I see the written representation of the equation and then manipulate it like I would with a pencil and eraser.

I actually like this question a lot. I've been practicing mental visualization and mental audio reproduction. I call it fidelity. My theory is by increasing my mental fidelity, I can increase my overall intelligence.

Interesting. It seems mine changes with the weather.

I can think of numbers as their symbols and manipulate them in my head like that. I can do small long addition, multiplication and division in my head this way but it's bounded by working memory I think. It seems like a skill I could improve on.

When I read numbers/equations first, however, it's audio. I sound the numbers out, much like I would words, since I hear the words in my head when I read. I can read without sounding out but I've noticed comprehension can suffer depending on the text.

None of this has any particular colour. The only things in my thought with colour are constructed imaginary scenes (for instance, walking on a beach) or memories. Both have a high degree of audio, visual and kinaesthetic information - that silly learning style categorisation test never seemed to work on me.

My head has more auditory stuff going on than visual. I "say" things in my head as I read them. It makes me a slow-ish reader (compared to most smart people, who typically read faster than I do, though I read faster than average).

But I was good at math in my youth. One of the reasons for that is I make all kinds of weird mental associations. For example, I relate to single digit numbers in a paired or complementary way, such that 3 and 7 are oppsite sides of the same coin and it helps me with addition or subtraction. To me, those mental categories are part of how I percieve numbers, though it doesn't correlate readily to perception in the vein of sight or sound.

There's no geometric complement in my mind. At times it's useful to think of "wholeness" base 10, so 4 and 6 fit together, and when adding numbers, I sometimes find the nearest multiple of 10 and then go lower from there. I don't picture numbers in the abstract (e.g. 178); when they refer to something else (e.g. 2 coins), they're a description or a property.

I lack the ability of attaching numbers to an image or anything, I just make them float in the air, but they don't keep up long, since my short-term memory is not that great. That's why I need to be extra fast to calculate anything before they go out 'poof!' :)

The way I visualize numbers is actually based on a picture from my maths book in primary school. In that picture numbers were visualized as coloured, spiralling band / number ray with the integers from zero to 1000 written on it. The band started somewhere in the centre and spiralled rightwards and outwards from there.

The funny thing is that I continued to use this image till today and for larger and more complex sets of numbers. Negative numbers for me are a continuation of the spiral to the left. For larger numbers the spiral continues to the right and in upwards in the 3rd dimension. Each number that's larger by an order of magnitude of 3 (10, 10000, 10000000 ...) roughly lies on the same 2-dimensional vector.

When I see numbers in passing, for example a telephone number on an advertisement, I'm obsessed with adding them up to see if the total is divisible by 9. I do the same with letters of company names. Done this for years, a compulsive tick.

Any reason why the number 9? Where did the habit start from?

Spelling. I was really bad at spelling. So one day on the bus, about 20 years ago, I decided to count the numbers of letters in words, assigning them different weights depending on being capitals or not. Why 9? It seemed fun at the time to have a secondary goal.

I kind of see equations with a video in my head of numbers moving around. so 11 might become 10+1 or 9 might become 10-1 depending on how I need to put the numbers together. X might move to one side of the equation etc.

physics ruined me. whenever i encounter a number now, i round it to the closest order of 10. you'll likely not encounter a number outside the range [10^-34,10^40], so you really just need to know how to subtract -34 from 40 and everything in between. no need for semi-autistic interpretations of numbers.

I don't get any kind of image, they are just numbers.