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Tips for mathematical handwriting (2007)
- fitsumbelay 2y agoused to write like this in high school, like my physics and calc teachers. always wondered why the style was so common
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- jovas 2y agoPretty much exactly how I write. (My lower case zeta is prettier)
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- thaumasiotes 2y agoHe doesn't mention that many people have identical handwritten "u" and "n" (both written "и"). 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.
- sfpotter 2y agoThere's nothing wrong with l.
- mkl 2y agoℓ is totally fine.
- adrian_b 2y agoThe original Carolingian lowercase "l" from 1200 years ago had a small right hook at the inferior end, which made it easily distinguishable from other similar letters, like uppercase "I". 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.
- thaumasiotes 2y ago> The original Carolingian lowercase "l" from 1200 years ago had a small right hook at the inferior end, which made it easily distinguishable from other similar letters, like uppercase "I". It might help a little. "Easily distinguishable" is dramatically overselling things. https://upload.wikimedia.org/wikipedia/commons/b/bd/Minuscule_caroline.jpg 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.
- card_zero 2y agoThat reminds me of mimi numinum niuium minimi munium nimium uini muniminum imminui uiui minimum uolunt. https://en.wikipedia.org/wiki/Minim_(palaeography) https://en.wikipedia.org/wiki/Minim_(palaeography) I guess blackletter script was an obstacle to mathematics (and reading).
- sideshowb 2y agoWhile we're here, any suggestions on what works better for writing math to a digital whiteboard? Wacom type tablet with no display, or an ipad type tablet with pen?
- Unearned5161 2y agoI would imagine something where you can see the immediate results of what you're writing under your pen to be better, otherwise you'd have to be facing the board somehow and not the class
- sureglymop 2y agoI use an iPad with pen and goodnotes. I can airplay it to linux with uxplay. All my notes are digital.
- mkl 2y agoA display is much better for this. I recommend Samsung tablets, as the Wacom stylus they use can't awkwardly run out of battery on you in the middle of a class.
- fph 2y agoThis is a very personal choice. A Surface-style tablet pc works best for me. The accuracy difference between an "active" Wacom pen and a glorified piece of plastic is noticeable. Display-less Wacoms have a much steeper learning curve before you produce good handwriting.
- fifilura 2y agoMy tip is to use white paper. No lines no squares, just white. In Sweden you were expected to use paper with squares but it adds a lot of clutter.
- Unearned5161 2y agoI've gone through so many phases on this. But lately, the more writing intensive my math classes get, I've settled on lined paper. Blank paper is too... Blank! And I'm more prone to write big and messy and waste pages (even though it's all digital)...
- annzabelle 2y agoAgreed. I did a pure math degree where most of my classes involved copying down 2-3 pages of axioms/proof per lecture, and I settled on mead letter size college ruled spiral notebooks, and yellow note pads for scratch work. Wide ruled led to too much wasted space, graph paper was visually busy and led to awkwardly spaced letters, dot paper just didn't really work. Smaller paper sizes didn't end up holding enough information per page, spiral binding was best for being able to rip out and toss pages, the perforation was nice for the occasional hand in sheet, and I had no need for a nicer quality paper. 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.
- fujinghg 2y agoI used plain for a few years but it has some problems. I now use faint lined paper. Usually Muji notebooks. I leave a blank line between each statement. The lines are handy if you want to make something readable and well aligned which is fairly important. Scans fine. 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.
- RandomBacon 2y ago
- Unearned5161 2y agonice writeup, I always like taking extra care to view the actual motions a professor uses when writing the symbols on the board as it can sometimes be non-obvious the order or direction to start in. 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!
- dieselgate 2y agoCrossing the Z is a good one. I cross 0 but understand the point with phi. Something that’s not mentioned is y (lowercase) and how they can look like a 4
- gilleain 2y agoI never used to cross my zeros until I was spending some time writing a lot of diagrams that had both O (for oxygen) and 0 (for the numeric label). It got very confusing!
- annzabelle 2y agoI never took a real chem or bio class, but more than half my degree involved classes where 0 and ∅ were extremely frequently used. Thankfully no math or CS professor is stupid enough to use the letter o as a symbol or variable name.
- dieselgate 2y agoDoes ∅ relate to diameter, zero, or something else?
- dvdkon 2y agoThe empty set, I presume. As a computer science student, I have to juggle O, o, 0, and ∅. (capital/lowercase letter, zero, empty set. Fun!
- ddq 2y agoI started crossing Z, 7, 0 and generally writing in as unambiguous a manner as possible in my math homework because I was afraid of the teacher reading it wrong and taking points off.
- mkl 2y agoI always cross my zs in maths to distinguish them from 2s, but in some countries they cross the 2s to distinguish them from zs, so the ambiguities remain.
