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michael_nielsen

@michael_nielsen

Searching for the numinous. Co-purveyor of https://quantum.country/ 

San Francisco, CA
michaelnielsen.org
Vrijeme pridruživanja: srpanj 2008.

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    michael_nielsen‏ @michael_nielsen 18. ruj 2019.
    • Prijavi Tweet

    Imagine you're a designer or mathematician living in ancient Rome. Being a curious & imaginative sort, used to exploring wild ideas, you ask yourself: is there some way you can improve on the roman numeral system? Might it be possible to find a better way of representing number?

    13:28 - 18. ruj 2019.
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    75 proslijeđenih tweetova 399 korisnika označava da im se sviđa
      1. Novi razgovor
      2. michael_nielsen‏ @michael_nielsen 18. ruj 2019.
        • Prijavi Tweet

        Now, I'm pretty certain this would have seemed a wild idea to most of your contemporaries. But from our modern vantage point we know that in fact a much better system is possible: arabic numerals.

        3 proslijeđena tweeta 32 korisnika označavaju da im se sviđa
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      3. michael_nielsen‏ @michael_nielsen 18. ruj 2019.
        • Prijavi Tweet

        So a modern phrasing of the question might be: how to invent arabic numerals, assuming you only know roman numerals?

        1 proslijeđeni tweet 39 korisnika označava da im se sviđa
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      4. michael_nielsen‏ @michael_nielsen 18. ruj 2019.
        • Prijavi Tweet

        To be clear, I'm not talking here about delving into the actual history of arabic numerals. That's long and complex and fascinating, but this is a different kind of challenge.

        0 proslijeđenih tweetova 17 korisnika označava da im se sviđa
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      5. michael_nielsen‏ @michael_nielsen 18. ruj 2019.
        • Prijavi Tweet

        What I'm talking about is a kind of discovery fiction, a plausible line of reasoning that might have led you to the discovery of arabic numerals, with roughly the set of raw materials on hand in ancient rome.

        5 replies 1 proslijeđeni tweet 56 korisnika označava da im se sviđa
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      6. michael_nielsen‏ @michael_nielsen 18. ruj 2019.
        • Prijavi Tweet

        I spent quite a bit of time trying to find such a discovery fiction. Originally, I started from the question "If we changed notation [from roman numerals] might there be an easier way to multiply?"

        0 proslijeđenih tweetova 22 korisnika označavaju da im se sviđa
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      7. michael_nielsen‏ @michael_nielsen 18. ruj 2019.
        • Prijavi Tweet

        But there's a different starting point, another rather natural question, that is more promising. It's to ask: how many possible distinct _words_ are there of any given length?

        1 reply 0 proslijeđenih tweetova 20 korisnika označava da im se sviđa
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      8. michael_nielsen‏ @michael_nielsen 18. ruj 2019.
        • Prijavi Tweet

        Now, the roman alphabet was much like our modern alphabet, but with a few differences. It had 23 characters, mostly the same as ours today: ABC.... Z (but with a few differences, eg no J).

        0 proslijeđenih tweetova 8 korisnika označava da im se sviđa
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      9. michael_nielsen‏ @michael_nielsen 18. ruj 2019.
        • Prijavi Tweet

        How many words are there of length 2 in this alphabet? Well, the possible words are AA, AB, AC... AZ, followed by BA, BB, BC, ..., BZ. Then CA, CB,..., CZ.

        1 reply 0 proslijeđenih tweetova 7 korisnika označava da im se sviđa
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      10. michael_nielsen‏ @michael_nielsen 18. ruj 2019.
        • Prijavi Tweet

        Obviously, that means there are 23 times 23 = 529 possible "words" of length 2. Of course, multiplication was actually pretty tricky for the romans (mostly because they didn't have arabic numerals!) But figuring this out was well within their ken.

        1 reply 1 proslijeđeni tweet 11 korisnika označava da im se sviđa
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      11. michael_nielsen‏ @michael_nielsen 18. ruj 2019.
        • Prijavi Tweet

        Stated in roman numerals: there are DIXXX possible words of length II in the roman alphabet.

        0 proslijeđenih tweetova 11 korisnika označava da im se sviđa
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      12. michael_nielsen‏ @michael_nielsen 18. ruj 2019.
        • Prijavi Tweet

        What about words of length III? Well, we can just enumerate again: AAA, AAB, AAC, ...., AAZ, ABA, ABB, ... ABZ. Keep going, you get to BAA, BAB. And so on. It's not too difficult to see the answer is 23 x 23 x 23.

        1 reply 0 proslijeđenih tweetova 8 korisnika označava da im se sviđa
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      13. michael_nielsen‏ @michael_nielsen 18. ruj 2019.
        • Prijavi Tweet

        Unfortunately, 23 x 23 x 23 (= 12,167) was starting to stress the roman numeral system. They had ways of dealing with such numbers, but it got pretty convoluted. The system was mostly designed to work up to a few thousand.

        1 reply 0 proslijeđenih tweetova 12 korisnika označava da im se sviđa
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      14. michael_nielsen‏ @michael_nielsen 18. ruj 2019.
        • Prijavi Tweet

        The reason was that the basic strategy behind roman numerals is to keep introducing new symbols as you get to larger and larger numbers. That's okay for a while, but eventually breaks down.

        1 reply 0 proslijeđenih tweetova 16 korisnika označava da im se sviđa
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      15. michael_nielsen‏ @michael_nielsen 18. ruj 2019.
        • Prijavi Tweet

        So a curious thing about the word-counting problem is that you get large numbers of different possible words, without needing to introduce new symbols. That's interesting if you've been bugged by the need to introduce new symbols to describe large numbers.

