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One thing that's emerged from our ramblings is that we view maths in completely different ways. I like to think about things in terms of english wherever I possibly can; he has an incredible functional understanding of maths (which stands him in great stead in exams) but thinks about everything that isn't explicitly geometrical (which he also has a great capacity to picture) in computational terms.

To take a clear cut example, say you're solving simultaneous equations using matrices. I've come to understand it by regarding the matrix of coefficients as a linear map that takes an unknown point (x y z) to some definite point (a b c), and see inverting the matrix as finding the inverse of this linear map and applying it to the point (a b c). The way he sees it, when you multiply both sides of the equation by the inverse of the matrix of coefficients it's inherently obvious that you're left with (x y z) equal to some definite vector, and he figures I just make my life difficult for myself.

I was just curious as to how everyone else thought about this sort of thing?