What shape are the numbers?
August 18, 2016 12:11 PM   Subscribe

When you think of numbers, do they have some specific, reliable spatial organization in your mind?

I'm fascinated by how many experiences we have that are largely internal and rarely discussed, but which can differ from person to person and shape the way we interact with the world. Inspired by this recent excellent question, I'm curious to know how people's internal experience of numbers varies.

I know that some people have a very strong "number line" sense: a specific shape that the numbers always are organized along. This is a somewhat foreign experience to me; my numbers have no specific spatial organization on their own. I can easily relate numbers to space (with coordinate axes for graphs and whatnot), but that feels like a metaphor or structure that I'm imposing on numbers, rather than something intrinsic to them. I've spoken with at least two people about their number lines who have described their specific spatial organization as being much more intrinsic to their concept of number than what it feels like it is to me.

As an example, a friend of mine described her number line as a straight line going left to right for the numbers 1-10, then a helix spiraling downward from 11-100, making a cycle every 10, and then shooting off into the distance from 101-1000. Numbers above 1000 did not have the same strong spatial organization. For her, numbers always have this specific organization, the only exceptions being numbers associated with time, which are organized like the face of a clock, or dates, which have a calendar-like organization.

Because this isn't the kind of thing that tends to come up in conversation much, I suspect this kind of number-space sense is a lot more common than it seems. Since this is an experience very different from my own, I'm curious to know how many people have strong number lines like this, and what kinds of shapes they take. Or, are there other spatial (or non-spatial) organizations to the numbers that some people have in their heads?
posted by biogeo to Science & Nature (50 answers total) 12 users marked this as a favorite
That thing your friend describes sounds like a number form, a type of synesthesia. I think there's a chapter about it in one of Oliver Sacks's books.
posted by theodolite at 12:16 PM on August 18, 2016 [1 favorite]

I don't have that, but I do see the numbers from 0-9 as having personalities. The number 2 looks "mean" to me, and 3, 6, and 9 all look "happy." I don't know why.
posted by xingcat at 12:23 PM on August 18, 2016 [2 favorites]

The numbers 1-9 for me all have shapes like they'd be on a domino and when I am counting them I do it physically like touching invisible dots on a piece of paper.
posted by jessamyn at 12:27 PM on August 18, 2016 [3 favorites]

That thing your friend describes sounds like a number form, a type of synesthesia.

I agree. To clarify, I actually know a fair bit about the science on this topic (I have an academic paper about the neuroscience of numerical cognition on my CV, though it is not my primary area of expertise). What I don't know is the diversity of individual people's personal experiences of number, because it's rarely discussed openly. I'm very interested in hearing how people personally experience numbers, spatially in particular, but really anything "unusual".
posted by biogeo at 12:29 PM on August 18, 2016

I have something similar to your friend. I described it once here on mefi.

Numbers for me are a clockwise logarithmic spiral. I should draw it sometime.
I add numbers as geometric segments on this spiral. The reason I often have trouble with 8+5 (sometimes I say 12) is that there is a discontinuity right above 8, but before 10, which always throws me off.

If it is relevant, I have always been very good at math and majored in Physics in college.
posted by vacapinta at 12:30 PM on August 18, 2016 [1 favorite]

Oh, this is interesting. For me, numbers are linear on a stacked line that I am on. Like, the best way I can think to explain it is you know how dry cleaners have big rods of hanging clothes going in a line back basically forever? Each one is a number sitting in a chain of numbers, obscuring the others. If I'm at 5, I can't get to 10 without mentally shifting the 6, 7, 8, and 9 out of the way; can't get to zero without shifting the 4, 3, 2, and 1.

I think this is part of why I have a really hard time with addition and subtraction and am a slow counter if I have to sift through a specific quantity of things. I can do division and multiplication relatively easily, because at that point the numbers-on-a-line thing is more abstracted and I'm dealing with groups instead of individual numbers.
posted by phunniemee at 12:30 PM on August 18, 2016 [1 favorite]

You might find the literature on "subitizing" to answer some of your questions. It is an early, foundational developmental skill that underlies the ability to do more complex mathematical calculation and number manipulation.
posted by goggie at 12:32 PM on August 18, 2016 [1 favorite]

This is eye-opening. I have pretty much no picture of the structure of numbers in my head at ALL. Looking at my past grades and experience, I strongly suspect I have a math learning disability, so this makes sense.
posted by thebrokedown at 12:34 PM on August 18, 2016 [3 favorites]

xingcat! I do the same thing!

Zero is patient. One is boring. Two is practical and common sense. Three is a friendly, happy-go-lucky number. Nine and six both have a crush on five, who is very outgoing. Seven is very mellow. Eight is mean.

They also have colors:
Zero is clear, one is white, two is dark brown, three is bright green, four is cardboard brown, five is red, six is yellow, seven is blue, eight is a very dark purple and nine is orange.

