[HN Gopher] Tips for mathematical handwriting (2007)
___________________________________________________________________
Tips for mathematical handwriting (2007)
Author : susam
Score : 270 points
Date : 2025-02-08 19:20 UTC (1 days ago)
(HTM) web link (johnkerl.org)
(TXT) w3m dump (johnkerl.org)
| jovas wrote:
| Pretty much exactly how I write. (My lower case zeta is prettier)
| thaumasiotes wrote:
| He doesn't mention that many people have identical handwritten
| "u" and "n" (both written "i").
|
| 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 wrote:
| There's nothing wrong with l.
| mkl wrote:
| is totally fine.
| johongo wrote:
| It sticks out. I prefer a subtle hook to the right at the
| bottom.
| layer8 wrote:
| That's awfully close to an iota though.
| Timwi wrote:
| You say that like it's a bad thing. Sticking out is good
| for distinctiveness and hence readability.
| adrian_b wrote:
| 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".
|
| 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 wrote:
| > 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/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 e. 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.
| adrian_b wrote:
| Perhaps this is because I am used to it, but for me the
| lowercase "l"s from your example are impossible to confuse
| with any other letter. For me the hooks are very visible
| and clearly distinct from any other kind of stroke
| termination.
|
| 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.
| thaumasiotes wrote:
| > The serif at the lower end of capital "I" is bilateral
| symmetrical, besides having a flat termination
|
| No, look at the capital "I" that I indicated in my
| comment. It has neither of those qualities.
| adrian_b wrote:
| It is true that unlike the capital "I" that starts the
| row immediately below, your "I" happens to have a shorter
| left serif, either because it has become erased due to
| the age of the manuscript or because it has been poorly
| written originally.
|
| 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.
| card_zero wrote:
| That reminds me of _mimi numinum niuium minimi munium nimium
| uini muniminum imminui uiui minimum uolunt._
|
| https://en.wikipedia.org/wiki/Minim_(palaeography)
|
| I guess blackletter script was an obstacle to mathematics (and
| reading).
| sideshowb wrote:
| While 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 wrote:
| I 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 wrote:
| I use an iPad with pen and goodnotes. I can airplay it to linux
| with uxplay. All my notes are digital.
| mkl wrote:
| A 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 wrote:
| This 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 wrote:
| My 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 wrote:
| I'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 wrote:
| Agreed. 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 wrote:
| I 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 wrote:
| Engineering Paper FTW (for the win):
|
| 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.
| ramenbytes wrote:
| I love engineering paper too! My weapon of choice is this
| pad: https://www.rspaperproducts.com/products/95182/ I have
| not tried the Tops yet, I think because I saw somewhere it
| was thinner? I like the slightly thicker paper the buff
| pads come with, and it plays nice with my fountain pens.
| fujinghg wrote:
| Worth nothing you don't need squared for mathematics. Not
| even slightly. Most of the stuff is written in English with
| a few lines of other stuff between.
| RandomBacon wrote:
| I don't think I needed squares for any of my notes.
| Graphes that I drew were just approximate. The squares
| are great for indentations, lists, tables, etc. My notes
| always looked sharper on engineering paper than regular
| ruled paper.
| Jtsummers wrote:
| Pencil boards can help. An example:
| https://www.jetpens.com/Hobonichi-Accessory-Pencil-Board-for...
|
| 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.
| sn9 wrote:
| You can always do dot grid paper.
| fifilura wrote:
| Not in Sweden. In the 90s...
| fifilura wrote:
| Also, when I look at the prices on amazon.se it looks like
| they cost around 5-10c each, a bit hefty for a student.
|
| 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.
| sn9 wrote:
| Even today for students constrained financially, they can
| print out a dot grid on blank printer paper, perhaps in a
| faint blue or grey to recreate the effect.
|
| You get a nice mix of benefits of blank paper and graph
| paper.
| lynguist wrote:
| I switched to plain white when I was ca 20 years old and never
| looked back. I need to use plain white since then and I am so
| used to it that I find lines or grids slightly offensive to me
| and it throws me off from layouting. You can get used to
| anything.
| grandempire wrote:
| Yes. the lines become distracting and constraining.
| a-dub wrote:
| that's right with mechanical pencils written on a proper
| writing surface with plenty of arm/elbow space with real
| erasers for completely removing noise from mistakes.
| pizza wrote:
| I was once taking a real analysis class and there was a very
| gifted student in my class. She pulled out her notebook one day
| at a study session and I noticed it was kind of unusual - the
| pages were very large, and made of a somewhat thicker material,
| with a slightly rougher texture. Ink also seems to set onto its
| surface slightly nicer, and it doesn't really bleed through
| onto the other side either.
