[HN Gopher] How I See Numbers ___________________________________________________________________ How I See Numbers Author : igpay Score : 127 points Date : 2022-03-03 10:24 UTC (12 hours ago) (HTM) web link (www.csun.io) (TXT) w3m dump (www.csun.io) | Mo3 wrote: | That does sound like synesthesia. I have sound -> touch | synesthesia. There's some discussions about whether a pretty big | percentage of people have some form or another actually | yeetard wrote: | Not necessarily synesthesia. I think what OP describes is | closer to Ideasthesia which indeed isn't as uncommon. | Enginerrrd wrote: | Welp... now i understand why I've only encountered like one | other person that seemed to immediately understand what I was | talking about when I used the word "thought-shapes". | Mo3 wrote: | TIL. | seiferteric wrote: | Almost like a chemistry of numbers. | laszlokorte wrote: | There is a really distinct feeling I have about the fact that 2 | times 8=16 and 3 times 6=18. Really hard to describe but | something like 8 and 6 being siblings fighting about who is | stronger/bigger. | moron4hire wrote: | And then 6 x 8 = 48 is like 6 graciously helping out 8 only to | get left behind by his vindictive brother. | yeetard wrote: | So, a few question to all the number-as-shapes-in-head- | representators out there: What happens in front of your inner eye | when you do more complicated operations like exponentials, | modulo, ...? Do you have distinct visualisation for certain ways | to represent a number (roots, fractions and so on) too? And do | these representations help you when you solve a problem where you | don't have to "count" anything, like when you have to write a | proof or something? | hamaluik wrote: | I don't see things the exact same way as the author, though | similar (simplest way I could describe it is things like | addition are filling up tanks of liquid of [usually] 10^n size, | though a little more amorphous and yet "jelly" than what you | would normally think of as liquid? I'm finding it hard to | describe). | | Exponentials get represented as a third dimension; where basic | arthimetic is 1 or 2d depending on the context, exponentials go | into a third dimension if that makes sense. | | Modulo is the leftovers / splash-out when I pour one number | into several smaller containers. | | Fractions are simply fractional amounts of a tank of liquid | (i.e. 2/3 is simply a measuring cup filled to the 2/3 line type | of thing), but I can't ever picture them very accurately for | weird fractions. "Improper" fractions are basically the same as | modulo.. almost as if they're unstable in my head and | automatically "pour" themselves into more tanks that fill as | needed until some remainder is left. | | I don't have a visualization for roots, which is probably why | I'm generally so bad at them. | | The representations helped in engineering school for getting a | "feeling" about a formula; it was often very easy to notice if | an equation I was massaging had gone off the rails. For a pure | proof however (not that I did much of that), it was useless. | rplnt wrote: | What else do you see besides generic numbers? | | I see time differently, days of the week, yearly calendar, | distance units, temperature. All these, maybe more I can't recall | now, visualize different from just numbers. E.g. the year is a | loop. If I want to recall a month name, I always see a part of | that loop, and the camera is not fixed. I'm fairly certain my | mind didn't come up with this on its own, but there were some | visual that got paired with it. Same with numbers I suspect, but | this one is more obvious. | picometer wrote: | I have this as well, including the loops for time units (day, | week, year) and varying "camera" perspectives. Sometimes it's | labelled as "space-sequence synesthesia" (or "space-time", | which sounds cooler but doesn't cover the non-time sequences | like temperature). Always fun to find someone else with it in | the wild! | one-more-minute wrote: | Cool! If the author is reading, have you looked at synaesthesia, | and do you think it applies here? The idea of addition having | both a visual appearance and a kinetic feel is alien to me | (perhaps partly because I avoid mental arithmetic like the | current plague). But it's apparently reasonably common to have | colour and spatial associations with numbers. | | I'm also curious if you are an unusually quick calculator | compared to others you know. Synaesthetes can sometimes turn | their condition into a talent, like the famous Shereshevsky who | had a photographic memory; every experience was utter sensory | overwhelm, making mundane information very memorable. | gumby wrote: | The "numbers" (rationals, mainly) definitely have a "shape" in my | mind, at least up to 800 or so. When I do simple arithmetic I | feel like I simply glance over to the right place and "see" the | answer. | | I remember as a child trying to draw the shape of the number | "line" (it curls and twists) and being surprised that I was | unable to do so. | | This has never seemed to have given me any advantage or | disadvantage in learning more complex maths than one gets