- dang 2y agoRelated. Others? Tips for Mathematical Handwriting (2007) - https://news.ycombinator.com/item?id=32665846 https://news.ycombinator.com/item?id=32665846 - Aug 2022 (2 comments) Tips for Mathematical Handwriting (2007) - https://news.ycombinator.com/item?id=22983274 https://news.ycombinator.com/item?id=22983274 - April 2020 (76 comments)
- moi2388 2y agoI personally prefer to use dotted paper. 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
- Kwpolska 2y ago> Make a point come out of the top of the p, to distinguish it from a rho. 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.
- mkl 2y agoThere are two other phis with the (non-mathematical) Greek letters earlier, but unfortunately fonts vary in which one they display as \phi or \varphi: φ (U+03CD), ϕ (U+03DC).
- lagrange77 2y ago> Put a hook on the x to distinguish it from a times sign [...] In 3rd-semester calculus and onward you’ll be using the times sign quite often. I wonder if he's talking about the cross product.
- JohnKemeny 2y agoThe \times sign is × in latex. To get • you use \cdot.
- lagrange77 2y agoYes, but it is rare to see \times used for scalar multiplication, isn't it?
- Izkata 2y agoIn the US it's how we're taught as kids in gradeschool. We only switch to dot later on during algebra once "x" shows up as a variable.
- xigoi 2y agoThe × sign is also used for the Hartesian product, which comes up often when building up Lebesgue integration.
- pansa2 2y agoIIRC at university-level we used the wedge symbol [0] for cross product - not sure how common that is. https://en.wikipedia.org/wiki/Wedge_(symbol) https://en.wikipedia.org/wiki/Wedge_(symbol)
- SOTGO 2y agoIn my experience it would not be typical to use a wedge to represent a cross product. Typically a wedge is used to refer to the outer/exterior product, which in three dimensions would correspond to a bivector as opposed to the vector you get from a cross product.
- mturmon 2y agoI used to work in a field that used \Sigma for covariance matrices, and pervasively needed discrete summations which also use \Sigma (and often with an understood index set, so the \Sigma appears without clarifying adornment). 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.
- rhelz 2y agochuckle I'm teaching high school math, and I have students who are jedi knights of making ambiguous mathematical signs. They make their "1"'s with a long top serif, which make it look like it could be either a "1" or a "7". A "4" written with a curly hook on the bottom can be mistaken for an "8", etc etc. 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 ;-)
- boothby 2y agoI'm probably too much of a stickler for high school. If I had students try to trick me with T/F confusion I'd immediately announce a "round towards the wrong answer" policy and watch 'em tidy up right quick. 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.
- madcaptenor 2y agoI remember true-false questions on tests asking us to circle the word TRUE or FALSE, I guess to make that trick not work.
- joshdavham 2y agoIt's incredible to realize how many of the habits mentioned in this post that I've unintentionally picked up while studying applied math. Even after graduation, I still follow a lot of these 'conventions'.
- gnatolf 2y agoMost important to me was something not mentioned here: to make i/j les ambiguous, I but effort into explicitly add the right swoop at the bottom of the i, which allowed for the j to be a straight longish line dotted at the top.
- buescher 2y agoFor Greek I like this: https://www.covingtoninnovations.com/pens/GreekChartLarge.jpg https://www.covingtoninnovations.com/pens/GreekChartLarge.jp... 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.
- bradrn 2y agoMeanwhile I always liked this one, even though it’s not specifically focussed on mathematics: https://foundalis.com/lan/hw/grkhandw.htm https://foundalis.com/lan/hw/grkhandw.htm
- paulluuk 2y agoMostly unrelated, but I wish the field of math would step away from greek letter notatation and just make variable and function names readable as programmers do. I know there are historic reasons, and I'm sure that mathematicians' time is so valuable that they can't be bothered to write more than 1 character, but it's a real barrier to entry in my opinion. def α(β, γ): δ = β ** 2 + γ ** 2 ε = δ ** 0.5 return ε ζ = α(3, 4) print(ζ)
- queuebert 2y agoIt's only a small part of the barrier to entry, though. I could write Einstein's equations out in words, and most people still wouldn't be able to do GR.
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- fxwin 2y agothe vast majority of mathematical equations/terms would become completely unreadable if you replaced single symbols with descriptive terms. You are going to have to internalize what the symbols refer to anyways to understand the formula, and once that is accomplished any additional description is a waste of space and cognitive bandwidth.
- card_zero 2y agoNah, try it, you'll be fine. The real problem is, everybody else will complain, because its unconventional. Then you look foolish.
- jordigh 2y agoWe did try it. We tried it for a couple of millenia. It was much harder to understand and our collective mathematical output as a human species was much slower than it is now. Shorthand is better.
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- lynguist 2y agoI came up with similar rules over decades (I still write sometimes mathematical symbols), and I have to only take care of my "u" vs "n" and "r" vs "v", and write these pairs especially slowly because in my handwriting (especially u and n) look the same. But for all the others I’ve developed similar disambiguations.
- xyproto 2y agoDotted 0 (number) and a regular round O (letter) should also be possible, instead of a round blank 0 (number) and an O (letter) with a top curl.
