• • Thread: pseudo-number theorist ramblings....

1. pseudo-number theorist ramblings....  Reply With Quote

2.  I can mail you a document about number theory if you'd like, which introduces modular arithmetic. The key to modular arithmetic is that you consistently define a finite number of elements which together represent all integers. By which I mean, in mod 9, for instance, you only need to consider the symbols 0, 1, ... , 8. The rest are superfluous. What 8 here actually symbolises (hopefully UM won't enter this thread and discover mathematicians using the same symbol to mean different things, the concept is clearly too much for him) is  = { ... , 8 - 2*9, 8 - 9, 8, 8 + 9, 8 + 2*9, ... }, which is the set of all numbers with remainder 8 after division by 9. After a small amount of work, one can define addition and multiplication on such sets. So, for doubling, we start with 1, then 2, then 4, then 8. But when we double 8 (once the small amount of work is done), it is completely legitimate to say the new element we get is 7 (inb4 modular arithmetic is a crock of shit). Then when we double 7 to get 5, and finally we double 5 to get 1. In this interpretation, it becomes clear that all repeated multiplication sequences such as yours must eventually start repeating: there are a finite number of symbols, so eventually you must hit a previous symbol, and the process repeats. The fact you can also do this backwards indefinitely is essentially because 2 is what we call a 'unit' in mod 9, which means there is a number u such that 2*u = 1. That number u turns out to be 5. This means to divide by 2, one may simply multiply by 5; say we want to divide x by 2. This means we want to find y where x = 2*y. Then 5*x = 5*2*y = 1*y = y. Not all numbers are units. For example, 6 in mod 9. 0*6 = 0, 1*6 = 6, 2*6 = 3, 3*6 = 0, 4*6 = 6, 5*6 = 3. So we can't in general divide by 6.  Reply With Quote

3.  That would be great Xei, thanks. I'll first try to grok the rest of your post. Its interesting seeing how these simple rules create such interesting dynamics. How does this explain the repeating pattern found in the fibonacci sequence though? I would think that given its nature, a repeating pattern is suprising. I understand that since the fibonacci sequence is infinite, All numbers on it would share their digital roots (I think thats the right word) with an infinity of other numbers. I just don't understand why these repeating links are placed precisely every 24 places. I guess the interesting mirror symmetery of the number pattern itself can be explained away by the innate laws of modular arithmetic. I don't think I understand everything in your post so I'll probably practice creating my own examples other then the 8 to see if I can grok it.  Reply With Quote

4.  I don't actually know that, although I imagine it's quite well studied and not so tricky to figure out after a bit of work. See if you can do it under the 'finite elements' interpretation, where you literally don't worry about what the Fibonacci number is in the integers, but just what it is in terms of the symbols 0, 1, ... , 8. In fact by writing that I just figured it out, it's quite easy given what I tried to explain above. See if you can get it. PM me your email address and I'll send you the document. The relevant bit starts at section 4.1. But I really recommend reading it all the way through (except maybe section 3 first time), it's a perfect introduction to what it's like on a university maths course (it's from a first term lecture series), the pure half anyway; and you need literally zero prior knowledge, but once you're done you'll know more number theory than your average mathematics graduate. The stuff in section 6 is still probably the coolest thing I have seen in my degree, it's about the different sizes of infinity.  Reply With Quote

5.  What y'all are discussing isn't real. Sorry, I couldn't resist the opportunity. Carry on. Have a nice day.  Reply With Quote

6.  I enjoy thinking about modulus as well, but not in the "clock arithmetic" sense.  Reply With Quote

7.  To be honest 'clock arithmetic' is a pretty perfect analogy for modular arithmetic and a good thing to have in mind when you're doing it. What's your beef?  Reply With Quote

8. Originally Posted by Xei What's your beef? Medium rare, with steak rub on a skillet.  Reply With Quote

9.  Me too actually.  Reply With Quote

10.  Xei I sent you my email but you still haven't sent me the papers.  Reply With Quote

11.  Oh sorry I was expecting a PM. I'll check my mails now.   Reply With Quote Posting Permissions

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