in | primary school. But since so much arithmetic is done in numbers | less than 100 (or scaled down to that range) it does make a lot | of things easier. | darreld wrote: | As an aside, my grown daughter recently told me that to her | numbers have always been gendered. I ran through them quickly and | she matter of factly told me their gender. She doesn't work in | tech but healthcare. She said it's always been like that. | glouwbug wrote: | Non-binary, right | johnatwork wrote: | This is similar to how I see it. I have a hard time explaining | how 7 is a tipsy number that's about the fall into 3 and 4, and | how 9 has a voracious appetite to take away a number from another | and you can't stop him. It all started when I was younger and my | mom told me to bring every number down to 2s and 3s, and to | always be adding or subtracting idle numbers (just numbers | without any operators). | | I explained it (poorly) to my wife once and she made fun of me | about it. Well until our son told us years later out of the blue | that it's how he sees numbers. | igpay wrote: | That description is eerily close to how I feel. Seven for me is | definitely a very "loosely bound" number, and it wants to | separate along the three and the four. | generalizations wrote: | Wonder if he learned his numbers with math manipulatives. [1] | | [1] | https://4.bp.blogspot.com/-Et6_8IvPOW0/VEPMsOiyVAI/AAAAAAAAP... | igpay wrote: | I recognize these, but they're not what come to mind when I | think of my early math education (mostly times table | worksheets, or seeing a teacher do long division on the | whiteboard.) Definitely possible that they influenced my | thinking, though. | Quai wrote: | As a person "self-diagnosed" with aphantasia, I feel cheated | knowing that other people have a built in cheat-sheet. No wonder | why I was struggling with memorizing things like the | multiplication table in school. | dkarl wrote: | FWIW, I don't have aphantasia and consider myself a very | visually-oriented person when it comes to math, but doing | precise arithmetic by visualizing the quantities is mind- | boggling to me. I memorized the multiplication tables using a | song we learned in school. | hwers wrote: | This is something I feel like we're going to have to realize | more and more in society over the next decades. That a lot of | people simply have genetic cheats that others are missing. At | the moment we kind of pretend it's nurture to a large degree. | Should we 'unbias' the world to make it more equal for everyone | regardless of genetic cheat? (If so how?) What's the correct | adjustment? | pavlovskyi wrote: | Awesome interpretation! I did not think about numbers | interpretation for a long time, but numbers were always my | passion, especcialy as a child. The fun fact is that I gave any | number some kind of uninterpretable personality, some kind like | information about its historical behaviour. and its allows me to | like them more or not, give them positive or negative judging. So | in case of addition, multiplication, etc (which I can do | blazingly fast from my youngest) I see it as some kind of story | in which numbers really meet each other and produce some results. | | There is no other feeling in my mind which give me that amount of | understading,but it is an understanding which cannot be formulate | properly to other person. I feel it as an phenomena which origin | started in my mind and grow there for my whole life (which is | highly correlated with my introvert pov). Thanks for that | thoughts, all best. | chrisstanchak wrote: | Watch Number Blocks. | | https://www.youtube.com/watch?v=OPTOCwQoYR4&t=29m23s | | This is how my toddlers are learning. It's really good. | serverlessmom wrote: | I show this to my kids constantly. I love how it plants the | seeds of concepts like square numbers and divisibility in a | show that is ostensibly about just addition and subtraction | feoren wrote: | We just watched their episode on zero and it was fantastic. | It's a hard concept to explain and I think they did it | beautifully. My toddler is addicted. | hwers wrote: | I would be more worried about my kids becoming addicted to | youtube from this (at such an early age). | dls2016 wrote: | Damn, I let my kids watch Alphablocks as toddlers but I never | noticed Numberblocks! At 7 and 8 they're very good readers but | the older one gets bored with arithmetic homework very easily. | I've failed as a parent and mathematician. | feoren wrote: | > the older one gets bored with arithmetic homework very | easily | | Is he/she getting bored because it's too easy, or frustrated | because it's too hard? I was crazy bored with arithmetic | homework and I later took 2 years of math from a local | university while I was still in high school because my high | school ran out of high-enough-level math classes for me. Look | up the story of Gauss in elementary school for a much more | extreme example. | | Don't assume your kid is behind at math because they don't | like arithmetic homework! They could be too far ahead! Useful | links if you suspect that might be the case: | | https://beastacademy.com/ | | https://www.singaporemath.com/ | | https://ssddproblems.com/ | alchemyromcom wrote: | I have a similar freakish ability, but mine has to do with | writing. I can basically ~see~ approximately three pages of prose | in my mind's eye while writing. It only works under certain | conditions, but it feels just like I'm transcribing something | rather then doing any kind of deliberate thinking. People are | shocked what comes out of me, and even more so when they see how | quickly it happens. You would need to see me in person to | experience the full effect, but my body does not match my words. | Imagine the biggest lumberjack you've ever seen describing the | petals of a flower with such high precision that it takes your | breath away. That's me. I've started to slowly nurture this | talent, because it finally occurred to me that it might be | special. | drittich wrote: | I would love to see an example - are you willing to share? | usgroup wrote: | A while back I got into thinking through allocation problems that | we'd typically use numbers for without using numbers. Things like | "how much what do you need to store in order to get a village of | N people through winter given the following consumption pattern | ..." . When you decide not to use numbers for the problem, you | end up writing algorithms. Every person gets a bucket ... a | ration is allocated to each bucket round-robin, and so on. You | end up writing logic and proofs for why your algorithm has to | terminate at the expected conclusion. That may sound fancy, but | its just what you end up doing as a regular person, without even | trying to be fancy. It is somehow inherent it what happens when | you avoid numbers. | | Your sort of just build everything you need out of analogs. It | makes me think that if we were not indocrinated into numbers from | an early age, we'd end up inventing them as an abstraction to the | sort of thing you have to do when you're trying to avoid numbers. | | Another one I suggest trying is expressing and exploring linear | regression without reference to probability theory. | calculated wrote: | What about +10 -7? Do you have a mental model for negative | numbers? | yesenadam wrote: | > Beyond the first ten natural numbers, some have unique forms | | Such a fascinating read, thank you! I'd love to read/see more | about those other numbers with unique forms, and also features of | the way numbers combine. (like the way you described 7+3 or 9+x), | I want a part 2! Thanks again. | hateful wrote: | I do the same thing! Though my shapes are different, it's the | same. My wife is tremendously bad at math and I kept telling her | you have to picture things and she said I don't think of it that | way, I just see the writing of the number itself and I say "well, | that's why you're bad at math!". | | This is the same thing as map reading or what we do in | programming. The thing that's disappearing in this comic: | https://heeris.id.au/2013/this-is-why-you-shouldnt-interrupt... | | I also realized early on that I could count way faster if I | fought the urge to say the numbers in my head because the idea of | the number would still be there. I started by saying (eh eh eh eh | eh) in my head instead of (one two three four five). Eventually | you can do things like run your finger across a comb and | instantly know how many bristles you passed - that gives you a | tactile response for each number rather than the words | themselves. If you count by 2s 3s or 5s you can go even faster | (which is what the circle is doing in the article). Shortening | the "time" axis of the counting. | drBonkers wrote: | whoa, any additional practice recommendations for whetting this | skill? | zesterer wrote: | I do something very similar, but with power-2 numbers (and have | done since my childhood, long before I knew what power-2 | numbers were). | | There's something very rhythmic about counting beats that line | up with powers of 2 and I'm able to count things extremely | quickly and precisely without even thinking about the numbers | I'm counting. When I want to remember how much I've counted, I | simply think back at where I am in the rhythm and come up with | the results in a strange vibes-ey way I'm not really able to | describe (for example, I'll just intuitively 'know' the | difference between having counted 32 beats and 64 beats, and | then I can use that knowledge to hone in on the precise number | I'm at using a sort of mental binary search). | | I'm sure someone with more knowledge of musical theory or | neurology could provide a better explanation, but it feels like | I'm somehow taking advantage of whatever part of the brain | keeps track of beats and rhythms in music, then using it to | count. | | Edit: I just tried this technique while listening to music and, | as expected, I completely lost the ability to count in this | way. Almost immediately I lost track, before I even hit 16. | gfody wrote: | in my 20's I went through a numerology phase and began taking the | digital roots of everything, it became a habit and now I can't | not do it. I developed a really similar sort of visual mechanical | sense for the digits 0-9 where the digits click together as if | they were magnets and the closer they are to 5 the more they | repel their own parts (eg two 5's might easily disintegrate to | snap into a nearby pair of 3's). it's really interesting to hear | about other versions of this sort of thing. | cecilpl2 wrote: | Wow, this is utterly alien to me. I have always had a head for | numbers but they are never shapes. | | The unique thing I think I have is that I visualize long strings | of digits as notes on a musical scale. 