|
| 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
| rossant wrote:
| Interesting. Do you have links, with pictures,
| specifications, purchase options etc?
| pizza wrote:
| I bought some in person at a store but try looking up 9x12
| mixed media heavyweight art notebooks. Typically they're
| spiral-bound, too.
| ykonstant wrote:
| This is famously how Maryam Mirzakhani worked as well. Huge
| thick A2 sheets where she doodled and did computations; she
| said she disliked the cramped style of normal notebooks.
| ziml77 wrote:
| Squares are pointless unless you're drawing graphs, but lines
| are very handy to have. Without lines my writing moves on an
| angle and the size of everything becomes inconsistent and
| usually too large.
| jszymborski wrote:
| I like squares because they allow me to align things
| vertically.
|
| This is especially useful when maintaining two margins to
| write in as well as for indenting, blockauotes, or sometimes
| maintaining parallel lists or columns.
| fifilura wrote:
| Yeah that is quite obviously the strength of a paper with
| rulers.
|
| 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.
| drsopp wrote:
| I agree. However, I have had some good experience using weakly
| dotted paper for math. Then it is easier to draw graphs by
| hand.
|
| 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.
| Unearned5161 wrote:
| nice 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 wrote:
| Crossing 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 wrote:
| I 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 wrote:
| I 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 wrote:
| Does [?] relate to diameter, zero, or something else?
| dvdkon wrote:
| The 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 wrote:
| I 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 wrote:
| I 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.
| ziml77 wrote:
| I started crossing the Z in college because I was tired of
| messing things up because I confused a Z for a 2. I only
| crossed the 0 when Os showed up and it could be confusing.
| Similarly my 1s would get the hat and feet if I felt like it
| wasn't clear what it was.
|
| 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
| dang wrote:
| Related. Others?
|
| _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)
| moi2388 wrote:
| I 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 wrote:
| > Make a point come out of the top of the p, to distinguish it
| from a rho.
|
| Or make it \varrho (r).
|
| > 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-PS-sign-without-the-line-through
| territory.
| mkl wrote:
| There are two other phis with the (non-mathematical) Greek
| letters earlier, but unfortunately fonts vary in which one they
| display as \phi or \varphi: ph (U+03CD), ph (U+03DC).
| lagrange77 wrote:
| > 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 wrote:
| The \times sign is x in latex. To get * you use \cdot.
| lagrange77 wrote:
| Yes, but it is rare to see \times used for scalar
| multiplication, isn't it?
| xigoi wrote:
| The x sign is also used for the Hartesian product, which comes
| up often when building up Lebesgue integration.
| pansa2 wrote:
| IIRC 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)
| SOTGO wrote:
| In 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.
| pansa2 wrote:
| Wikipedia says it's more common in physics, and we mostly
| used it in that context (e.g. fluid mechanics) rather than
| pure math. It was pronounced "veck", IIRC.
| wazdra wrote:
| \times denotes the cartesian product (to my knowledge)
| universally. If 3rd-semester calculus is when you introduce a
| general definition of continuity (I am not from the US,
| wouldn't know how the programs usually work there) on either
| metric or topological spaces, the cartesian product starts to
| appear quite a lot I guess ?
| annzabelle wrote:
| Typically in the US, the calculus sequence is one semester
| differentiation, one semester integration, and a third
| semester of three dimensional and vector calculus. The x
| symbol is used a lot for vector cross products in the third
| semester. Typically these courses don't involve proofs.
| Serious students frequently take a portion of this sequence
| +/- matrix algebra in high school as AP courses or dual
| enrollment where the school cooperates with a local college
| to share their exams and get official credit. They are
| technically considered to be college level courses in the US.
| I think a lot of the content in them is covered in A level
| further maths or IB HL math or whatever your local equivalent
| is.
|
| 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.
| mturmon wrote:
| I 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 wrote:
| _chuckle_ 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 wrote:
| I'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 wrote:
| I remember true-false questions on tests asking us to circle
| the word TRUE or FALSE, I guess to make that trick not work.
| joshdavham wrote:
| It'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 wrote:
| Most 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 wrote:
| For Greek I like this:
| 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 wrote:
| Meanwhile I always liked this one, even though it's not
| specifically focussed on mathematics:
| https://foundalis.com/lan/hw/grkhandw.htm
| paulluuk wrote:
| Mostly 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
| a(b, g): d = b ** 2 + g ** 2 e = d ** 0.5
| return e z = a(3, 4) print(z)
| fxwin wrote:
| the 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 wrote:
| Nah, try it, you'll be fine.