        0 proslijeđenih tweetova 20 korisnika označava da im se sviđa
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      16. michael_nielsen‏ @michael_nielsen 18. ruj 2019.
        • Prijavi Tweet

        It's also interesting that you get very large numbers of different possible words, even with very short strings.

        0 proslijeđenih tweetova 6 korisnika označava da im se sviđa
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      17. michael_nielsen‏ @michael_nielsen 18. ruj 2019.
        • Prijavi Tweet

        Now, if you were playful, you might wonder a bit about using strings of letters to _represent_ numbers. One way would be to identify I with A, II with B, and so on, through XXIII -> Z.

        1 reply 0 proslijeđenih tweetova 8 korisnika označava da im se sviđa
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      18. michael_nielsen‏ @michael_nielsen 18. ruj 2019.
        • Prijavi Tweet

        You could just continue from there: XXIV -> AA. XXV -> AB, and so on. Symbolically: yz -> y*23 + z, where y and z are just single letters.

        1 reply 0 proslijeđenih tweetova 5 korisnika označava da im se sviđa
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      19. michael_nielsen‏ @michael_nielsen 18. ruj 2019.
        • Prijavi Tweet

        Of course, this isn't notation the romans would have used. But the ideas - multiplication, equality, addition, are all things the romans understood. It's not too far a stretch.

        0 proslijeđenih tweetova 7 korisnika označava da im se sviđa
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      20. michael_nielsen‏ @michael_nielsen 18. ruj 2019.
        • Prijavi Tweet

        What about with three digits? Well, with two digits you get up to 23+23*23. So we have: xyz -> x*(23+23*23)+y*23+z. I must admit, that seems slightly ugly to me, and it gets uglier with longer words.

        0 proslijeđenih tweetova 5 korisnika označava da im se sviđa
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      21. michael_nielsen‏ @michael_nielsen 18. ruj 2019.
        • Prijavi Tweet

        Let's try something slightly different - a tad less obvious, but also prettier. Let's imagine that instead of numbers of variable length, we have just a single universal counter, starting at: .....AAAAAA

        0 proslijeđenih tweetova 4 korisnika označavaju da im se sviđa
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      22. michael_nielsen‏ @michael_nielsen 18. ruj 2019.
        • Prijavi Tweet

        We increment by 1 and get to: ...AAAAAB. And then by 1 more and get to: ...AAAAAC. And so on.

        1 reply 0 proslijeđenih tweetova 3 korisnika označavaju da im se sviđa
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      23. michael_nielsen‏ @michael_nielsen 18. ruj 2019.
        • Prijavi Tweet

        Of course, in practice we don't want to write out all the A's on the left. So we'll just adopt the convention of taking those as given. In other words, something like: BD really means ...AAAABD, but the A's on the left are implied.

        1 reply 0 proslijeđenih tweetova 6 korisnika označava da im se sviđa
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      24. michael_nielsen‏ @michael_nielsen 18. ruj 2019.
        • Prijavi Tweet

        Now there's a bit of a problem, which is this convention makes ...AAAA be written as just a blank. Obviously that would be unreadable. So we'll introduce an exception: for that, we'll write A, just as a kind of placeholder.

        1 reply 0 proslijeđenih tweetova 6 korisnika označava da im se sviđa
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      25. michael_nielsen‏ @michael_nielsen 18. ruj 2019.
        • Prijavi Tweet

        I don't claim, by the way, that this "universal counter" approach is necessarily _more_ obvious than what we were doing earlier. But it's at least plausible as an alternate. And it has the benefit that it's an attractively unified system.

        1 reply 0 proslijeđenih tweetova 6 korisnika označava da im se sviđa
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      26. michael_nielsen‏ @michael_nielsen 18. ruj 2019.
        • Prijavi Tweet

        Okay, so the way counting goes is: A, B, C, ..., Z, BA, BB, BC,...., and eventually on to BAA, BAB, and so on.

        1 reply 0 proslijeđenih tweetova 3 korisnika označavaju da im se sviđa
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      27. michael_nielsen‏ @michael_nielsen 18. ruj 2019.
        • Prijavi Tweet

        In this system, xy -> (x-1)*23+y. And xyz -> (x-1)*23*23+(y-1)*23+z. Those seem satisfyingly neat, and the pattern continues.

        1 reply 0 proslijeđenih tweetova 3 korisnika označavaju da im se sviđa
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      28. michael_nielsen‏ @michael_nielsen 18. ruj 2019.
        • Prijavi Tweet

        There is still a slightly ugly thing, which is the -1 terms, which show up over and over again. We could get rid of those if we decided to start counting at 1 -> B, 2 -> C, etc.

        0 proslijeđenih tweetova 2 korisnika označavaju da im se sviđa
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      29. michael_nielsen‏ @michael_nielsen 18. ruj 2019.
        • Prijavi Tweet

        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.

        1 reply 0 proslijeđenih tweetova 4 korisnika označavaju da im se sviđa
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      30. michael_nielsen‏ @michael_nielsen 18. ruj 2019.
        • Prijavi Tweet

        At this point a clever designer might be bugged by the use of the alphabet in this number representation. After all, the alphabet is already being used for words! So it'd be better to switch to different symbols to prevent confusion.

        1 reply 0 proslijeđenih tweetova 7 korisnika označava da im se sviđa
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      31. michael_nielsen‏ @michael_nielsen 18. ruj 2019.
        • Prijavi Tweet

        The alternate symbols we use are pretty arbitrary. Let's go for A -> 0, B -> 1, C -> 2, and so on. Of course, we could introduce 23 new symbols. But the 23 is actually pretty arbitrary. So let's use just 10 symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

        1 reply 0 proslijeđenih tweetova 8 korisnika označava da im se sviđa
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      32. Još 26 drugih odgovora

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