735 is high-low-middle. I | have found I can retain strings of up to 15 or so digits in | short-term memory by chunking them into triplets and memorizing | them as arpeggio chords, or by their relative positions. | porkphish wrote: | Uh... That's rad! And now I feel even dumber. | encoderer wrote: | Holy moly. Wow. | | I am a successful software developer and I'm terrible at math. To | me, 6+3 is not an interaction between two different anything, | rather, it's a key in a hash table where I've stored "9" as the | value. All arithmetic is rote memory recall for me. I work with | complex numbers by just breaking them down into multiple steps. | | Now I'm wondering if I should challenge my brain to do this | differently. | omarhaneef wrote: | I do what you do -- we are symbol manipulators. | | OP does what they do in number blocks. | | I guess with enough practice they are both fine for solving | known problems. I think our way is better for programming, and | his way is "better" for physical building. | igpay wrote: | Author here - like one of the other commenters said, I don't | think there's anything wrong with your approach, or any way of | thinking for that matter. I don't think there's anything | particularly "right" or advantageous about the way my brain | works either. I don't have any reason to believe I'm better at | math than the average engineer - definitely not a math prodigy | or super genius or something like that. | | With that being said, trying to think a different way for the | challenge of it is definitely interesting. Reading through some | of the other comments here and trying to taste words or | replicate other people's minds is a weird, fun exercise :) | mftb wrote: | I think the really great thing you did here, was just lay it | out. So little is said/shown on this topic that it's really | valuable to just get people conscious of their own process, | so that they can compare and contrast. | hosh wrote: | I mean ... just as an example, what happens if what you are | adding are not numbers? | | For example, a string concat can be understood as an addition | operation: | | 1 + 0 = 1 (identity) | | 1 + 1 = 2 | | 1 + 2 = 3 | | 2 + 1 = 3 (communitive) | | "a" + "" = "a" (identity) | | "a" + "a" = "aa" | | "a" + "b" = "ab" | | "b" + "a" = "ba" (non-communitive) | | There's this whole intuition about addition itself that can be | applied to something other than integers, and being able to | reason about that is applicable to how you design software, | particularly function interfaces. | | Just as a note, my mother made me memorize the multiplication | table when I was a kid, and I had ended up memorizing additions | just through sufficient practice. I was able to intuit what | additions and multiplications meant, but for the purpose of | taking tests in school or doing homework, additions just pop | out as answers because of the memorization. It wasn't until | much later in life that I started encountering ideas such as, | what if you were adding something other than numbers. | davchana wrote: | In India we learn tables (multiplication tables, but we just | call them tables) from 1 to 10, and later till 20. Each one | has this format, 1x1=1 1x2=2 | | First number is 1, so its table of 1. Then x as multiplier | sign. Then a count from 1 to 10. Then = sign. Then the | result. We kids are supposed to write each line in left to | right direction, then move to next line. | | We use paper with square tables or graph on it. Most of the | time, kids simply write 1, move to next line, again write 1, | all the way till 10th line. Then we move to next column, | write x, then move up, x all the way till 1st line. Then | 1,2,3, in next column, = in next column coming up. Then the | answers going down. | bckr wrote: | This is how I learned arithmetic in the US 2 decades ago. | | edit: But actually after that I used something called | "Math-U-See", which used physical blocks to develop | intuition. That was pretty cool. | cyberbanjo wrote: | "Young man, in mathematics you don't understand things. You | just get used to them." -- John von Neumann | andai wrote: | Some of my favorite articles on von Neumann: | | https://sites.google.com/site/steveyegge2/math-every-day | | https://www.cantorsparadise.com/the-unparalleled-genius- | of-j... | deltaoneseven wrote: | I don't see his way of viewing numbers as particularly | efficient. It's very inn-efficient. It's an anomaly for sure | but I would hesitate to call it a talent or super human | ability. | | I would argue his way of thinking of numbers makes him slower | at doing calculations. | | When you create a 2D visual representation of a number system | you want