Lots of interesting details here and there. Like Chinese counting rods, which were used somewhat like an abacus. Not so interesting maybe, but the thing that tickles me is that the rods change orientation in each column for a power of ten, presumably as an error correction aid. So 1's go vertical, while 10's go horizontal, then 100's are vertical, and so on.
The Babylonians had a place-value system for their base-60 mathematics, but it generally fell into disuse. By antiquity, most numbers were written using a coded alphabet. The early Greeks had a system much like Roman numerals that repeated (as necessary) the letters for 1, 5, 10, 100 and so on; this acrophonic system used the first letter of the number it stood for. So pi for penta was 5, delta for deka was 10. The later Greek system used all the letters of the alphabet (and a few obsolete ones). The first nine letters of the alphabet stood for 1-9, the next 9 for 10-90, and the final 9 for 100-900. It's sort of a place system, but it's a crappy one, and it's miserable how long it lasted, both in Greece, and in borrowed form throughout the Old World. One of the peculiarities is that it was borrowed by the Arabs, but later the Arabs rearranged their alphabet (putting signs that look similar together) but they didn't reorder the number assignments! (You only moved the headstones!) So the alphabetical order for dictionaries and phone books goes 1, 2, 400, 500, 3, 8, 600 ... they had to devise mnemonics to remember how the numbers go. And it's also interesting that you can sure hear the old Phoenician-derived alphabet if you read the Arabic alphabet in numerical order: alif, ba, jim, dal, ha, waw, zay...
Once we finally get to Hindu numbers, I was surprised that in addition to just having names for the numerals, they also had 'poetic' substitutions that often related to mythology. So instead of 'two' one might say 'eyes', or instead of 'four' one might say 'yuga' for the four cosmic cycles. Zero was sky, space, atmosphere, or void.
Oh and if you like the multiplying with crossing lines thing, maybe you'll like multiplication per gelosia.