If we did that, then we'd have xy -> x*23+y, xyz -> x*23*23+y*23+z, and so on for longer strings. That's even simpler, though A is now playing pretty much the role of a placeholder.
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Still, those things perhaps don't seem that important. Certainly not worth replacing an entire piece of intellectual infrastructure with!
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But where this number system really shines is in simplifying certain other things you might want to do. For instance, consider addition of the numbers wx and yz.
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We have wx + yz = (w*10+x) + (y*10+z) = (w+y)*10 + (x+z). Fiddle with this for a while - it's more work than I want to go through here - and you eventually recover the grade school algorithm for adding two-digit numbers. Do some more work, & you get the n-digit algorithm.
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There's a miracle going on here, one we don't notice b/c it's so familiar: I pointed out above that the numerals have very different meanings, depending on their location. But despite this, in the grade-school algorithm we use the _same rules_ for addition, regardless of place!
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E.g., in computing 27+38 at some point in the computation we'll use 2+3 = 5; in computing 72+83 at some point we'll also use 2+3=5. That's despite the fact that the 2 and the 3 in the first sum have a very different meaning than in the second sum!
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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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