to choose a shape that has the same properties as | numbers. Namely the shape must be monoidal under composition. | This allows you to keep one type of shape | | For example (int + int = int). When you compose two triangles | together you get a parallelogram, so triangles are actually | kind of bad as you would need to classify several different | types as numbers. (triangle + triangle = parallelogram) The | only shape that I can think of that is monoidal under | arithmetic composition is rectangular quadrilaterals with at | least two parallel sides. | | Examples: Rectangles, parallelograms, and trapezoids each can | be composed to form another shape in its own class. With | rectangles likely being the most efficient representation as | they are fully symmetrical (to compose two trapezoids to form a | new trapezoid one trapezoid has to be inverted, this does not | happen with rectangles). | | So his even number visual representation is quite good (it uses | blocks) but his odd number representation is all over the place | and seems arbitrary. Just look at 9. It involves "orange | peeling" another number just to shove it into the little dent. | His system involves mutating, rotating and changing the shape | of each "number" in order to perform composition. This costs | more "brainpower" to do and is the main reason why I don't | classify his ability as a "gift". | | It's highly inefficient. I think many HNers are mistaking it | for a super human ability. I don't agree. This is more of an | interesting ability then it is a talent. | | But that's just a guess. Would actually like to see a | quantitative measure of how fast he is at adding numbers under | his system. This would definitively answer the question. | cupofpython wrote: | I relate to the OP on a fundamental level although the | literal expression would be different for me. I do not think | it has any relation to speed. It is not a deliberate step. It | would be slower to mimic this behavior, but if you have it by | default it's just kind of there. | | Certain calculations are actually faster because i begin to | have faith in my feeling of the math over doing an actual | calculation - with the same type of confidence i have when | recalling a times table for example. Still, it usually doesnt | get me all the way to an answer | | There are certain mathematical rules that you can probably | identify that are related to my internal expressions and how | they "fit" together. For example, I do not know without | calculating what "25 x 15" is, but I have an idea of what the | answer feels like. anything below 100 or over 1000 feels | outright OCD level out-of-place. Numbers like 114, 201, etc, | feel dirty and incomplete. we can identify in this scenario | that the shape / feeling of the answer for me is related to | an intuition for the mathematical principle that the product | of two numbers that are divisible by 5 is also divisible by 5 | - but at no point did I deliberately evoke that rule when | conceiving of a possible answer. Also this is a simple | example, this intuition runs beyond my knowledge and ability | to formally explain the principles. In reality, many such | principles (learned or inferred) come together at once to | feed my internal expression of the answer. A calculator says | 375 is the answer, though 325 and 475 feel about the same | | I do not think it makes me better at getting correct answers, | but it does help me accept an answer as being correct when | looking at it also feels right. It's most useful when | identifying errors. There is a big help when you see "15 x 25 | = 356" and without thinking you can feel internally like | something is out of place, dirty, needs attention (this | applies to advanced topics as well). As I said above though, | more than the correct answer can have the same or similar | feeling - so it is prone to false negatives | | With something like math, intuition based guess work that has | room for false negatives is hardly that useful overall. So | maybe the only real edge it can provide is in working with | novel concepts where you have to guess a direction to explore | and hope you uncover something useful. That is an unfounded | hypothesis though. | greggsy wrote: | I don't think there's anything wrong with your approach - you | don't have to 'think' about the solution because it's already | there. I don't know if that translates to an actual reduction | in mental fatigue, but if it works for you then changing it | will no doubt cause at least short term strain. | | I also think there's no need for people to feel like they need | to be some math or grammar prodigy to get by in life. It's | perfectly fine to outsource your mental functions, including | memory to a calculator, notebook or PKM system like Obsidian. | ChrisKnott wrote: | I find it interesting that in the UK a primary school child (say | aged about 7) would trivially know that "80 + 4" is 84, but for | the problem "4 x 20 + 10 + 7 = ?", might require quite a lot of | effort to work out that the answer is 97. | | In France, "97" is said "Quatre vinght dix sept", i.e. 4x20+10+7. | This is apparently acceptable to the