
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
Random side-comment - high school is probably the time to do it too. I'm in the middle of teaching my kids how to read/write (the older one) and recognize letters (younger one). I reflexively crossed the "z" when writing out the alphabet the other day and thought to myself, "should I be teaching this?" I decided to erase it and write how they normally see it.
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.
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.
X written as )( is the absolute worst. It always looks like two parenthesis when I'm grading student writing and I have no idea how to parse what they write.
Do you grade that lack of distinction and teach the student a better orthography? Obviously x is a single symbol that's joined in the middle, so with proper orthography it would seem to be better than using a × as an x?
What's your subject?
What's your subject?
In high school my teacher preferred that to distinguish it from the multiplication symbol. These days I write it as a times with a hook at the top-left to distinguish (a la most cursive styles, but disjointed), with the added bonus that I write uvwxyz all with the same top-left hook that helps visually group them together, matching their typical role in mathematical writing.
at some point in math you'll never use x for times/multiplication -- you'll always use a dot, unless you're specifically doing a cross product. And that is a good stage to reach. (I still have trouble using x for multiplication though.)
And the cross product is used with vectors, which are either symbols like u or v, or written with an underscore / arrow on top, so there's zero risk of confusion.
(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.)
(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.)
> you'll always use a dot
No, often you will use nothing - simple juxtaposition.
No, often you will use nothing - simple juxtaposition.
At which point it becomes ambiguous with function application.
Which leads to simple mistakes: http://zerooutoffive.blogspot.com/2008/10/1n-sin-x_21.html
> 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.
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.
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.
Why are people so against it? It literally ressembles x in CMU
In my school we always used a middle dot (· U+00B7) for multiplication since 6th grade.
That works great until the note that \times returns for cross product, curl, and cartesian product.
(And in both cross product and curl, if you substitute a \cdot, then that's an entirely different operation!)
(And in both cross product and curl, if you substitute a \cdot, then that's an entirely different operation!)
It’s not entirely true that context can’t help disambiguate mathematical notation. The situations in which it’s common to use a lowercase omega are quite distinct from those where it’s common to use a lowercase w. Even if both appear in the same formula, they will tend to have very different roles. Good notational choices also try to avoid these ambiguities. You probably shouldn’t use v and nu at the same time, even in typeset math.
Reminds me of my first analysis professor. He managed to explain the proofs while writing them with chalk on the blackboard, all equations nice an clean like the Latex in our scripts - while we struggled to keep up with his pace, was impossible for us to copy it all. He came across a bit dry, overstructured, exact, but also kind and humble - maybe like you imagine a German mathematician.
Don't waste time in lecture taking notes. Pay attention and think.
Save your rote copying for homework time.
Save your rote copying for homework time.
In my good old maths undergraduate days, in many courses the notes were what was written on the blackboard in lectures. No handouts or slides. There might have been a relevant textbook, but its contents would not be exactly the same as the course.
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.
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.
Different things work for different people. The opposite to your approach worked much better for me.
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.
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.
Often people take notes because it actually helps memorise and pay attention what's being said, without any intention to read the notes at all afterwards.
I'm like this, and many I know. I suspect the actual memorising comes when writing, creating the muscle memory. If not written, I can't ensure I'll remember.
> "Pay attention and think"
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."
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."
"Put a loop on the q, to avoid confusion with 9"
yet the given example looks like a (badly drawn) 8 to me
yet the given example looks like a (badly drawn) 8 to me
Loop your l's, and cross your sevens and your z's to avoid confusing them with ones and twos. Exaggerate the differences between your i's and your j's by looping your j's. Curve the base of your t's to avoid confusing them with f's. Math is a lot harder to do when you cannot distinguish your variables from each other and numbers.
I go a little further on my i's. i for the imaginary unit looks very different from i for indexing.
And also make sure to distinguish between your pi's and your n's! I can't count the number of times I've mixed the two up while finding Fourier coefficients.
This is why I exaggerate the pi's top cross (like a capital T with two legs). A lot of people write pi in a way such that the top cross has no dangling edges[0]. I always try to exaggerate the unique features of symbols. One can avoid a lot of these problems by first exhausting the least overlapping symbols.
[0] See left variant: https://pm1.narvii.com/6331/cb50e106b141735ad714b934eb18ea55...
[0] See left variant: https://pm1.narvii.com/6331/cb50e106b141735ad714b934eb18ea55...
"Put a loop on the capital O so it doesn’t look like a zero"
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.
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.
But now it looks like a Theta!
My printed or cursive thetas have horizontal or tilde like bar in the middle.
The most maddening for me was using letter pairs such as u and v, or zeta and xi, that are just handwriting death traps. No matter how good you are, you're going to screw up once, and kiss your derivation goodbye.
Plus, as good an idea as it sounds, telling my hands to make the same letter the same way twice is hopeless.
Plus, as good an idea as it sounds, telling my hands to make the same letter the same way twice is hopeless.
In my teaching, I tend to write a lot of Greek letters on the board. Students from physics and mathematics are comfortable with this and can make out my squiggles easily. To make things easier for students coming from biology, I write out the names of the symbols, the first few times I use them. This helps a lot, because they are not just unfamiliar with the symbols, but also the words I use when I talk about them.
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.
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.
Don't loop the 2 or you'll have the problem with alpha, as they show a few lines later.
Crossing the Z resolves the need to loop the 2.
Crossing the Z resolves the need to loop the 2.
I can tell the design is ad-hoc. There is a science to manual writing, the author appears to be unaware of it (I guess by virtue of being born into an English speaking nation which afaict never saw fit to follow other ones who had already developed relevant standards in the 1960s). I see a couple improvements so that the proposal falls more in line with those standards:
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.
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.
[deleted]
I guess the tip with the "hooked x" is not so perfect because the result can be confused with a chi (which is in fact rather a hooked x than a slashed wave like in the article).
I was a math major and started doing all of this very quickly because I couldn’t read my own work. I would put this in the realm of common sense.
My cursive “x” looks quite similar to my cursive “n”. In my Real and Abstract Analysis exam, IIRC, I remember writing a limit expression, as n goes to infinity, that involved no “x” variable. The professor read it as an “x”, despite the context, and promptly marked it incorrect. It taught me to make my handwriting unambiguous, but I’m still bitter about this, some 15+ years later :P
My biggest gripe in school and now isn't legibility, but wondering if the professor / author is using his Greek letters as some magic, universal constants or just throwing them in as variables because it's "convention". Spend more time hunting down context and the "With {} as {}" section than understanding the equation.
> Put a loop on the q, to avoid confusion with 9
I already put a loop on g. I put a little "serif" bar on the stem of the q.
I already put a loop on g. I put a little "serif" bar on the stem of the q.
Do we need everyone to memorize these guidelines or do we need an alphabet that is designed to be naturally distinguishable despite stylistic differences? I feel like with cursive and handwriting focus going out of style, it isn’t practical to expect convention to be followed more strictly than it already is(n’t).
Your idea is great but impossible.
Since my penmanship isn't all that great, I use quite a few of these guidelines when I'm writing out math problems. At any rate, you can ignore a lot of them if, like most people, all you need is A-Za-z0-9πθ+.
The lowercase zeta isn't right, and the lowercase chi looks just like the xes.
I have somewhat strong opinions on this. I uploaded a similar set of symbols here: https://imgur.com/a/gKJ3KO2
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.
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.
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.