Now our exams for the year are done my housemate and I are free to wallow in esotric nerdishness without recourse to computation or the relevance of our discussion matter to.. well, anything really 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?