Lord only knows what all that means, but it's been stable all my life. I blame the multiplication songs from Schoolhouse Rock.
posted by winna at 12:42 PM on August 18, 2016 [4 favorites]

In my mind's eye, 1 to 20 are in a line from left to right (1 is on the far left, 20 on the far right.

After that, each group of ten is clumped together with the lowest at the bottom rising up to the highest. For example:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20



And so on until 100. The 30s are also stacked that way but NOT on top of the 20s; they just float around in a block on their own, like all the other groups of 10. However, 100 is definitely above all the "clumps" of 10, which all hover above the horizontal line of 1-20.

Zero is by itself, floating around to the left of 1, but quite far away.

This is so interesting. I have never ever thought about this in detail or articulated it in any way before.
posted by hurdy gurdy girl at 12:46 PM on August 18, 2016 [2 favorites]

Like jessamyn, I see dots. On some of my old school jotters you could make out faint pencil tap marks where I'd counted. Numbers above 6 give me a problem; seven is murder, as it could be any one of:
 * *    * *        * *    * *                                           
 ***    * *        * *    * *                                           
 * * or * * * or * * * or * *        
… but it can't be the last one as everyone knows you can't stack four dots up or it would fall off and you might lose it, or it might hide behind another dot and pretend to be a six.

I was an anxious child, and atrociously bad at arithmetic.
posted by scruss at 12:47 PM on August 18, 2016 [8 favorites]

This is eye-opening. I have pretty much no picture of the structure of numbers in my head at ALL. Looking at my past grades and experience, I strongly suspect I have a math learning disability, so this makes sense.

I don't have any such map or spatial phenomenology such as others are describing either, and I was always good at math (although nothing special by mathematician standards).
posted by thelonius at 12:49 PM on August 18, 2016 [8 favorites]

And xingcat and winna, I also feel that the numbers under 10 have personalities! Not all of them, just some.

9 is very mean. 8 is nice to everyone but intimidated by 9. 4 is sad and feels left out a lot. 5 is very businesslike. 6 is kind and quiet. 7 is very rowdy, the life of the party.

Some of the numbers have associated colours, also:


I have mild synaesthesia and have similar thoughts about letters of the alphabet and days of the week (which all have specific colours).
posted by hurdy gurdy girl at 12:50 PM on August 18, 2016 [3 favorites]

   … They also have colors

There's a fair chance that some mefites might see 1–10 as White, Red, Light green, Purple, Yellow, Dark green, Black, Brown, Blue & Orange. Though our school was damp and the tens went mouldy, so that number will always be orange with black splodges for me.
posted by scruss at 12:59 PM on August 18, 2016

If I look at a row of five or six zeroes printed on something, I don't have enough peripheral vision to count them unless I physically cover some of them up or touch them as I count. I can picture four items in a row, but after the fourth one I lose track and can only guess how many I am looking at, or use a crutch or take it on trust.

In order to picture five of something I have to arrange them in a pattern, like the one found on an ordinary die. This is also the dot visualization source I use if I try to picture six or twelve.

This means that I really can't visualize an array of 100 objects or numbers. On some level any array bigger than a four by four is really the number "many". It means that a printed number like 100,000 is emotionally indistinguishable from the number 1,000,000. I love scientific notation because it makes it possible to see the difference between those two numbers and I don't have to stop and consider which one of them is bigger since I can tell the numerals five and the six apart and know which one of them is bigger.

Six is really three plus three to me. Five is really four plus one to me. Seven is three plus four and eight is four plus four. Nine is really three times three to me and ten is five plus five, or a different kind of a one or four plus one plus four plus one.

I cannot usually remember a string of seven digits long enough to dial a telephone number. I have to move back and forth really quickly by dialing the first three, checking and dialing the next four and I learned to find making phone calls really irritating when the phone companies switched to software that disconnects you if you dial part of a number and pause more than a second or three as now dialing numbers is stressful. I have to chant them to myself because I can't picture them.

The magnitude of numbers is meaningful to me. "It is the mark of a civilized man that he can see a number and cry." is a meme supposedly going back to when the Times of London recorded the latest British casualty figure in the top right hand corner of every edition during the First World War. Big numbers can give the the chills. The more I know it is beyond my capacity to picture the deeper the emotion it can evoke. I see a picture of one drowned child, like Alan Kurdi and I feel sad. I hear about 155,587 Syrians being killed and I don't picture anything, I just feel the bottom of my belly drop out and I feel wretchedly sick and hopeless. There isn't a spatial dimension to that amount of woe. It's a stronger experience, the bigger the number appears. It's on a scale of more nausea.

The numeral three is always apple green in my mind due to my first apple green crayon and a colouring book with a numeral three in it.