brain as a final answer, | there's no way to collapse it to "90+7". | zeropoint46 wrote: | sure there is, speak swiss french :) | honksillet wrote: | This reminds me a little how some people with perfect pitch | describe each not as having a color. | teaearlgraycold wrote: | One thing that helps at lot with programming is my tendency to | visualize branches and dependencies as graphs/trees as I | read/write code. This makes aberrations and code smells extremely | obvious. A dirty hack makes you go from something that looks like | a beautiful fine-toothed comb to a comb with a cancerous tumor on | it. | sjosund wrote: | Sounds like something that would make an interesting blog post! | teaearlgraycold wrote: | That's a good idea. | | Sometimes the domain is already graphical - and I take every | opportunity to make the code match the visual layout, ex: | | https://github.com/danthedaniel/gameoflife- | rs/blob/master/sr... /// Count living cells | adjacent to a cell in the matrix. #[inline] | #[rustfmt::skip] fn alive_neighbors(&self, x: i32, y: | i32) -> u8 { [ self[(x - 1, y - | 1)], self[(x + 0, y - 1)], self[(x + 1, y - 1)], | self[(x - 1, y + 0)], /* selected cell */ self[(x + 1, y + | 0)], self[(x - 1, y + 1)], self[(x + 0, y + | 1)], self[(x + 1, y + 1)], ] .iter() | .fold(0, |total, &neighbor| total + (neighbor != 0) as u8) | } | pjacotg wrote: | @author - is there anything special about the way you visualize | prime numbers? I'm wondering if there are indicators for you that | a given number would be prime. | bufordtwain wrote: | This blew my mind. I would never have guessed that this was a | thing. I wonder if the mathematician Ramanujan had a visualizing | ability similar to this. | virtualwhys wrote: | I see nothing; for the most part there is no mind's eye, but | there is a mind voice, and that's what performs mathematical | operations (and everything else for that matter). | agumonkey wrote: | Talking about inner representation, I'd really like to know how | people computing nth root of large number operate :) | visviva wrote: | What a fascinating and delightful read. It's something that's | totally alien to me, explained in a very satisfying way. | robofanatic wrote: | I can taste words. Meaning some words immediately remind me of | something I have eaten before. I can logically understand why | some words taste like the food because they sound like the name | of a food but some words don't even come close still they remind | me of a certain food. I guess I am alone because I haven't found | anyone who feels this way. | meowface wrote: | Sounds like synesthesia. | feoren wrote: | I caution against looking at numbers in _any_ single way. The | more different ways you can visualize math concepts, the better. | Practice seeing them in different ways. | | Sometimes numbers are for quantifying a pile of things, and 255 | and 256 are basically the same. | | Sometimes numbers are for cryptographically signing things, and | 255 is extremely secure while 256 is completely vulnerable. | | Sometimes numbers are for arranging tournaments, and 256 is a | tremendously useful number while 255 is super annoying and you | should look for another. | | Sometimes numbers are stored in a single byte, and 256 (=0) is | the friendliest number you will ever know, while 255's words are | BACKED BY NUCLEAR WEAPONS. | | Sometimes infinity is a useful number, sometimes it's not. | Sometimes 1/2 is a useful number (pies), sometimes it's not | (babies). Sometimes sqrt(-1) is a useful number, sometimes it's | not. Sometimes the sum of all positive integers equals -1/12; | sometimes that's stupid. | | All of these situations may call for visualizing numbers | differently. | wedn3sday wrote: | This is one of the best/most insightful comments I've ever seen | in casual internet discourse, I approve and applaud you. | tarentel wrote: | While I don't think this is bad advice I don't really think it | is along the same lines as what the author is describing. | | This sort of thing reminds me of an article I read a while back | about how some people don't have an inner monologue when | they're thinking which I assumed everyone did and found wildly | strange trying to think about how other people think. This | article is also equally confusing to me. | hackingthelema wrote: | I think I have aphantasia and no inner monologue. Mind you, I | can summon an inner voice to compose a sentence before saying | it, but when I'm thinking about something being discussed and | someone asks me what my thoughts are so far... I never have | any idea what to say. _My mind is blank!