John, do you know of any other number/logic system that yields harmonious or symmetrical results that compete with our current one? Perhaps an ancient one?
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All the truly cool number and logic systems are under active investigation by mathematicians now. There are lots! Topos logic, paraconsistent logic, surreal and hyperreal numbers, p-adic numbers, adeles, quaternions, octonions... I could go on all day!
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Any truly ancient or "ancient" one?
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No, sorry - ancient mathematics is what we learn in grade school and high school.
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Ok, thanks. Actually, I don't believe other base systems are covered at all. I meant mathematics that would have evolved from those other systems, like the Babylonian one (base 60).
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I learned about other bases in grade school. I don't consider those different kinds of numbers, just different ways of writing the same numbers. I thought you wanted something a bit more radical. Modern math is full of much more radical stuff!
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নতুন কথা-বার্তা -
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I wonder: does a similar picture exist for all elementary numbers, that is, to all solutions of all equations featuring field operations, logarithms & exponentials with integer only coefficients everywhere?
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Nobody has drawn that picture yet, so it's up to you! The real challenge is to decide a good way of measuring complexity, so you can make the simpler numbers look bigger. The picture is likely to be more complicated: in the algebraic case, you can see many simple patterns.
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indeed: for algebraic numbers, one can iterate over all possible integer-only polynomials either in fully expanded/braced or whatever canonical form & compute/approximate their roots, for elementary numbers, things will get a little more hairy, won't they?
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A lot more hairy.
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Gorgeous, but curious, this article is from 2013, yet there seem to be no comments on it over 5 yrs. time!? (or are they lost?) ...and any special reason for re-citing it at this particular moment?
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I'm slowly copying all the pictures from my blog "Visual Insight" to Twitter. Yup, there were no comments on that blog entry - maybe because only mathematicians read that blog, and they don't talk unless there's something new to say. They don't say stuff like "wow".

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Ooooooooomg I love this
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Thanks! It was made by David Moore and Stephen J. Brooks. There's software you can download to look around and zoom in, but I haven't tried it:https://sourceforge.net/projects/algebraicnumbers/ …
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Prof. I was wondering if there is simple description of symmetries w.r.t. the coloring scheme used in the picture. I think transformations z \to -z or z \to 1/z won't change the complexity?
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I didn't do the programming - you'll have to click on my link and dig back. With luck, the complexity will be invariant under those two transformations. It should be!
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Am I looking at a Cartesian plane? Y up and X right?
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Yes, but the points in this plane are "complex numbers" X + iY where i is the square root of -1. The brightest point directly above 0 in this picture is i.
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লোড হতে বেশ কিছুক্ষণ সময় নিচ্ছে।
টুইটার তার ক্ষমতার বাইরে চলে গেছে বা কোনো সাময়িক সমস্যার সম্মুখীন হয়েছে আবার চেষ্টা করুন বা আরও তথ্যের জন্য টুইটারের স্থিতি দেখুন।
