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Thread: Ten is the lovliest number...

  1. #1
    Member
    Registered: May 2004
    Location: Canuckistan GWN

    Ten is the lovliest number...

    ...or is it?


    What is it about base ten? Does it actually have a special symmetry or is it just dear to humans because we have ten digits?

    Orders of magnitude can be effected by moving the decimal place. To this matho-phobe, that is an elegant capability which seems unavailable to other number bases. True or false?

    There are doubtless many gaps, errors and misunderstandings even in how I frame this question, so please don't take my statements as a fixed position, just a starting place. If anyone cares to help unravel this mystery (if it is one), I would love to hear.



    Many ancient numbering systems are based on sacred numbers; the seven visible "planets", the six male deities of the Babylonians, (possibly morphing into the twelve apostles, etc.).

    The Imperial system is a mad mashup of base four, with base twelve used for quantity, weight and distance but base sixteen used for fluid measures. Except of course for British human weights measure being base fourteen (a stone). It wasn't until the 1800's that Britain attempted to systematically inter-relate these different measurement magisteria with reference to common materials (for example, a gallon becoming ten pounds of distilled water at 62 degrees Fahrenheit and 30 inches of mercury.

    Even though we derive our decimal terminology from the Romans, their military units were generally organised on multiples of eight, occasionally alternating with multiples of ten or six. A century was eighty men, a cohort was 480 men and a legion of 4,800 men. This despite their mercantile and engineering calculations being done in quinary (base five) and decimal (base 10) as evidenced by the pocket calculators of the day. Apparently, Roman and Asian abacuses were ultimately binary and other cultures used similar calculating frames based their own culturally informed number bases.

    The ancient Chinese organised their armies on multiples of five while the Mongols used strict decimal increments when scaling up from the basic squad.

  2. #2
    Member
    Registered: Aug 2004
    Quote Originally Posted by Nicker View Post
    ...is it just dear to humans because we have ten digits?
    Bingo.

    Quote Originally Posted by Nicker View Post
    Orders of magnitude can be effected by moving the decimal place. To this matho-phobe, that is an elegant capability which seems unavailable to other number bases. True or false?
    You can move a decimal place in any base. We use "order of magnitude" as "power of ten" because we use base 10.

  3. #3
    Moderator
    Registered: Jan 2003
    Location: NeoTokyo
    Just yesterday the YouTube channel 3Blue1Brown released a summary video of top SoME3 entries (3rd annual Summer of Math Explanations or whatever it stands for), which I watched (and recommend to anyone with even a passing interest in math), as well as some of the entries I liked.

    One of them brought up the arbitrariness of base 10 and was asking the question if our entire real number line is arbitrary, or is there something special about it no matter what base you use.

    If you're going to drop a thread like this, I may as well link to it, since it's in the neighborhood and gives you a sense of what is and isn't arbitrary about numbers.

  4. #4
    Member
    Registered: May 2004
    Location: Canuckistan GWN
    Thanks for the replies. I have often wondered about this matter but lack the knowledge to even construct the question properly, and therefore to research it effectively.

    OK, I'm lazy. Thanks for indulging me.

  5. #5
    Member
    Registered: Sep 2001
    Location: The other Derry
    That's a cool visualization.
    Would love to see that extended to complex numbers.

  6. #6
    Member
    Registered: Aug 2009
    Location: thiefgold.com
    The Pythagorean tetraktys, the sum of the first 4 numbers


  7. #7
    Member
    Registered: May 2004
    Location: Canuckistan GWN
    The visualizations are indeed amazing. I wish I had been exposed to more such visual aids in school. My friend, who was far more numerate than me, remembers his teacher using such aids, like a set of wooden blocks like these...



    The amount of understanding contained in this simple device is astounding. He said he remembers the instant of realization it brought him as a small child.

    I remember the instant I gave up on math in high school. After weeks of solving geometrical equations, I could not remember the last time I had seen a geometrical shape, either in the geometry text book, or on the blackboard or in my head. I was crap at calculation and all the joy just drained out of me.

    I hope this thread might reignite some of the curiosity I lost. And I am grateful for any enthusiasm that people share here.

  8. #8
    Moderator
    Registered: Jan 2003
    Location: NeoTokyo
    For complex numbers, which heywood asked about, visualization can be tricky. Every value is a place on a 2D grid x + iy. The x is the real line, but there’s every value ∞i to -∞i added to it to make a single value modeled as a single point on that grid, so that's what the complex number space looks like. You just can't "line them up" as easily to know what it means to move around in that space.

    I actually think it's often better to think of them like vectors (arrow with its tail nailed to the origin) in a radian circle grid with a magnitude (distance) from the origin and an angle from the x-axis, which has the advantage of the angle just being between 0 to 2pi, and you can use the exponential and Euler’s Business, which just means it really simplifies the math for certain kinds of problems. In particular, it's good for modeling things that go through regular cycles like waves, wheels, pendulums, orbits, walking, economies, etc., where the angle part of the vector models the place in the cycle a thing is at some point (what they call phase), and if you take that trick and spin it, when it reaches 2pi it automatically is reset to zero for the next cycle.