|
| The real problem is, everybody else will complain, because
| its unconventional. Then you look foolish.
| jordigh wrote:
| We 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.
| card_zero wrote:
| No, I don't believe you. Mathematicians love
| inaccessibility.
| fujinghg wrote:
| It's not that.
|
| 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.
| xigoi wrote:
| Consider the formula (a+b)2 = a2 + 2ab + b2. How would you
| name the variables to make it more "readable"?
| fxwin wrote:
| Even if i agreed with you that full words/phrases were
| easier to parse, any gains made here are easily offset by
| how much harder and laborious it is to manipulate them on
| paper
| fujinghg wrote:
| The use of the Greek alphabet is mostly driven by convention.
| You get used to it. They aren't just sprayed around randomly
| wherever you look.
|
| 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)
|
| That's pretty tight and clearer than the code is.
| card_zero wrote:
| Maybe tight is harmful.
|
| 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._
| xigoi wrote:
| How would you name the variables in the code to make it
| clearer?
| fujinghg wrote:
| No they didn't use them because they ran out. You know
| nothing about mathematics clearly. There are conventions to
| keep it concise. Some definitions of simple things would be
| unreadable if you used longer names. Consider the quotient
| rule: k'(x)=(g(x)f'(x)-f(x)g'(x))/(g(x)^2
|
| Or in your world apparently quotient'(exp
| ression)=(left_expression(expression)right_expression'(expr
| ession)-right_expression(expression)left_expression'(expres
| sion))/(left_expression(expression)^2
|
| Yeah, no.
|
| Surprisingly I've just gone through the last 30 pages of
| mathematical work I've done and there's a couple of p in
| [?] there and that's it.
| BeetleB wrote:
| I'm sorry, but you used ' to denote differentiation.
|
| To have it his way, you'd right:
|
| (derivative(right_expression))(expression)
|
| and so on.
|
| Clear as mud!
| fujinghg wrote:
| Good point. Urgh.
|
| Perhaps he's a LISP programmer and has some spare
| brackets floating around :-)
| queuebert wrote:
| It'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.
| xanderlewis wrote:
| It's for the same reason that almost everything in Unix is like
|
| 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.
| The_suffocated wrote:
| Personally I don't find greek letters problematic. In contrast,
| I was traumatised by upper-case Fraktur font in older abstract
| algebra texts.
| jks wrote:
| Fraktur is still used in model theory, but I don't believe
| that is easier to write than A' or A or something like that.
|
| Set theory has aleph and beth which are refreshingly not
| Greek but Hebrew letters.
| jks wrote:
| Heh, U+1D504 MATHEMATICAL FRAKTUR CAPITAL A seems to have
| been filtered out of my comment, perhaps because it's
| outside the BMP.
| layer8 wrote:
| Handwriting doesn't have autocomplete.
| jordigh wrote:
| > they can't be bothered to write more than 1 character
|
| 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.
| annzabelle wrote:
| A massive amount of day to day pure mathematics work is still
| done by hand, on paper, whiteboard, or even chalkboard (and
| there's a preferred brand of chalk). Of course it will all be
| typeset before sharing, but mathematicians typically think by
| writing by hand, not think by typing.
| lynguist wrote:
| I 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 wrote:
| Dotted 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.
| xigoi wrote:
| In math, zeros are more common than Os by several orders of
| magnitude, so it makes sense to use the lower-effort symbol for
| them.
| xyproto wrote:
| You are right! I didn't think of that.
| mysteria wrote:
| For distinguishing between the lowercase "a" and "a" without
| switching to cursive you can also use a double storey a.
| jebarker wrote:
| People that use xi should lose their math credentials
| clircle wrote:
| But i need to signal that i am smart
| dogmatism wrote:
| learned this by experience in...1998
|
| thanks for the insight
| queuebert wrote:
| The 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 wrote:
| I 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 wrote:
| This 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 wrote:
| > 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 wrote:
| > 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 wrote:
| > 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.
| manvillej wrote:
| I found this little pocket mathematics notation book when I
| first studying undergrad in this used book store in Boston. it
| literally carried me through Calc, Linalg, stats, dynamic
| programming, stochastic processes, game theory, economics, etc.
|
| 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.
| temp3208 wrote:
| Sounds great! Do you have the title/author? Thanks very much.
| milesrout wrote:
| Why 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.
| layer8 wrote:
| I'm European and write it as a cross as well, although others
| don't. For some reason I feel that ")(" is just an entirely
| different shape topologically, not an "x".
