Tips for mathematical handwriting (2007)
johnkerl.org
johnkerl.org
This is counterproductive IMO, because it makes it look like a chi. (The article notes the problem.) That seems more likely to cause issues than the possible ambiguity with the “times” symbol (“×”). If you need a multiplication symbol, use the middle dot (“·”) instead.
No one in the target audience is using × for scalar multiplication.
Funnily enough, when I write for the purpose of math, my numbers are more legible than when I just write down a number. For some reason, I code switch in my handwriting. Kinda obnoxious when I'm filling out forms.
Anyway, I usually use simple cross as x, because it's only very rarely confused with cross product (and you can always parenthesize).
× is the cross product.
• is the dot product.
In dimensions higher than 1, they are different.
This made the x problem much smaller.
(at best you might use a .x subscript for 3D graphics but when did anyone ever do that by hand)
There’s an old joke about “mathematical maturity” meaning being able to write a lowercase zeta, but I took enough classes from professors that would confuse xi and zeta that it probably doesn’t matter that much.
Students with math and physics backgrounds are fine with Greek letters and other mathematical symbols, but the biologists in the class are mystified. They also get terribly confused when I reuse symbols for different purposes.
What I've discovered is that the students who have trouble with mathematical notation and reasoning got derailed when a teacher, in an early grade, said "let x be the unknown". That is a phrase that never comes up in other contexts, and I think it throws them off track. Many find it difficult to get back on-track later, so they memorize and sleep-walk their way through other mathematics classes until the system no longer insists that they take them. A shame, really.
I don't have the experience to know myself, but I imagine that there are various triggers of early mathematical derailment. It would be interesting to see a list of common causes.
Personally I find it hard to internalise canonical notation. Like f and F in probability theory, which is which again?
Probability theory's notation isn't very canonized. The typical usage, f for PDF and F for CDF, is easy to remember from the calculus notation of uppercase being an integral of the lowercase.
I have come to believe that the main trigger by far is the attitude of society. Of parents, family, friends, tv stars, heck even many (non math) teachers. "I wasn't good at math haha" is such a standard phrase to hear, and parents telling their kids that they don't need to worry if they "don't get it" as if it's some mystical topic that only a few gifted can unlock. Plus the uncool stigma attached to "math nerds", folks who simply have an open mind to try to "get it", turns out that it isn't actually that hard. At least when talking high school math or some basic college classes.
I ended up copying it by hand along with every exam and test notes over my entire degree into one little moleskine notebook. its a god send any time I have to remember how to do something or learn something new.
Or make it \varrho (ϱ).
> Keep the slash in the phi vertical; keep the slash in the empty-set symbol slanted.
Again, \varphi (U+1D711 which HN doesn’t seem to like) is easier to distinguish.
The author silently chose \varepsilon in their TeX, but chose to ignore the rest of the variants.
My grammar school math teacher used a very large ascender for the alpha, almost into serif-£-sign-without-the-line-through territory.
In Sweden you were expected to use paper with squares but it adds a lot of clutter.
She explained to me that it was actually a kind of notebook specifically for artists, and that she much preferred it to the normal plain paper notebooks you typically get.
I bought one myself, and I had to agree with her - it was a much 'nicer' experience to write on it - diagrams could be way less cramped, branch out without hitting the edges. The tactile feedback due to the thickness of the paper was also nice - in a way, it felt like the "mechanical keyboard" of paper notebooks.
Never switched back again after that, and many people I work with have found it curious and nice to work with too.
Part of me feels that there may be more than just a gimmick to this. In the way that it's been shown that pen and paper help for understanding versus typing, I wonder if "the niceness of the pen and paper experience" would have an additional tangible positive effect, too
My kids were forced to use heavily marked blue squared paper. They had problems writing and reading. I pointed this out to their mathematics teacher and said that it may be detrimental and got a diatribe of “what do you know”. Such a bad attitude. I had an answer to this which was embarrassing to him.
https://www.amazon.com/Tops-Engineering-Computation-Punched-...
I assume most middle school teachers specify how papers should be organized in binders and labeled with names, dates, etc; but below was my method I developed halfway through college:
Top Margin use: left - name, center - class, right - date
One binder per semester; a divider for each class; handouts, tests, etc were hole-punched and placed with my notes in chronological order.
If my notes didn't make sense as I was copying down what the professor did, I would rewrite my notes later that day while figuring the problem-solving process out and making the notes/arithmetic easier to follow.
This method also applied to non-math-related classes.
Blank paper is too... Blank! And I'm more prone to write big and messy and waste pages (even though it's all digital)...