_ It 's always blank! | There are never any discernable words or images in there to | give you. If I need to communicate my thoughts, I have to | spend significant amounts of time translating to words and | choosing words before I can actually summarise what I was | _thinking_ , which is much more nebulous to me than words or | images. | | My 'thoughts' are closer to a mouse cursor changed into an | hourglass while waiting for a computation to finish than | 'First we need to do <XYZ>, but to do <XYZ> we need <X>, <Y>, | and <Z>. To get <X>, <Y>, and <Z>, we need to ...' | | I find it really hard to operate in live/in-person | discussions because of this. I physically end up just as | silent and blank as my mind! | tarentel wrote: | I find this kind of stuff, including the authors article, | weirdly fascinating. I try to do what other people | describe, such as yourself, and it really is impossible. It | just makes no sense to me. I'm sure I have ways of thinking | as well that probably baffle other people. It's all very | strange. | | With that being said I wish my mind was blank sometimes, I | wish my inner monologue would shut up every now and then. | :) | feoren wrote: | The author describes thinking of 9 as floating around looking | for a 1 to chomp off another number. This is very clearly | designed to support good intuitions about adding in base 10, | but it produces bad intuitions about binary numbers, | multiplication, polynomials, etc. If faced with myriad other | problems that involve 9, like, say: "which is bigger: 2^9 or | 9^2?" or "how should we store words from an alphabet with 9 | characters in memory?" or "how can we distribute 9 things | equally?" or "for which n is 9^n + 2 prime?" or "how should | we expect an atom to act if it has 9 electrons?", a | completely different way of looking at the number 9 is | warranted. In that last case, the exact opposite is true: 9 | is desperately trying to rid itself of a 1, not find another | 1 to grab. | tarentel wrote: | I still think you're missing the point a bit. I don't think | the author is doing this as some sort of trick or by | design. I think they're literally describing how they | visualize numbers in their head. Reading through other | peoples' comments seems to support my conclusion on this. | | Maybe they visualize other number relations differently in | their head. To me, I could not do math in my head like this | and it makes very little sense to me. I don't even really | get what they're describing to be honest with you. I | visualize numbers in my head as the number symbol you'd | write down. | gouggoug wrote: | The author isn't describing a random system they came up | with to deal with numbers. | | They are describing how their brain naturally sees numbers: | which is commonly referred to as synesthesia. | hosh wrote: | Some people can think in numbers in a way that does not | require visual representation or any kind of representation, | and as such, it is also possible for such a person to express | the pure idea in different ways, including numbers as shapes | as the author is doing. | | 'Cause I am very curious how the author experiences imaginary | and complex numbers ... or even negative integers, | irrationals, and transcendental numbers. | igpay wrote: | Negative numbers are just like the positive ones, but kind | of... the opposite. Like the indentation formed if you | pressed the positive number into clay or sand or something. | It's like they want to be filled or take away from | something else rather than adding onto other forms. | | RE how I think about imaginary or complex numbers, in | short, I don't :) | | I've never studied much higher math, and don't have any | reason to think that I'd be particularly good at it. | igpay wrote: | Author here: was definitely not trying to frame this as a | tutorial or anything like that. I don't think that my "methods" | have any particular advantages. It's just how my brain works. | MisterTea wrote: | > I caution against looking at numbers in any single way. | | You misunderstand. This person is talking bout how they see the | numbers in their minds eye meaning this is how their brain | works. As a visual thinker I can relate to how there's an | uncanny ability to see things as shapes or things. | ajkjk wrote: | Neat. Some of us can't see things in our heads at all | (aphantasia), so we definitely can't do things this way. | | Although now that I think about it there is still some element of | what's described in this article. There's no visual shape | involved in the way I model numbers, but it resonates to think of | 7 as "10 with a 3 missing", but also as "5 with a 2 on it". The | concepts are built in reference to their closest multiple of 5, | and slide between different equivalent forms as necessary in | calculations. | | By the way, the way I do mental math without images feels like it | is using sounds and words for the short-term storage and recall. | The language brain seems good at putting something aside for a | minute and then bringing it back afterwards with a low chance of | error, like repeating something someone just said back to them | verbatim even though you weren't really listening. | | The one method I am sure _doesn't_ work well for mental math is | picturing the grade-school algorithms on an imaginary sheet of | paper. For whatever reason it is very error-prone. I once did an | informal (definitely unscientific) survey on this (30 or so | people IRL plus like 100 reddit users) and iirc there was a | strong correlation between "imagining the pen-and-paper | algorithm", "being bad at mental math", and "not liking math". | Wish I still had the data from that -- all I remember is roughly | confirming