When I went back to school I had to take math again, which wasn't my best subject in school. I was determined to figure out why I could understand the material, yet consistently never quite did as well as I thought I should have on tests. So I kept a journal of every mistake I made so I could categorize them and understand them. Most of the mistakes were very specific handwriting mistakes, but they were very particular to how I personally tended to write things. By changing how I wrote a few numbers and letters, I eliminated all of those mistakes. I didn't have to come up with an overly elaborate system that puts way too much thought (and not enough evidence) like in the OP. It was quite surprised that I tended to mistake 7s, qs and 9s (if the loop is too small and there's another character written above it, you can misread some of my more atrocious attempts at writing 9), or that I would fail to properly coil the 6 and make it look like an overly spiraled 0. Meanwhile, I never once mistook a 1 for a 7 and my solution to the letter O is just to never, ever use them in math.
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.
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!
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 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.
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 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.
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.
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.
> 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).
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.
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.
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.
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.
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.
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.
> 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.
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.)
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.
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.
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.
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.
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.
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.
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).
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.
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).
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+.
[+] [-] c3534l|5 years ago|reply
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.
[+] [-] _0ffh|5 years ago|reply
That seems reasonable. My solution was to slash my zeros, to the point that it takes conscious effort not to.
[+] [-] JadeNB|5 years ago|reply
[+] [-] exmadscientist|5 years ago|reply
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....)
[+] [-] unknown|5 years ago|reply
[deleted]
[+] [-] seepel|5 years ago|reply
[+] [-] supernova87a|5 years ago|reply
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.
[+] [-] filmor|5 years ago|reply
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.
[+] [-] wenc|5 years ago|reply
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.
[+] [-] knzhou|5 years ago|reply
0: https://physics.stackexchange.com/questions/488903/advantage... 1: https://physics.stackexchange.com/questions/533523/confusion...
[+] [-] diffeomorphism|5 years ago|reply
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).
[+] [-] patrickthebold|5 years ago|reply
Certain variables pick up a conventional meaning, and one quickly runs out of letters.
[+] [-] mnl|5 years ago|reply
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.
[+] [-] jks|5 years ago|reply
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.
[+] [-] buescher|5 years ago|reply
[+] [-] dan-robertson|5 years ago|reply
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.
[+] [-] rahimnathwani|5 years ago|reply
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
[+] [-] jackpirate|5 years ago|reply
[+] [-] samkater|5 years ago|reply
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.
[+] [-] gjvnq|5 years ago|reply
[+] [-] jacobolus|5 years ago|reply
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.
[+] [-] kaitai|5 years ago|reply
[+] [-] aesthesia|5 years ago|reply
[+] [-] phorkyas82|5 years ago|reply
[+] [-] lonelappde|5 years ago|reply
Save your rote copying for homework time.
[+] [-] xzcvczx|5 years ago|reply
yet the given example looks like a (badly drawn) 8 to me
[+] [-] mkl|5 years ago|reply
[+] [-] teabee89|5 years ago|reply
[+] [-] arketyp|5 years ago|reply
[+] [-] anonytrary|5 years ago|reply
[+] [-] 6gvONxR4sf7o|5 years ago|reply
[+] [-] h-cobordism|5 years ago|reply
[+] [-] mynegation|5 years ago|reply
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.
[+] [-] coopsmgoops|5 years ago|reply
[+] [-] teabee89|5 years ago|reply
[+] [-] analog31|5 years ago|reply
Plus, as good an idea as it sounds, telling my hands to make the same letter the same way twice is hopeless.
[+] [-] bluenose69|5 years ago|reply
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.
[+] [-] djcjr|5 years ago|reply
Crossing the Z resolves the need to loop the 2.
[+] [-] bmn__|5 years ago|reply
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.
[+] [-] unknown|5 years ago|reply
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[+] [-] yanestra|5 years ago|reply
[+] [-] cik2e|5 years ago|reply
[+] [-] Xophmeister|5 years ago|reply
[+] [-] huffmsa|5 years ago|reply
[+] [-] amelius|5 years ago|reply
I already put a loop on g. I put a little "serif" bar on the stem of the q.
[+] [-] throwawaysea|5 years ago|reply
[+] [-] adiabatty|5 years ago|reply
[+] [-] lonelappde|5 years ago|reply