Also I always kept Pentel Twist Erase III mechanical pencils with 0.5 mm lead, Hagoromo chalk, and a 4 color set of chunky expo markers in my bag.
Grid or lined, placed underneath thinner blank paper (heavier paper won't let the lines or grid show through as easily, if at all). Keeps the final presentation neat while giving the structure you may need to keep things aligned.
Here is an example, 4 different dot sizes:
http://trondal.com/p1.pdf http://trondal.com/p2.pdf http://trondal.com/p3.pdf http://trondal.com/p4.pdf
You need to print them to see which one is suitable.
This is especially useful when maintaining two margins to write in as well as for indenting, blockauotes, or sometimes maintaining parallel lists or columns.
But if you just slightly misalign your lines it will look even uglier.
You'll just have to learn to draw straight lines. Or trade one benefit for the other.
Another note, it was actually surprisingly difficult even to get a hold of blank paper notepads in Sweden in the 90s. At least cheap enough. The cheap ones were all lines or squares.
You get a nice mix of benefits of blank paper and graph paper.
When I was growing up in Bulgaria my first 7 (I would write the 7 crossed by default for example, the Z as well) grades were in a school where we learned German (and Russian as a secondary foreign language) and I remember distinctly handwriting practice in German using more or less the same ways outlined in this article. Is this way of writing cursive not common in the US/UK?
Reading Greek is easy for _most_ Bulgarians (inventors of Cyrillic if you didn't know that) as you can imagine. Them being a geographical neighbor and the close historical ties etc.
With the version I learned in the US in the late 90s, mostly not: https://en.wikipedia.org/wiki/D%27Nealian#/media/File:D'Neal...
The "z" in particular with the crossbar wasn't a thing for any version of writing I learned, and they're missing the leading/trailing tails for most of their versions of the lowercase letters. Even in printing - I learned to write an "l" as a vertical line with a tail to the right, so it's different from 1 or I without a cursive loop.
Funny enough in "digits" section: "Put a loop on the 2 so it doesn’t look like a z" - The looped version is identical to a cursive uppercase "Q".
[1]: https://en.wikipedia.org/wiki/Glagolitic_script
[2]: https://en.wikipedia.org/wiki/Preslav_Literary_School
EDIT: Sorry, I misread what you wrote. It's late and it's been a long day. You weren't saying the brothers created Cyrillic. As for whether they were Greek/Bulgarian I cannot say. I've read different opinions on that throughout the years. Definitely Byzantine, but anything else I cannot say.
Tips for Mathematical Handwriting (2007) - https://news.ycombinator.com/item?id=32665846 - Aug 2022 (2 comments)
Tips for Mathematical Handwriting (2007) - https://news.ycombinator.com/item?id=22983274 - April 2020 (76 comments)
As a computer science student, I have to juggle O, o, 0, and ∅. (capital/lowercase letter, zero, empty set. Fun!
I can't say I ever had trouble with y and 4 looking the same. I use the open 4 but the y has one less angle and it sits lower. If anything a y is more likely to look like a v if you can't see the descender clearly
I ended up writing my discrete summation \Sigma's with a little serif on the bottom, and ordinary \Sigma's as in OP, with 4 quick back-and-forth strokes.
I wonder if he's talking about the cross product.
This sequence is followed by differential equations courses for the physicists, engineers, and most mathematics majors. Then every college has a mechanism to generate mathematical maturity in their first or second year pure math majors - sometimes it's a proof focused version of linear algebra, sometimes it's a specific Introduction to Proofs course, sometimes it's a discrete math/set theory course, sometimes it's groups/rings or real analysis but slowed down a bit at first. This gates the upper level pure mathematics courses, where most programs require one semester each of algebra and analysis and some number of elective courses.
A general definition of continuity typically doesn't arise until a topology course or a second semester real analysis course. It is entirely possible to graduate from most mathematics bachelor's programs in the US without taking either of those courses.
I have always written and seen written the variable x as )( and I was very surprised when I met Americans and they didn't do this.
[0] https://openlibrary.org/books/OL18996361M/Handbook_for_spoke...)
This barely matters at all; context will tell you what symbol is being used, just like it does in prose, and most material is typeset.
It really is a very bad idea to use lowercase Ls, though.
Unfortunately after the invention of printing there has been a bad fashion in serif typefaces that made lowercase "l' too similar with other letters, by replacing the traditional hook with a serif identical to the serifs of all letters that end on the baseline, so that the only difference from uppercase "I" remained in a triangular left serif added to the upper end.