I like mathematics and do well with bookkeeping. The highest level of mathematics I have passed is the advance High School math course that comes before Cal. I stopped just after I reached the stage where certain number patterns started to fall into place and make me gasp with delight and awe and made me feel like I was starting to understand something essential about the universe. However, I think I probably have some kind of discalculia due whatever makes me go back to a three and a four to picture a seven.
posted by Jane the Brown at 1:06 PM on August 18, 2016 [3 favorites]

Pretty sure the space of numbers as a whole is just an infinite number line for me. Pretty much what you'd expect on the basis of schooling. Imaginary numbers float above/below the line as appropriate (although I'm not sure they're in their correct mathematical positions, just vague estimates thereof)

But some of the irrational numbers also float, even though they technically belong on the line. Pi is definitely in the upper right quadrant. I think that might actually be related to circles, even though the positions don't match the unit circle well at all - tau is at - 1, Pi is hanging around the 45 degree mark, and e is on the same line as Pi but closed to the origin.

The orientation of the line is not clear, ie it can be laid flat on the ground or floating at eye level, trying to pay attention which one it is currently doing makes it fluctuate between the two and gives me a headache.

I am always positioned at zero, which means smaller numbers are "closer" to me and larger numbers are "farther" away. (also for negative numbers, ie the closer to zero, the closer to me)

I feel like this is all super standard and boring though. It's kind of weird to think that other people don't see numbers in standard school graph form.

(as for the individual numbers - I got nothing. They don't have color and they don't have personality, they just... Are)
posted by Cozybee at 1:10 PM on August 18, 2016 [4 favorites]

I have synesthesia (touch->sight, touch->color) too, and single digits also have colors for me, though my color lineup is different than others listed here. For me:

0 is white
1 is also white
2 is light pink
3 is royal blue
4 is medium green
5 is red
6 is orange
7 is golden yellow
8 is purple
9 is brown

A multi-digit number has all of the colors in them (so 2016 is pink, white, and orange), though often some of the colors are stronger than others (I see more of the brown and the green in 1984 than I do the white or the purple, but I see more purple in 1948). I'm slightly dyslexic when it comes to individual digits within a four- (or five- or whatever) digit number, though I think I generally remember which numbers are involved, just not necessarily the correct order. Memorizing years/dates in school was always hard for me.

I also "like some numbers (and their colors) better than others. So for instance, I liked the color pattern for our wedding date, 9/3/11 (lots of blue and white, which happened to be our wedding colors as well) better than some of the other options we thought about (9/10/11 didn't have blue, 9/17/11 had yellow, etc.).

My favorite colors are blue and green, but my favorite single-digit numbers are 3 and 7, which don't quite line up. I have a preference against both 6 and orange, so that lines up, but I worry a little about the number 4 and I don't really like yellow as a color, so...

I have zero sense of numbers being spatially aligned, but I'm also really bad at spatial reasoning in general.

This is a fun question!
posted by bananacabana at 1:21 PM on August 18, 2016 [2 favorites]

My internal number line looks like the street I grew up on, and goes east to west facing south. My old house is located at around 18-20. It's not the same with large numbers and equations, so my kid brain probably connected the two when I learned numbers and how to navigate the neighborhood at the same time. Numbers at the larger end of the number line do a clockwise spiral. There's also a slight feeling of altitude, like going higher up a mountain as the numbers get larger.

Numbers also have colors. The impression isn't as strong as an adult, but 7 is still orange-yellow, 4 is green, 5 is an orangey-red, 8 is blue. I was probably picturing my crayon box during math class. Some number/color combinations get along better than others.

Negative numbers have a faint look of Tron about them.
posted by Orange Dinosaur Slide at 1:24 PM on August 18, 2016 [1 favorite]

Much to my husband's amusement, I have visualize all whole numbers below 100 as pips on dice since I was a small child. Sometimes I'll add the pips like 2 & 5, and 6 & 1 make 7, and 5, 5, 5, 4 for 19, but sometimes the pips are representative. 48 could be visualized as 4 for the 10s place and 6 & 2 for the ones place. I've never been a gambler or game player, so I have no idea where it came from. Fun question!
posted by defreckled at 1:25 PM on August 18, 2016 [1 favorite]

I apparently have a stronger feeling about this than I thought.

My number line/graph is a plane parallel to the ground. Bigger to my right smaller to my left, and "above in the graph (above x axis) is actually in front of me, not higher in height.

Trying to visualize it as perpendicular to the ground (ie, with pi dangling above my head instead of an armlength diagonally in front of me) makes me feel really twitchy and irritable and scratchy underneath my skin.

(oh, also, while graham's number is way too far away to be seen, infinity is hanging out smugly on zero. I don't know why. But that's where it is. Not the horizon. On zero. It is definitely hovering above the plane of the graph, though.

