[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).
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