Even worse, in the 19th century, after sans-serif typefaces became popular, most of them have simplified lowercase "l" even more, making it almost identical with uppercase "I".
Nevertheless, while many of the typefaces used today follow the bad styles of serif or sans-serif designs with hard to distinguish lowercase "l", there exists a decent number of typefaces which have returned to the original form of lowercase "l", with a right hook at the lower end, which is easy to differentiate from any other letters. I am using almost only this kind of typefaces. Examples are Palatino Sans and FF Meta, but there are many others. Also all recent monospace typefaces intended for programming have such lowerspace "l" (being a narrow letter, in monospace typefaces lowercase "l" also has a left slab serif at the upper end, to expand its width).
In handwriting, the best solution for lowercase "l" is not to make it cursive like suggested in the parent article, but to make it with a low right hook, like in its original ancient form.
It might help a little. "Easily distinguishable" is dramatically overselling things.
https://upload.wikimedia.org/wikipedia/commons/b/bd/Minuscul...
I, capital and lowercase, also has a serif hooking to the right at the bottom of the stroke. Look at the first line beginning with a big red I.
It says "Iterum autem pilatus locutus ẽ. ad illos volens".
The line above it says "Et homicidium missus In carcerem;".
How much difference do you see between the capital I in "In carcerem" and the lowercase L in "pilatus"?
It's easy to distinguish lowercase L from lowercase i because L is taller. Capital I is only slightly different.
(I should note that wikipedia's page on Carolingian minuscule also includes an image of a page of written Slovene. I couldn't really use that as an example, because without any knowledge of Slovene I can't reliably identify the letters. But I think that goes to show that the alphabet by itself doesn't reach the level of perfect unambiguity.)
I'm interested to see that all Ses look the same. At some point we got word-final "s", but this was clearly not that point.
The serif at the lower end of capital "I" is bilateral symmetrical, besides having a flat termination, so it looks very different from the asymmetric and curved right hook on lowercase "l".
Also in my handwriting, I write lowercase "l" with a right hook and uppercase "I" with serifs, making them impossible to confuse.
No, look at the capital "I" that I indicated in my comment. It has neither of those qualities.
Despite this defect of this particular letter, which is not a frequently seen defect, I still cannot see how the right serif could be confused with a right hook. The letters would have to be much more degraded for the difference between an orthogonal serif and a curved hook to become invisible.
I agree that the differences between the 6 narrow alphanumeric characters "1IJijl" are small so they are more likely to disappear when the letters are partially erased, but when the letters are intact and they have been written with the right distinct shapes they can still be distinguished easily, in contrast with some typefaces (usually sans-serif typefaces) where they are intentionally made more similar than they were traditionally, which has been a very bad idea, so such typefaces (e.g. Helvetica or Arial) should be avoided.
https://en.wikipedia.org/wiki/Minim_(palaeography)
I guess blackletter script was an obstacle to mathematics (and reading).
Not to mention a glyph which is maximally ambiguous between "T" and "F" so that when grading true/false questions, could stand in for either :-)
If the students would put half as much effort into learning the material as they do in trying to trick teachers they would get straight A's ;-)
Some european students threw me when I started grading in university though. Their 1s look like my 7s as you describe, but they cross their 7s to disambiguate.
And I write every Latin letter in its capital form, however actual capitals twice as large. Put a dot in zero.
That way everything looks distinct
Here's mine:
My u's have the little curved tail to distinguish them clearly from v's. And then, to further differentiate a vs. u, I deviate from the norms for fonts and make my handwritten lowercase a look like a half-height uppercase a with a curved top.
Honestly I only really use about 1/4 of the tips in this cheat sheet. But, the a vs. u trick is where I actually make the most impact by being intentional about my handwritten 'font' (with l/i vs 1, t vs. +, and x vs. × being the runner ups).
And the main reason I don't use the 'a' that fonts like this HN font use is that can look a lot like a 2 with the loop when drawn quickly (and I draw my 2's with loops so that they don't look like uncrossed z's or equal signs with an 'I failed to pick up my pen completely ' diagonal) ;-P
But for all the others I’ve developed similar disambiguations.
would also love tips on making right curly brackets not be the bane of my existence. I don't get it, I write a right one basically for every left and yet they feel so different!
thanks for the insight
def α(β, γ):
δ = β ** 2 + γ ** 2
ε = δ ** 0.5
return ε
ζ = α(3, 4)
print(ζ)As for your code, urgh. A mathematician would carefully define what the function and operands are and the domain of each. More, and I'm vasty simplifying here as I don't have LaTeX available. I don't know what your code does so I've invented some words.