Man this is weird stuff to actually consciously tune into.
posted by Cozybee at 1:35 PM on August 18, 2016 [1 favorite]

This is so fascinating!

I see numbers 1-10 like scruss. What color they are depends on what mood I'm in. I too, was a very anxious child and really struggled with math as well. I always excelled in the visual arts and I am quite sensitive to tones in music; I can often identify a song by its key in the first few seconds I hear it. But math? Forget it.

To scruss's response I'd add that all numbers, in my mind, are dots that are always spaced out in a manner that pleases me and feels spacially comfortable for me. With odd numbers, I have to even out the dots spacially so they all have a "home." This sounds absolutely ridiculous and I probably sound crazy, but that's what I do. The further past 10 I have to process, the more irritated I get because I run out of homes, or parking spaces, or spots, or whatever : ) Data point - I came quite near to failing algebra I and II and also geometry. I was better at geometry, but "better than failing" isn't all that great. This always disappointed me and I always wanted to be good at math.

Oddly, I love knitting and written patterns delight me (not charts - written), because I can see how the math turns into a tangible thing. Possibly related, at the point that I was unable to turn the shapes over in my mind in math class (like, after elementary school) it all went to pot and I was completely lost. I could follow along with what the teacher did and understand it, but I could never follow those same steps to produce a response. It's what it's like passively understanding a language versus actively speaking one - receptive is always easier than expressive. That's how I feel about math. I can understand the example, but to try to do it myself creates a logjam in my brain.

Side note, I almost typed "numbers which are uneven" instead of "odd" - this reflects how I "see" them.

I had a teacher in college tell me once that I'm really mathematically minded but never had a teacher teach me in a way that I could process. I always suspected I had a learning disability when it came to math but was never tested.
posted by onecircleaday at 1:40 PM on August 18, 2016 [1 favorite]

Oh! I forgot about negative numbers. Those are on a number line, with one dot representing where the negative number is. Only positive numbers are dots.
posted by onecircleaday at 1:42 PM on August 18, 2016 [1 favorite]

I normally don't have any visual associations with numbers. If prompted to, and with a lot of energy, I can make myself see individual numbers as digits, as they're represented typographically in text. E.g. the concept one just looks like "1". (I remember visualizing heard or spoken words like that, as text, also, as a kid; don't so much anymore.) I don't see any number as relating to other numbers, spatially or in any other way - a digit just appears and sort of hangs in front of a kind of white space, and then recedes. That's thinking of numbers qua numbers (under duress).

When I'm actually using numbers to perform (even simple) operations, I might do any of these
- think in a sort of abstract way (via the application of rules memorized by rote, if there's a cute or handy pattern - like when you're multiplying with 9, 10, 5, 3. Actually, just those times)
- refer to external physical entities that I can see or feel - my fingers, or a grid or point system that I have to draw (if e.g. grouping numbers).
- if I'm just straight counting a lot of things, may count out loud or tap things out with my finger. (I think these are all compensatory strategies. Was actually average in skill, but math-phobic. Had a really hard time with time.)

(I think it'd be neat to compare how people who were taught using different pedagogical strategies visualize things. I was taught using flash cards [for multiplication] and then with whatever books were around in the 90s. Bet the common core kids might see things a little differently.)
posted by cotton dress sock at 1:55 PM on August 18, 2016 [1 favorite]

My number form synesthesia has gentle curves in most places but also right angles at 10, 20 and 100. It has a black background that is covered in white points of light, similar to the night sky. The numbers are light yellow and three dimensional, like letter magnets. I also have a calendar form that is like 12 calendar pages laid out in front of me from left to right, but laying sideways with a white background and black text. The current year is closest, but all the previous years are behind it in descending order. The only right angle as the years line up is at 1900. The 24 hour clock is similar with just the hours going from left to right but sideways and repeating with no right angles. The color is not as strong but it seems red to me now that I think about it.

When doing basic arithmetic or date or hour calculations, these pictures pop up automatically for me. It took reading about the phenomenon for me to realize that this does not happen to everyone.
posted by soelo at 2:10 PM on August 18, 2016 [1 favorite]

In grad school (earning a math degree), I asked a classmate what he visualized for certain mathematical objects, and he replied "nothing".

"What about the number 2? Don't you see two dots, or a location on a number line, or the numeral written out?"

"I don't think of anything for the number 2, or the group Z/6Z, or the projective plane."

He finally admitted that he would, if necessary, think of a round object when discussing a circle.

I asked him because I was reflecting that as a child, I associated each digit with a gender. I don't do this anymore, but it still seems a natural choice. I could perform the operations by "abstracting" away the gender and just remembering various combination facts, but I always felt that each number had more defining characteristics. Today, I am so very grateful to not have to do this. Am I missing something strange and perhaps amazing about the way my brain used to work here? Maybe, but I'm no longer tied up in spontaneous stories and concerns about how the numbers feel.