- mysteria 2y agoFor distinguishing between the lowercase "a" and "α" without switching to cursive you can also use a double storey a.
- jebarker 2y agoPeople that use xi should lose their math credentials
- clircle 2y agoBut i need to signal that i am smart
- dogmatism 2y agolearned this by experience in...1998 thanks for the insight
- queuebert 2y agoThe blog writer has particularly bad handwriting, and maybe that's why he needs these tricks. I've seen a lot of math handwriting, and seriously his is among the worst. Maybe he was drawing the letters with a mouse.
- mkasberg 2y agoI changed the way I wrote my y's (now very curly) and z's (now crossed) in college while studying math after realizing I was making silly mistakes changing y's to x's and z's to 2's.
- bluenose69 2y agoThis is a good resource, and pretty much what I tell students in my classes. I take great care to explain how to write symbols, and I also give multiple pronunciations of the Greek letters. 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.
- RossBencina 2y ago> got derailed when a teacher, in an early grade, said "let x be the unknown" 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?
- brianstrimp 2y ago> I imagine that there are various triggers of early mathematical derailment 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.
- jampekka 2y ago> 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.
- milesrout 2y agoWhy do Americans write x as a cross instead of like )(? 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.
- gerdesj 2y agoThis is all very well but surely all mathematicians/engineers/scientists (n that) have considered this both for themselves and had it mentioned when being taught in relevant subjects, or simply noticed. 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 ...
- abdullahkhalids 2y agoMy rho and p identical. My Z and 2 look fairly close. My students have actually told me to just stop using rho as a symbol. I have probably filled 50-100 notebooks with just mathematical calculations. It's just that my handwriting is bad since childhood.
- gerdesj 2y agop is a vertical downstroke followed by a vertical upstroke and then the loop. rho is a curved upstroke and the loop all in one go. 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.
- wazdra 2y agoI found your comment quite agressive, and a bit out of place. I, for one, do have trouble getting others to read my handwriting (on boards mostly, on paper it's ok). I'm left-handed, and the system of traditional writing education you seem to value so much has left people like me unattended for most of recent history. And it is not the kind of mentality that transpires from your comment that would have helped anything change as --thankfully-- it did.
- ch3cooh 2y agoThis is remarkably thorough! But it's missing one very common case: lowercase a vs. u (or v). If you're trying to make your handwriting more legible, don't skip this case-- figure out a way to be intentional about it because it's so common. :) 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
- layer8 2y ago> Put a hook on the x to distinguish it from a times sign 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.
- lupire 2y agoThe article alludes to this when it mentions Calculus 3 (vector calculus): × is the cross product. • is the dot product. In dimensions higher than 1, they are different.
- RossBencina 2y agoI find it clearer to write dot products using inner product notation 〈,〉rather than \cdot .
- fifilura 2y agoIn Sweden, with scalars, we used vertically centered dot, but even that is pretty uncommon since most of the time, two letters or a letter and a number are mixed and then it is left it out altogether. This made the x problem much smaller.
- rossant 2y agoIn France we use the wedge ∧ for vector cross product.
- f1shy 2y agoSame in Spanish. That gets rid of the problem.
- SAI_Peregrinus 2y agoThen what do you use for the outer product?
- nxobject 2y ago
- dapperdrake 2y agoAnd as always Lawrence A. Chang's Handbook for Spoken Mathematics [0] [0] https://openlibrary.org/books/OL18996361M/Handbook_for_spoken_mathematics_(Larry's_speakeasy https://openlibrary.org/books/OL18996361M/Handbook_for_spoke...)
- jheriko 2y agomissing the cursive theta.
- jks 2y agoI've seen Greek computer scientists write Omega as an underlined O. Much easier than trying to approximate the printed letter Ω but still unambiguous, unless you use underlining to denote something in your formulas.
- powerawq103846 2y agoThat is amazing thanks
- _hao 2y agoI find these quite interesting and I would be very surprised if these were not the actual common way of writing in cursive learned in school? 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.
- croisillon 2y agoplaying oboe is easy for _most_ Frenchs (inventors of the oboe if you didn't know that)
- _hao 2y agoYour snark comment is out of place since if you knew something about Cyrillic you'll know that most letters have 1:1 mapping with their Greek counterparts. And since Bulgarian language does use Cyrillic the jump to Greek is quite short. You could argue that there's a bigger difference in pronunciation of letters between English, German and French which all use the Latin alphabet than between Cyrillic and Greek.
- croisillon 2y agoall i'm arguing is whatever "nation (2 guys) invented" something doesn't imply that every national know it better than everyone else
- tzot 2y agoISTR the origin of Cyril and Methodius (the two monks that created the precursor to the cyrillic alphabet, can't remember the exact name of that alphabet ATM) cannot be pinpointed as either Greek or Bulgarian because no such documentation has been found. They surely were Byzantine, though.
- jan_Inkepa 2y agoA friend related to me that at a maths seminar recently, the speaker "accidentally reused the letter z. and saved himself by erasing the crossbar from one set of z, and naming them zed vs. zee"