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    I can do maths visually without pen or paper. I just imagine the equation or expression and manipulate it graphically. It usually helps to close my eyes :P.

    But with pen and paper it's not very visual initially. If I come a cross a problem, I do use visualisations to make sure I'm on the correct track. For example if I have to differentiate a function, I can usually visualise the graph of the function, and from that I can visualise the graph of it's derivative, then I check the equation I have for my derivative, and see if the visual for that is the same for the previous visual. That's just an example though, with differentiating I just integrate to check, but you get the idea. Also, I tend to associate some functions with certain colours and stuff too. For example I associate sine with red, cos with blue... which is handy because I also associate tan with purple, which makes it easy to recall that tan = sin/cos, since mixing my associated colours would give purple, for tan :P.

    So, yes, my maths is quite visual.

    Then again I am an Engineering student, so maybe what you say about 'physicists' and 'rules of thumb' is true when it comes to visual mathematics... perhaps pure mathematicians aren't as visual.
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    I often visualise pieces of Maths when I'm not doing Maths, too. Sometimes, something comes into my head and sticks there. For example, the other day, I was trying to envisage solution curves for the differential equation \frac{dy}{dx} = -(\frac{x}{y})^n for different values of n, that was a strange one. Also, trying to differentiate and integrate the Lambert W function, proving the SHM formula and trying (and failing, even when I later tried it on paper) to derive a similar result for \frac{d^3x}{dt^3} = -\omega^2 t. The funny thing is, half the time when this is happening, it's all subconcious, and I don't even realise I'm thinking about it, and I'm often doing something else.

    I'm a ****ing weirdo.
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    (Original post by Simplicity)
    Anything with more then three dimensions and anything that involves infinity.
    Anything in more than three dimensions I will view as projections into three dimensions. "Anything that involves infinity" could mean anything, but I assume you mean limits, in which case I see number lines and sequences and graphs and stuff. Or a Riemann sphere.
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    (Original post by tommm)
    I often visualise pieces of Maths when I'm not doing Maths, too. Sometimes, something comes into my head and sticks there. For example, the other day, I was trying to envisage solution curves for the differential equation \frac{dy}{dx} = -(\frac{x}{y})^n for different values of n, that was a strange one. Also, trying to differentiate and integrate the Lambert W function, proving the SHM formula and trying (and failing, even when I later tried it on paper) to derive a similar result for \frac{d^3x}{dt^3} = -\omega^2 t. The funny thing is, half the time when this is happening, it's all subconcious, and I don't even realise I'm thinking about it, and I'm often doing something else.

    I'm a ****ing weirdo.
    Eh, I do similar things.
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    I find that I'm only visual with maths when I need to be. If there's a predetermined formula which computes the answer, I don't waste my "thought energy" and don't think about things visually. But in STEP say [I've been sad enough to try some questions, yes], I sit there for long periods of time to construct a method visually to the extent that the method is very precise; then I answer the question.

    (Original post by tommm)
    I often visualise pieces of Maths when I'm not doing Maths, too. Sometimes, something comes into my head and sticks there. For example, the other day, I was trying to envisage solution curves for the differential equation \frac{dy}{dx} = -(\frac{x}{y})^n for different values of n, that was a strange one. Also, trying to differentiate and integrate the Lambert W function, proving the SHM formula and trying (and failing, even when I later tried it on paper) to derive a similar result for \frac{d^3x}{dt^3} = -\omega^2 t. The funny thing is, half the time when this is happening, it's all subconcious, and I don't even realise I'm thinking about it, and I'm often doing something else.

    I'm a ****ing weirdo.
    I do that consciously, but not subconsciously lol.

    Seems like maths is a large part of your life
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    I think most mathematicians think visually. It certainly helps the memory and the intuition.

    OP. You're not likely to do much at A level. I mean, algebraic manipulation doesn't really have a picture. But at uni, there are loads of things. Most definitions and therems in topology (that's about 130 definitions and 70 theorems in a first course) have a picture. Proof of the existence of the Laurent expansion in complex analysis is probably best remembered by the picture. Most of linear algebra can be reduced to pictures in R^3. When you start group theory, quotient groups, the first isomorphism theorem and group actions all have pictures. Field theory kind of has a picture but it's not a very good one.

    What tends to happen is you use pictures initially, and then you develop a feel for when you need to use certain results and do away with the pictures. Especially when things get more abstract or intricate.
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    I was both a Maths and an art student so was expected to be able to link the two, but never quite got the hang of it until last year; I had a great Alevel FMaths teacher who stressed the importance of presenting even abstract mathmetics in a visual form...I think you need someone to point it out to you properly; but once the notion's stuck in your head it never leaves...I reckon it's significantly improved my understanding of maths I never got the hang of combining my artistic part of my brain with my mathematic side though...it's odd the way in which I visualise and think about both is completely different...sometimes I think my maths is aesthetically prettier than my art
 
 
 
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