Today, I guess the digits just occupy a certain distance on a line to 10, and then the tens are the same thing going toward 100. Do I actually picture these things? No, but I have to create pictures when explaining things to my classes. I keep the pictures people seem to appreciate, and I only think about them if explaining a topic.
posted by klausman at 2:16 PM on August 18, 2016 [1 favorite]

This is a very odd and interesting question. I have a math degree but I'm not super amazing at it. Anyway I never really just see numbers. I think it sounds really neat that a lot of you have fun personalities or colors or shapes to your thoughts about numbers. It's only if I'm using them that I seem to have any real form they take and in that case I tend to visualize myself writing things down and doing the math with a pencil and paper in my head. I've always done things that way whether it's addition or division or calculus or physics. I've never thought of numbers as pips or dominoes or anything cool like that in my head. To me they just, are? I feel boring now hah.
posted by FireFountain at 2:25 PM on August 18, 2016 [2 favorites]

For me, single digit numbers are mentally paired with their "antithesis", sort of like this:

5 (5)
4 6
3 7
2 8
1 9

0 10 obviously

So, if I need to add or subtract, I can "flip" the numbers to make the process easier. So, if I need to add 15 to 7, I can flip that 7 to be 2 one stage up, thus 22.

I also mentally "pair" all kinds of mathematical things. Like addition and multiplication are similar in my mind. Plus, I make a similar "ladder" of going up one side and down the other when I am seeking the least common denominator and I use primes to cut that process down.

I also count in binary on my fingers, a think I learned at age 14, and no doubt do all kinds of weird things mathematically. I seem able to hold more digits in my head than average, according to some data I have seen and I have a few other mathematical quirks.

But I am not very visually oriented. I make most of these mental associations without any big visuals and it certainly isn't colorized.
posted by Michele in California at 2:37 PM on August 18, 2016 [3 favorites]

My number visualization is pretty boring. I have a vague mental image of numbers in a line going from left to right. The standard image has zero at the far left and 100 off on the right. I'm closest to and paying most attention to the numbers from 1-10. If I need to, I can pay more attention to other parts of the line or picture the parts with negative numbers or numbers above 100 but that's not what's in my standard mental image. There's some vague sense of the numbers going up as well as to the right, especially when I'm doing tasks like figuring out 37 + 6 by thinking "add 3 to get up to 40, then 3 more makes it 43." The numbers aren't represented by dots or marks on a line, they're just numerals that are lined up. (As I explained in this thread, my "visual" images aren't really very visual, so I just have the sense that they're numerals. It's not like I'm literally seeing them.)

The months of the year are on a similar left-to-right line with January at the far left.

My daughter thinks of each number as having a gender. (She thinks of many other things as having gender also.) They don't feel at all gendered to me. If I ask myself, "Does 2 feel more male or female?" no answer comes to mind and either possibility seems equally valid. They really seem completely neutral, like dust or rubber bands. (But I wouldn't be surprised if my daughter said dust or rubber bands had a gender in her mind.) The one exception is 9, which seems more female to me. And now that I think about it, I guess 10 seems male.
posted by Redstart at 3:06 PM on August 18, 2016 [1 favorite]

Wow, really interesting to read these responses. I don't visualize numbers at all. I do give them some personality traits, though:

1 is individual (duh)
2 is paired and bound
3 is ... huh, I guess I do visualize some numbers, because I keep seeing 3 dots in a row, like on a die, or ellipses. But 3 is unique, a little edgy and fun
4 is stable, but a bit boring
5 is wealthy and desirable
6 is my favorite number (because I loved being six years old)
7 is dangerous and a bit full of itself
8 is ... I don't know, I don't have a strong feeling about 8
9 is untrustworthy and domineering
10 is ... ubiquitous. It can mean so many things that it doesn't mean anything.

I've never consciously thought about this before, and now, having typed it out I'm not sure I like what it says about me...
posted by widdershins at 3:18 PM on August 18, 2016 [1 favorite]

on consideration: when e.g. working out a restaurant bill, in the absence of a fully charged phone & therefore calculator, I tend to draw the equation with my finger [on my palm or a piece of paper or the table]. So there's probably no visualization going on re numeric concepts per se, but maybe some sort of secondary visuo-kinaesthetic associations going on with symbols & operations (as taught), driven by a more abstact and general process.
posted by cotton dress sock at 3:32 PM on August 18, 2016

I would tend to imagine each digit as being a series of dots arranged how they would be arranged on a domino or a kid's poster (so 3 is a diagonal line or maybe sometimes a little pyramid, 5 is a square of dots with a dot in the middle, etc).