    Complex numbers add like vectors. But they multiply weirdly. For real values, multiplying extends or shrinks the input as normal. But multiplying by i works like you're spinning a vector 90 degrees CCW by definition: 1*i = i, i*i = -1, -1*i = -i, and -i*i = 1, which normally wouldn't seem useful, but again if you're modeling anything that cycles, that's really useful.

    Anyway, it's a little easier to imagine "lining the numbers up in a sensible way" with that idea, though still a little tricky.

    And that's just imagining the imaginary number space itself. Trying to visualize functions, which map from one number to another number, is a whole other bag of worms!

    For real functions, you just imagine a line on a Cartesian grid, giving you a y output for every x input. But for complex functions, for every one-number-two-value input you have to give a one-number-two-value output, plus that business that multiplying by i spins the whole thing CCW.

    You can put both the 2D input and 2D output onto the same 2D grid with arrows from the input to output, or by redrawing the input grid on to the output grid, or you could use color or the third dimension to give you an extra dimension each. Actually I watched a video that I liked recently walking through the different visualizations.
    Last edited by demagogue; 9th Oct 2023 at 13:49.

  9. #9
    Member
    Registered: Aug 2004
    Quote Originally Posted by demagogue View Post
    Complex numbers add like vectors. But they multiply weirdly.
    I feel like complex numbers multiply more normally than vectors, lol. Complex multiplication is just... Multiplication. (With i.) But there's two different vector products (dot product and cross product) and neither of them are really all that much like normal multiplication at all. The dot product of two vectors isn't even a vector, and cross products aren't commutative (or rather they're "anti-commutative" so V1 X V2 = - V2 X V1).

  10. #10
    Moderator
    Registered: Jan 2003
    Location: NeoTokyo
    As I was saying, they don't multiply like vectors. I was just going from the previous sentence when I worded it that way, though, when like you say it's probably not worth bringing it up in the context of vectors at all... So it's good you pointed all of that out.

    I really shouldn't be explaining complex numbers or analysis at all, since I haven't really digested it yet, and I bet I misworded or got a good amount of stuff outright wrong.

    But I'm talking about it because I'm studying it these days, since I really want to finally understand quantum mechanics this year (as in be able to actually solve homework problems), and complex analysis is its bread and butter. And I'm always looking for intuitive ways to ... well it's really too ambitious to think you can visualize this stuff, but at least have some good working metaphors and mental images that can help.

    But it's such a high when I get a homework problem right, or anyway I do the steps correctly and an answer pops out. I was starting to assume I'd get all of the homework wrong, but a few days ago I was going through the motions for finding the expectation value for something called the infinite square well, assuming as usual it was all wrong and I'd have to see where I went off, when suddenly the right expectation value just popped right out, length/2. I mean I knew it was right because it's easy to intuit it before you even start the problem. But to see it actually pop out of the mess gave me a real high at the time.

    It's really like magic, since in the thick of it you think none of this is going to end up anywhere but a big mess, and then like magic all these terms start suddenly canceling and simplifying and a meaningful solution pops out and has a real meaning.

    I was one of those liberal arts students that still took calculus up to Calc3 in university & a few other advanced math classes. There were a handful of us doing that. It's a specific type of personality that wants to be rigorous and flexible (as in understanding that history and politics are messy and come as they go) in their thinking at the same time. But I kind of hated that it was just pointless number or symbol pushing. I had an intuition if you were doing it to actually answer a real world problem, that would make the symbol pushing really come alive like a kind of magic, and doing physics homework helps with that feeling. That's how I felt about game programming too (like the work I did for Dark Mod); where you're trying to solve a puzzle that actually does something interesting to the world. (I was working on the sound system, which you all know is a big part of stealth gameplay.)

    Any kind of physics can do it, I guess, but I kind of hated high school physics because my teacher didn't bring any of the magic out. It all seemed so arbitrary or not very enlightening. It got a little better with electromagnitism, which the math helps make some sense of it and you can understand some modern tech. But quantum physics is on such a different level of having bewilderment and magic in the math while still following rules in a logical way. Well I don't want to derail this thread when we're starting with the magic of base ten. XD But maybe it will help on the inspiration end.
    Last edited by demagogue; 9th Oct 2023 at 15:46.

  11. #11
    Member
    Registered: Jun 2001
    Location: under God's grace
    Quote Originally Posted by Pyrian View Post
    But there's two different vector products (dot product and cross product) and neither of them are really all that much like normal multiplication at all. The dot product of two vectors isn't even a vector, and cross products aren't commutative (or rather they're "anti-commutative" so V1 X V2 = - V2 X V1).
    Not to mention the cross product exists only in three dimensions. Also, there's a whole host of other products that come with vectors, which is expected: more structure -> more things that can be done with that structure.

