We're very _used to_ this kind of representation. But there's nothing obvious about it at all. Every single thing in the representation can be questioned, and possibly changed. It's fun to try some experiments!
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What's responsible for this astonishing fact? If you look back at the reasoning above, you see it's a consequence of associativity, commutativity, and distributivity. That's a pretty huge set of things! And it makes addition _really_ nice in this representation.
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In particular, it means that although the numerals have different meanings in different locations, in many ways they continue to _behave_ as though they are in some sense "the same". They're _not_ the same. But they retain many of the same affordances, in modern interface lingo.
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You can go through the same kind of thing, trying to figure out how to multiply two numbers: wx*yz = (w*10+x)*(y*10+z) = x*y*10*10 + ... I won't go through it, but you can guess what happens: you get the grade-school algorithm for multiplication.
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And, again, even though the numerals have very different meanings in different positions, they have a lot of the same affordances, and so behave in some ways as the "same" object.
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You can go through this with long division. Same story.
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At this point this new numeral system looks vastly superior to roman numerals. It's exploiting all this structure to get compact representations, which don't require new symbols, & which exploit deep properties of numbers to simplify addition, multiplication, & division. Amazing!
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I'm far from an expert on the history of mathematics or its representations. The story above is unrealistic in many ways. Still, I think it gives at least a hint of the incredible mathematical and design insight required to invent arabic numerals.
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And, of course, this story requires many incredible earlier insights: words, the alphabet, various ideas about multiplication and addition, and so on.
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To finish, a fun question: is there some way to improve still further on arabic numerals? I believe the answer is almost certainly yes! But that's a thread for another day.
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A few addenda: (1) as lots of people have pointed out, it'd have been more accurate to name them Hindu-Arabic numerals; (2) the ancient Greeks seem to have known much of this (which I didn't know);
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(3) the ancient Babylonians ditto (which I did know - it was meant as a thought experiment about discovery and lines of insight, not history); (4)
@DavidDeutschOxf has a lovely discussion of number systems in chapter 6 of "The Beginning of Infinity".Show this thread -
(5) Via
@CXGonzalez_, a paper arguing that for educated romans, the computational difficulty of working with roman numerals was comparable to us working with Hindu-Arabic: http://csjarchive.cogsci.rpi.edu/Proceedings/2008/pdfs/p2097.pdf …Show this thread -
(6) I haven't published anything specifically on improving Hindu-Arabic numerals. But here's some related work inspired in part by that problem: on "Magic Paper" (new interfaces for mathematics) http://cognitivemedium.com/magic_paper/index.html …
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"Toward an Exploratory Medium for Mathematics" (on developing a logic of heuristic discovery, to underly creative exploration) http://cognitivemedium.com/emm/emm.html
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And "Thought as a Technology" (about the idea that we internalize the interfaces we use as part of our thinking; interface designers actually help us think new thoughts): http://cognitivemedium.com/tat/index.html
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End of conversation
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