I guess if I had to give numbers personalities, what I'd think of is connected to they symmetry and balance on a domino as well
10 feels like a family with too many kids.
9 feels like a very flexible, versatile, balanced thing (it makes a nice cube)
8 feels balanced, symmetrical, simple, and lucky to me.
7 feels unbalanced and a bit precarious (that 3 dot line can't hold up the 4 dot square!)
6 is like a junior 8, trying to be stable and solid
5 seems a bit cluttered and mobile, like a van full of stuff
4 is nice and stable
3 is a nice stable little pyramid.
2 is a pair in comfortable tension.
1 feels solitary and a bit final but also kind of badass

I don't imagine sequences of numbers as having any arrangement at all.
posted by pseudostrabismus at 4:39 PM on August 18, 2016

I don't really visualize numbers! I think of them more as words/sounds in my head, or just abstract concepts. I will say that working a job where I made change (even though the register told us the amount, we had to select the correct number of coins) means that adding or subtracting is now change making in my head, more than a year after leaving that job.
posted by MadamM at 4:49 PM on August 18, 2016 [1 favorite]

I do not do this with numbers, but I do with the upcoming days of the week. Tomorrow and the following days always are imagined as moving upward and to the left, as if following a path into the future. Only about 5-6 days at a time.
posted by yclipse at 5:25 PM on August 18, 2016 [1 favorite]

same as michele here, plus the carry is like a typewriter return zinging the carriage across. so 123 + 877 interlocks except at the end, which goes bading across to the left. if that makes any sense at all (it's very satisfying).

also, 1 is brown, 2 is red, etc not from synaesthesia, but resistor colour code.
posted by andrewcooke at 6:02 PM on August 18, 2016 [3 favorites]

I picture the numbers 1-10 on a plain number line that continues on infinitely, but really only "see" the spatial relationship between numbers up until about 100. Above 100, I have no spatial concept of them.

Similar to others above who see "dots" as visual representations of numbers, I always see the "corners" of each digit representing the number of each units in numbers 1-6, and 8. So, the two end-points of the physical shape of number 2 represent that there are two units; 5 has its two end points, the corner where it has a right angle, the corner where it meets the curve, and a mid-point on the curve, for a total of 5 points/dots. This doesn't work for 7 or 9 though.
posted by samthemander at 6:59 PM on August 18, 2016

For simple arithmetic, they're little bits of things that happen to like to self-assemble in base 10. So two clumps of 5 are much happier as a clump of 10. They're really just a formless blob though, 10 marbles or sesame seeds together. 100 is a small pile that holds together in a satisfying way, 1000 stops being messy and makes a lovely small pyramid. 10k and 100k are very sloppy giant piles, hard to tell apart, but 10e6 (and 10e9, 12, etc) makes a big pyramid, shinier the bigger it is but also hard to differentiate. I often have reason to think in 10e-3, 6, 9, and those are spheres or circles that live inside one whole unit, a bit like an atomic model in a child's textbook.

If I'm knitting and counting stitches it's a physical number flipping over like a just pre-digital alarm clock, and it continues as I knit whether or not I pay it any mind up to about 150. I'll have a conversation while casting on and stop and check and find I'm within 2 stitches of the right spot.

Compared to folks I've had reason to compare with, I'm particularly good at fast order-of-magnitude guesses in my head, but pretty bad at making change or similar problems requiring accuracy without pen and paper. When adding a long column of numbers, I clump any 10's I can find first, which my family finds very strange.

I read bits of this thread to my husband who apparently is in the "numbers have personality" camp. He says prime numbers have a Dread Pirate Roberts roguishness, and long numbers ending in 01-49 are more innocent and 51-99 are more malign.

In one of Richard Feynman's two memoirs, he discovers that some of his physics/mathematician colleagues can count while reading and others can't, and the difference seemed to lie in how they pictured the numbers. So if you're looking for published work that touches on this topic you might seek that out.
posted by tchemgrrl at 7:36 PM on August 18, 2016 [1 favorite]

I'm unable to visualize numbers. Can't do math in my head except what was put there by painful rote memorization, e.g., multiplication tables up to 9. Others, especially coworkers, have trouble believing me because, for them, it contradicts the other evidence of my level of intelligence.
posted by Homer42 at 7:41 PM on August 18, 2016 [1 favorite]

39 38 37 36 35 34 33 32 31 30
20 21 22 23 24 25 26 27 28 29

... and so on. They keep zigzagging in groups of ten up to 100 and then the pattern repeats. I can't think of a number (or a year) without automatically seeing it in this context.