    I recently started studying geometric algebra, and even though I still don't understand much of it, I'm absolutely stunned that for example in R3 the cross product pops out of the geometric product as the exterior product.

    And Nicker, ten absolutely IS the best number:


  12. #12
    Member
    Registered: May 2004
    Location: Canuckistan GWN
    WEll! That's settled then.

  13. #13
    Member
    Registered: Dec 2001
    Location: Marlboro, MA, USA
    I like using binary to count to 31 on one hand or 1023 and two hands, very useful to keep track of larger numbers that way!
    The Keep for Thief 1 and 2 FMs, Shadowdark for Thief 3 and Dark Mod FMs

  14. #14
    Member
    Registered: Jun 2001
    Location: under God's grace
    Quote Originally Posted by Komag View Post
    I like using binary to count to 31 on one hand or 1023 and two hands, very useful to keep track of larger numbers that way!
    132

    EDIT: Sorry, that wasn't very nice of me. Anyways, I've been wondering if anyone else did this too. Cool that you do! 513
    Last edited by Qooper; 11th Oct 2023 at 08:57.

  15. #15
    Member
    Registered: Dec 2001
    Location: Marlboro, MA, USA
    Yeah, my 11 yr old daughter was like "Daddy, I can never show 4!"

  16. #16
    Member
    Registered: Aug 2002
    Location: Point Nemo
    69 is the loveliest number.

  17. #17
    Member
    Registered: Apr 2003
    Location: Mossad Time Machine
    "Ten" would always exist, wouldn't it, even in a different system? I mean, whenever I've tried to visualise a different base, I've never been able to do so without 10, 20, 30 - although obviously these numbers would have different values, proportionate to the relative base being used.

    So a base 8 system may go

    1 2 3 4 5 6 7 10
    11 12 13 14 15 16 17 20
    21 22 23 24 25 26 27 30
    etc

    Is there any other way of doing it? I can't see one.

  18. #18
    Member
    Registered: Aug 2004
    You'll always have 10 in any base digit system. 20 and 30, not so much; e.g. binary has neither.

  19. #19
    Member
    Registered: Dec 2020
    Quote Originally Posted by Pyrian View Post
    The dot product of two vectors isn't even a vector
    The dot product is an extension of matrix multiplication however, so the output is actually a 1x1 matrix using the rules of matrix multiplication. It's also a 1-dimensional vector.

    Quote Originally Posted by SD View Post
    "Ten" would always exist, wouldn't it, even in a different system?

    ...

    Is there any other way of doing it? I can't see one.
    I think you might not be thinking outside the box here. Many systems have been invented to write numbers down other than the positional digit system. For example in this number system you can express any 4-digit number as a single rune:

    https://frisellagiuseppe.medium.com/...s-23bed7491537
    Last edited by Cipheron; 13th Oct 2023 at 21:03.

  20. #20
    Member
    Registered: Apr 2003
    Location: Mossad Time Machine
    Quote Originally Posted by Cipheron View Post
    I think you might not be thinking outside the box here. Many systems have been invented to write numbers down other than the positional digit system.
    Right, I was thinking about what the most simple, straightforward, logical way of doing it would be. It stands to reason that there would be many less intuitive ways (the Yautja language numerals in Predator sprung to mind).

  21. #21
    Member
    Registered: Feb 2002
    Location: In the flesh.
    It was motherfuckers like you who designed the alien algebra crap in Omicron that took so long to figure out wasn't it?

  22. #22
    Member
    Registered: May 2004
    Location: Canuckistan GWN
    I think I am having a similar difficulty to SD trying to understand this.

    Moving the point in base 10 is the same as dividing or multiplying by 10.

    But moving the point doesn't provide the same elegant effect in other bases, does it?

  23. #23
    Member
    Registered: May 2004
    Location: Canuckistan GWN
    On a side note...

    "The Nobel Prize in mathematics was awarded to a California professor who has discovered a new number! The number is bleen, which he claims belongs between 6 and 7." --George Carlin
    Given the almost universal authority granted to Carlin on all matters, I would like to know why bleen is still being suppressed.

  24. #24
    Moderator
    Registered: Jan 2003
    Location: NeoTokyo
    Moving the point would be the same as dividing or multiplying by the base number, which is what you'd want if you're working in a different base system. It'd be the elegant thing in that base.

    The thing that I want to wrap my head around is not only are the complex numbers you map to a 2D plane, but there's quaternions numbers you map to 4D, and octonions numbers you map to 8D, and there are all kinds of crazy rules to apply in those contexts.

  25. #25
    Member
    Registered: May 2004
    Location: Canuckistan GWN
    Thanks for the first paragraph, Dema. That makes sense. It doesn't entirely alleviate my confusion but it's one fewer mole to whack.

    After that you are just flexing over by the free weights while I plod on the treadmill though. LOL

    I am finding the discussion fascinating and inspiring, never the less.

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