| Timwi wrote:
| If you do it right, )( looks a lot more like the printed
| cursive x than a cross does.
| layer8 wrote:
| It's writing the strokes that feels wrong, not simply the
| appearance. If I wanted the curvy appearance, I would still
| want draw the lines as two roughly diagonal crossing lines.
| saagarjha wrote:
| It keeps the lines collinear.
| gerdesj wrote:
| This 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 wrote:
| My 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 wrote:
| p 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 wrote:
| I 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.
| lolinder wrote:
| > 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).
|
| 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?
| ch3cooh wrote:
| This 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. x 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 wrote:
| > 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
| ("x"). If you need a multiplication symbol, use the middle dot
| ("*") instead.
| lupire wrote:
| The article alludes to this when it mentions Calculus 3 (vector
| calculus):
|
| x is the cross product.
|
| * is the dot product.
|
| In dimensions higher than 1, they are different.
| RossBencina wrote:
| I find it clearer to write dot products using inner product
| notation <,> rather than \cdot .
| fifilura wrote:
| In 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 wrote:
| In France we use the wedge [?] for vector cross product.
| f1shy wrote:
| Same in Spanish. That gets rid of the problem.
| SAI_Peregrinus wrote:
| Then what do you use for the outer product?
| nxobject wrote:
| I think the two example images are actually the same. (I've
| always done an italic X.)
| js8 wrote:
| In the handwriting I learned, x had hooks on all four ends,
| while greek chi only on two.
|
| Anyway, I usually use simple cross as x, because it's only very
| rarely confused with cross product (and you can always
| parenthesize).
| caturopath wrote:
| I make my x's with a backwards c and a c, like Computer Mondern
| and lots of fonts https://i.ibb.co/8LPsJKsj/image.png - doesn't
| look much like a chi or a times sign
|
| No one in the target audience is using x for scalar
| multiplication.
| brianstrimp wrote:
| Not for scalar, but certainly for vector multiplication which
| a large part of the audience certainly needs a lot.
| TRiG_Ireland wrote:
| My x looks like a cross in normal handwriting, but like two
| back-to-back c's in algebra. I developed this habit very
| young, and have kept it.
| FungalRaincloud wrote:
| I think it's important to consider audience. If I'm working
| with the intent that what I write is legible to folks who only
| have a basic understanding of math, I'll usually use the
| multiplication symbol (NOT the letter x, but x, intentionally
| in the middle). Someone with more advanced knowledge of math,
| who may be more inclined to think it's an x, because my
| handwriting is shit, I will typically use the dot operator. But
| then there's the whole other audience where I need to define
| what the dot operator does. At that point, I'm probably pulling
| up something like LaTeX, because, again, my handwriting sucks.
|
| 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.
| layer8 wrote:
| Audience is important. I'm not American and didn't learn the
| cross symbol for multiplication in school. We used the middle
| dot.
| TheRealPomax wrote:
| I can't remember the last time I actually wrote a times sign.
| floodle wrote:
| I did often in my degree, not for multiplication but the
| cross product
| TheRealPomax wrote:
| sure, but in that context there's no ambiguity (and
| technically, it's not the "times" symbol, that's the cross
| product symbol). We don't really use "x" in vector maths,
| it's all "a", "b", "v1", "v2", etc =)
|
| (at best you might use a .x subscript for 3D graphics but
| when did anyone ever do that by hand)
| patrick451 wrote:
| Depending on the context, dot and times may have different
| meanings.
| dapperdrake wrote:
| And as always Lawrence A. Chang's Handbook for Spoken Mathematics
| [0]
|
| [0]
| https://openlibrary.org/books/OL18996361M/Handbook_for_spoke...)
| jheriko wrote:
| missing the cursive theta.
| jks wrote:
| I've seen Greek computer scientists write Omega as an underlined
| O. Much easier than trying to approximate the printed letter O
| but still unambiguous, unless you use underlining to denote
| something in your formulas.
| powerawq103846 wrote:
| That is amazing thanks
| _hao wrote:
| I 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 wrote:
| playing oboe is easy for _most_ Frenchs (inventors of the oboe
| if you didn't know that)
| _hao wrote:
| Your 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 wrote:
| all i'm arguing is whatever "nation (2 guys) invented"
| something doesn't imply that every national know it better
| than everyone else
| tzot wrote:
| ISTR 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.
| fitsumbelay wrote:
| used to write like this in high school, like my physics and calc
| teachers. always wondered why the style was so common
| jan_Inkepa wrote:
| A 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"
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