I always assumed everyone else had some arrangement of their own, until a few years ago when my wife overheard me comparing mine with my son's.
posted by wps98 at 8:38 PM on August 18, 2016 [1 favorite]

I tried to get my 13 year old daughter to describe how numbers are laid out for her. It sounds like maybe she just doesn't visualize numbers except one at a time. About all she could tell me was, "The teen numbers are over to the left." I asked her to close her eyes and point to numbers as she saw them in her mind and that was weird. Some bigger numbers were to the left of smaller numbers and some were further out ahead of her. In general, the bigger numbers seemed to be farther away, but not farther to the right or left. She couldn't decide whether a negative number was down or way off to the right.

She sees all the numbers as having colors, gender and personality. She says they're all big, chunky, colorful block numbers in her mind. Here are her descriptions:

1 - white, kind of boring, little personality, male
2 - green, male, kind of good-natured, likes games and stuff, doesn't have super complicated personality, gets along well with other numbers
3 - maybe like 6's adopted sister. Also kind of rusty-pink but more rusty. Likes playing tricks a little bit, but good-natured, friendly.
4 - male, green but more colorless somehow than 2. It participates. It's relatively good at stuff. Friendly and good-natured. It doesn't have hair, though the others would if they were people. Kind of looks up to 5.
5 - enthusiastic, dull orange, male, happy all the time, really into sports, gets along great with the other numbers, kind of the center of attention and likes that. Would probably have short light brown hair, spiky and messy.
6 - friends with 7. (6 is younger, obviously.) Rusty-pinkish. Female. Tomboyish, happy, enthusiastic.
7 - female, mellow, laid back, straight long blonde hair
8 - male, dark blue or some dark color, serious. 8 and 9 are kind of buddies. They're almost teenagers. They hang out and drink or whatever and are kind of separate from the other numbers. They're both guys.
9 - kind of friends with 10 but 8 is pretty much just friends with 9. 9 is almost colorless but at the same time some dark color.
10 - pink, female, really cheery, maybe into soccer or something but not too hard-core. Hangs out with the other numbers but probably feels a little superior to them. The bigger numbers are older and more mature.
11 - probably 10's best friend, yellow but almost cream-colored, thinner than 10 and probably a little older and acts a bit more mature. Probably has pretty short hair that's maybe dirty blonde.
posted by Redstart at 8:40 PM on August 18, 2016 [3 favorites]

My 10 year old son says he also sees numbers as having gender:

1 - male
2 - female
3 - male
4 - male
5 - male
6 - female
7 - female
8 - male
9 - male
10 - male
posted by Redstart at 8:49 PM on August 18, 2016 [3 favorites]

When I think of numbers, I just see the numerals written out. In middle school I gave significant thought to how each one should look when I wrote it.

For instance, I decided 7 always has to have a crossing bar. If I'm trying to be fancy, it gets a vertical serif on the left end as well. Each number has a regular form and a fancy form.

If I can't seem to do a simple arithmetic problem in my head, I often find myself writing out the numbers in the air or tracing them on my palm or the table.

No number line or other spatial metaphor for me.
posted by danceswithlight at 9:09 PM on August 18, 2016

As sometimes happen on MeFi, I stop to wonder if a thread is all an elaborate hoax to see how I'll react. Like now. I know about synesthesia, but yeah, no.

I don't visualize numbers at all. Not shapes. Not dots. Not genders. Not colors. Not personalities. Depending on whether it's math (calculations) or mere numbers (like a phone number), I can see a calculator (7-9, 4-6, 1-3, then zero, going down) or the phone (with 1-3, 4-6, 7-9, then zero, going downward), but that really only happens if I'm going to be calculating or dialing. But numbers have no shape, size or anything to me. Nor do letters.
posted by The Wrong Kind of Cheese at 10:05 PM on August 18, 2016 [3 favorites]

The number 24 makes me very happy. 12 makes me almost as happy. Odd numbers make me very uncomfortable, especially in addresses. I love binary numbers because they challenge me to convert them to 10-base.
posted by bendy at 10:26 PM on August 18, 2016 [1 favorite]

I don't visualize numbers at all. Not shapes. Not dots. Not genders. Not colors. Not personalities.

That's exactly like me, which is why I find this question so interesting! I'm actually pretty good at and enjoy math, but to me, numbers are just numbers, abstractions which I can bind to symbolic representations (sounds or numerals), collections of objects, list ordinals, dimensions, etc. It seems like having so many additional perceptions and associations with numbers must make working with them a much richer experience!

The people I know who have a strong number line tend to be particularly good at arithmetic, which seems to echo what several people are saying here. Personally, I don't have a particular talent for arithmetic, though I'm also not bad at it and I've learned some tricks over time to improve my mental calculations. (I do have a strange difficulty adding up the pips on two or more dice sometimes, often declaring that a 4 and 5 rolled together make 10, for example, which is one of my wife's favorite facts about me for some reason.)

Thanks everyone for the answers so far! I love the diversity of experiences. The idea that numbers could have gender is something I never would have considered.
posted by biogeo at 10:55 PM on August 18, 2016

I think of numbers primarily as words and cultural functions. There's no visuality. I have a math degree because I liked the words used to describe structures derived from numbers, and there were no lab reports or pre-meds to compete against.