Let f(x,y) represent the rate of change of doodads where x is the number of doodads and y is the amount of doodads per gronk.
f(x,y) = ( x^2 + y^2 ) ^ (1/2)
The code is unclear because it has single-letter variables instead of names, exacerbated by being in Greek, which mathematicians reached for because they were using single-letter variables, and ran out of them, and weren't bold or imaginative enough to break from convention. Then they used up the best of the Greek letters too and moved on to things like Gothic. This is stupid.
k'(x)=(g(x)f'(x)-f(x)g'(x))/(g(x)^2
quotient'(expression)=(left_expression(expression)right_expression'(expression)-right_expression(expression)left_expression'(expression))/(left_expression(expression)^2
Surprisingly I've just gone through the last 30 pages of mathematical work I've done and there's a couple of π in ∫ there and that's it.
To have it his way, you'd right:
(derivative(right_expression))(expression)
and so on.
Clear as mud!
Perhaps he's a LISP programmer and has some spare brackets floating around :-)
You try handwriting all of your code and let's see how long until you start abbreviating everything.
Mathematical notation is all abbreviations. We used to write mathematics without abbreviations. It was absolutely horrid. Try reading some 13th century mathematics, translated to your language (e.g. Fibonacci https://archive.org/details/laurence-sigler-fibonaccis-liber... ), and see how much of it you understand without the benefit of symbolic notation. We would even write aaa instead of a^3.
The point with mathematical notation is that it can all be sounded out and it's extremely general and abstract. Generally, x is not a measurement, a quantity with a unit, a meaningful anything. It's just a number, and x is a better name than front_server_count or whatever thing you're programming about.
ls, man, wc, ps, grep, …
instead of
list, manual, word count, process(es?), <actually, what the hell does that stand for again>, ….
You get used to it, and then it’s much quicker. And being able to write things down quickly is very important when you’re in flow.
Set theory has aleph א and beth ℶ which are refreshingly not Greek but Hebrew letters.
The real problem is, everybody else will complain, because its unconventional. Then you look foolish.
Shorthand is better.
You know when you go to a foreign country, ask for a particular thing in a restaurant proudly in their language that you picked up on Duolingo the week before and the waiter starts talking to you quickly in their language and you lose it completely.
We are just that waiter. You didn't learn the language or get the prerequisite skills. So do that or don't bother. It's not easy and there are no shortcuts.
So much more readable! /s
Who on earth screws up writing rho with a p? Yes, I know that someone will but they need to write the symbol properly.
Zed and two? RLY? (Crossing your Z is a Germanic thing for me, which is why I do it - I grew up in W Germany). Despite that, you would be in no doubt if I wrote a 2 or Z, because I know how to use a fucking pen and I am able to write.
To be fair, I come from an age where cursive means something!
We are heading into an era where being able to use a pen or pencil will border on arcane skills. Calligraphy isn't hard nor is ensuring you get your message across.
Cross your zeros but leave your Os alone (or dot them if you like - I do sometimes). Cross your zeds if you like. Write Greek letters as they should be - a lower case rho starts from the bottom right and is written in a single stroke - it never looks like a "p".
x (eks) is two curves and not a diagonal cross. Multiplication is . or adjacency of symbols, however it might be x depending on context.
The given advice is don't slash the zero. I disagree. Phi is fatter than tall - its not hard to be precise, which I would hope a mathematician might manage.
In the end can you write or not? If not, use a keyboard. Nothing wrong with that per se ...
I have probably filled 50-100 notebooks with just mathematical calculations. It's just that my handwriting is bad since childhood.
Z - try putting hooks on the head and tail of the letter - I'd stick with crossing your zeds/zees.
Funky calligraphy is not for everyone - you can always explain things with prose. The important thing is the message.
And it is not the kind of mentality that transpires from your comment that would have helped anything change as --thankfully-- it did.
Crossing z's was something I picked up naturally in the states in High School Algebra, for the reasons listed. It's not a matter of "can you write or not"—that a very weirdly dismissive line of argument—it's a matter of "how can you provide the most information redundancy so that when you're scribbling notes quickly you and others can decipher them later".
This advice in this article dates to 2007, so it's not a modern cope for the terrible handwriting of modern college kids like you seem to think.
> Who on earth screws up writing rho with a p? Yes, I know that someone will but they need to write the symbol properly.
That's literally the point of this piece: to provide help to people who are new to these symbols so they can both recognize the difference between them and write them distinctly.
Are you somehow under the impression that every just intuitively picks up the distinctions between these symbols?