> And addition and subtraction don't require thought, as far as I can tell. Give me a couple numbers and I add them up. Unless there are digits that need carried, in which case I think about it for a split second.

I do it that way, too. For me, borrowing in subtraction tends to take a split-second longer than carrying in addition.

> I don't *think* it's possible to actually visualize a 4D sphere for example, but if you just see the important parts as invisible streaks of movement, it's not too hard.

There is where the way we do math diverges. I can't do it that way. I see the shape in my head.

Incidentally, when multiplying in my head, I tend to visualize it as if I were doing it on paper. I've learned some tricks for quick mental math from a book, and I now use some of the techniques described. For example, when I multiply a 2-digit number close to a multiple of ten by another 2-digit number, I tend to multiply the other number by the multiple of ten, then multiply the difference by the other number, and then do the requisite addition/subtraction to get the answer.

The book (actually, it was a pair of volumes in the same series) had 114 tricks total. I just use the ones that I found practical. One thing I've learned to do is that, when I look at a string of numbers that needs to be added, I usually find groups of 2 or 3 that add to multiples of ten.

Lucky "I've also memorized the first 12 powers of 2 and the first 6 powers of 3, and so on... 11 is the largest number for which I've memorized the third power" Wizard