Like, 269 is special because it is the page of a song called Bear Creek, and it also is the (now defunct) bus line that went to Bear Creek Park and Ride. HTTP response codes, obviously, have those associations.

8 is very cool because it spans computers, music + dance, chemistry (because earlier chemists were also musicians, apparently), all things I care about. 6 is kind of lame. It's sandwiched between 5 (which is very strong -- V-I, and also, being half of 10) and 7 (which is jester-like, with the repeating .142857 in its reciprocal -- double, double, double plus one). Sure, it's perfect and the size of the smallest non-Abelian group, but like, it's so _composite_ without being a perfect square like 4 or 9, or a cube like 8.

I've always thought 19 was a bit frightening for having 361 as its square; it turns itself upside down, and square roots its child-most digit? I also think of 13 and 14 as grudgingly friends: their squares are 169 and 196 -- same digits, different order, but the roots are as different as can be: one is a prime, and significant in western culture, whereas the other is even, and has a similar significance to Chinese people who care (or maybe just me, who occasionally makes things up by overthinking): you can spell 14 as yao1si4, which sounds like yao4si3, "about to die".

1 has no particular meaning to me, nor 0. Negative and complex values are purely abstract.
posted by batter_my_heart at 2:25 AM on August 19, 2016 [1 favorite]

I usually visualize a number line in empty black space, and I'm floating above and around it. If I'm zoomed out looking at a relatively large range, the numbers go from right to left, except if I'm looking at negative numbers, then I orient them left to right. Basically, larger absolute values are on the left. However, if not zoom in to the span of just a few integers the number line in my head reorients so the larger values are on the rights. As a point of reference, I have an advanced degree in math, and have always tended towards the types of math that can be more easily visualized (e.g., graph theory, topology, etc).
posted by noneuclidean at 3:54 AM on August 19, 2016 [1 favorite]

  (that 3 dot line can't hold up the 4 dot square!)

inorite?!?! Primary school teachers didn't know what to make of me and my concerns. Though they did, every year without fail, manage to summarize me as “lazy” on my report card.

Didn't stop me loving mathematics and getting exemptions to every final throughout university in engineering mathematics. To this I owe endless gratitude to one exceptional grammar school teacher — Miss Anderson — whose mix of confidence, threats and very dry humour got me past my fear of sums.
posted by scruss at 9:39 AM on August 19, 2016

Numbers have volume to me. Like the blocks as a kid where you have a single block for 1, a row for 10, a square of 10x10s for 100. So '362' if I want to know what to add to it to get '500', I'm just looking for the matching shape to fit together with the 362.

The digits also have a hint of colors. I realized recently it's the same color set as Super (Number) Munchers from when I was a kid. So like 167 is white and blue/green. 852 is brownish red yellow. It's only a hint of colors, though.
posted by Lady Li at 2:40 PM on August 19, 2016 [1 favorite]

I'm fast at mental calculation. Quasi-freakishly fast, to be honest. Like just now I multiplied 23 by 59 to observe my own thought process; it took one second and then I grinned really big because the answer was 1357.

Nowhere in that process did I imagine a number line. The actual process was roughly "23 times 3 is 69, doubled is 138, multiply by ten and subtract 23"; if there was any visualization, it was of the digits involved. I did, however, irrelevantly notice that 23 and 59 are both one less than a multiple of 6, which led me to very briefly imagine 6-by-4 and 6-by-10 rectangles of stars (colored like the stars on the U.S. flag) with one star missing from a corner. And while typing this out, I thought of two dozen eggs with one egg missing, and a clock showing 11:00 (which is 23 hours past midnight), and so on. I think some of my calculating ability comes from having an abundance of visuals available, with no one of them dominant; I can flip through them very fast to find one that's relevant, or ignore them and just imagine pencil and paper. Phunniemee's comment about having to pass through 6, 7, 8, 9 to get from 5 to 10 made me feel a little queasy!

Biogeo, I find it interesting that your friend's number line has very distinct sections for the one-, two-, and three-digit numbers. Sounds like it has aspects of both a linear and a logarithmic scale.
posted by aws17576 at 5:07 PM on August 19, 2016 [2 favorites]

I see numbers very physically - like almost like coffee beans set on a table - and they somehow 'feel' warm to the touch. and they are always broken down into smaller, specific groups e.g.
4 is 2 & 2
5 is a 2 & 3
7 is 3 & 4
8 is 4 & 4
9 is 4 & 5
45 is 4 rows of ten & a group of 5 etc. etc.

..needless to say, i'm ridiculously bad at math.
posted by speakeasy at 1:17 PM on August 20, 2016 [1 favorite]

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