my hunch that those were related. I also wrote a blog | post about this a few years ago | (https://alexkritchevsky.com/2019/09/15/mental-math.html) but I | wish I had included the survey information in there, it would | have been much more interesting. | igpay wrote: | While writing this article, I learned that Ed Catmull has | aphantasia. It's amazing to me that someone with a Turing award | for work on computer graphics can't mentally "see" those | graphics when he closes his eyes. It'd be really eye-opening to | somehow get his (or anyone else's) mental state into my own | brain, just to try it out for a little. | | Interesting that we share some conceptual similarities in how | we think about numbers, but they're expressed through different | pathways (language vs. visual.) I wonder if the people who | imagine pen-and-paper stuff when doing mental math just don't | have these pathways set up, and instead recall memories of | math-adjacent experiences in lieu of another internal | representation of numbers. | bricemo wrote: | Since Stephen Hawking's movement was limited for much of his | life, he claimed that he had learned to do more math quickly in | his head via visualizing geometry. Seems similar. | busyant wrote: | I view the digits as having genders and personalities. | | "5" and "6" are definitely guys. | | The evens tend to be a bit kinder than the odds. Hasn't helped me | with arithmetic, though. | function_seven wrote: | Heh. For me all the odds are male and the evens are female. | Except for 10; he's a dude. | | My weird thing is that they all have a color. 0 is gray, 1 is | blue, 2 is yellow, 3 is red, 4 is green, 5 is blue again, 6 is | purple, 7 is red, 8 is orange, 9 is yellow. After 9, it's just | the last digit that typically "colors" the number in my head. | | And I get what you mean about numbers having a personality: | https://news.ycombinator.com/item?id=30365289 | kderbyma wrote: | interesting. I Wonder what their numbers look like in different | radix. | | 7-3 I found interesting because those are modulus complements in | base 10 | deltaonefour wrote: | What a highly inefficient way to represent numbers. | | Was this learned by him or is this some sort of synesthesia | condition? | jkingsman wrote: | It seemed to me that the author was describing his instinctive | mental representation of numbers, and not that mental math is | only achieved by using shape-analogs. | | I've never thought about it before, but while I definitely | don't have as distinct models as the author, I do understand | and agree with an instinct around numbers "fitting" together to | make tens, and it definitely informs how I break down e.g. | triple digit mental addition. | deltaonefour wrote: | Why not make all the integers square blocks? Then everything | fits together. It seems strange what's going on with the odd | numbers. Especially adding something to 9 is even stranger. | Seems arbitrary rather then instinctive. | kderbyma wrote: | not necessarily. only if you think in terms of cou ting | does your sense make priority. if I were to think in terms | of multiplication - circles are more useful for a lot | things. | ryanklee wrote: | You seem to be carrying the impression that this is | deliberately constructed -- it's not, this is just part of | the author's intuitive representational system. He's not | "making it" one way or the other. It's made; he's | perceiving it. | mynameisvlad wrote: | This is like telling an anxious person "not to worry" or a | depressed person to "cheer up". | | Yes, it would be grand if our minds worked in a rational, | logical fashion. But that's not even remotely | representative of reality. | simion314 wrote: | I don't have a visual representation for numbers, but | numbers with a 9 at then end like say 29 in my mind is | always transformed in to 30-1 , so instinctively (nobody | teaches me this) computations like 29 + 15 = (30 -1) + 15 = | 30 +15 -1 =45-1. This makes it more easy for | multiplications 29 _15= 30_ 15 - 15 . I could apply this | for numbers ending with 7 or 8 but it does not fill natural | for me as 9. | swah wrote: | I'm pretty sure he's faster at 7+3 with his system... | educaysean wrote: | I think it's more of the latter. The shapes are not there to | help him do arithmetic in a more efficient way. The shapes are | there just because that's how numbers are represented in the | author's brain. | | I experienced similar things growing up. For my case, it was | usually colors. Each number was associated with a specific | shade of color, but in my case it was less about the numbers | themselves; it was more contextual. Eg. The number four | represented different colors depending on whether it was | describing the time of day, the number of floors on a building, | or amount in currency. | | I had brought this up in my youth only to be met with derision | and threatened with being labeled "abnormal" by the authority | figures, so I worked to suppress and hide this aspect. (South | Korean society had a lot of backwards ideas in the 90s). ___________________________________________________________________ (page generated 2022-03-03 23:00 UTC)