Ten is the lovliest number... - by Nicker
Pyrian on 12/11/2023 at 10:53
Quote Posted by demagogue
I especially get a kick out of base e, which is useful for exponential functions & natural log but I doubt it's clean in any other base you'd want to use. But as a base it's far from arbitrary.
How do you even use a non-integer as a base in the first place? Like... Base refers to the number of symbols. You can't have a 0.71th symbol or whatever.
Quote Posted by Qooper
Very true, S is not a number. But I think we can assign mathematical meaning to such algebra in certain cases, as long as the path is well defined.
Since in this case everything depends on Grandi's series, we must get a proper value to it in some way.
The problem with an indeterminacy isn't that you can't assign it a value, it's that you can assign it two or more different values. Any given value is therefore wrong. They can't be rigorously simplified. If you're doing Physics and you encounter something like that, it's another matter; it can be approximated easily enough as 1/2 +/- 1/2.
Quote Posted by Qooper
I think you meant 1/5 instead of 1/7?
Yes, thank you for the correction.
demagogue on 12/11/2023 at 16:00
Quote Posted by Pyrian
How do you even use a non-integer as a base in the first place? Like... Base refers to the number of symbols. You can't have a 0.71th symbol or whatever.
But when you use it as the base for log, things magically clear up: ln e = 1, ln e^x = x, etc.
That's what I was referring to.
Qooper on 12/11/2023 at 16:48
Quote Posted by Pyrian
How do you even use a non-integer as a base in the first place? Like... Base refers to the number of symbols. You can't have a 0.71th symbol or whatever.
A more general meaning of 'base' is a value b to the power of the digit's index (the first digit left of the decimal point is index 0 and the first one to the right is index -1) that each digit gets multiplied by. So for example 45100 would be 4*b^4 + 5*b^3 + 1*b^2 + 0*b^1 + 0*b^0. For base e we could have the symbols 0, 1 and 2, and as such, 10 = e. So 122 would be 1*e^2 + 2*e^1 + 2*e^0 = e^2 + 2e + 2. Though, having an irrational base means you can't express almost any value easily :D
By the way, base doesn't necessarily have to be the number of symbols. We could have a base-10 writing system where we have 20 symbols, spaced 0.5 apart: 0, a, 1, b, 2, c, ..., i, 9 and finally j. This would mean that there's redundancy in expression, for example a0 is the same as 5, but the system works. It's just not a very useful or interesting system.
Quote:
The problem with an indeterminacy isn't that you can't assign it a value, it's that you can assign it two or more different values. Any given value is therefore wrong. They can't be rigorously simplified.
Unless it is gotten analytically, like from the eta-function. And that can then be used in other divergent series for getting a value. You can use η(0)=1/2 to get a valid value for 1 - 2 + 3 - 4 + ..., which would be the same as directly evaluating η(-1), is all I'm sayin'.
Quote:
If you're doing Physics and you encounter something like that, it's another matter; it can be approximated easily enough as 1/2 +/- 1/2.
But the calculations in physics still follow from rigorous math, even if physicists aren't concerned with the proofs. The math has to work, and it can only work if it's true. Approximations are of course on purpose not precise, but they usually come only at the very end of a general calculation. A precise equation is used for a system, and then a specific case is finally approximated to sufficient precision.
Cipheron on 13/11/2023 at 14:34
Quote Posted by Pyrian
How do you even use a non-integer as a base in the first place? Like... Base refers to the number of symbols. You can't have a 0.71th symbol or whatever.
Actually not quite. What about base -10 for example? Presumably that would need -10 symbols?
With base -10, each odd power of the base is negative so we need to use a system of subtraction similar to Roman Numerals to get all numbers. However, 10 symbols is both sufficient and required to make everything:
50 in base -10 = 1 * (-10)^2 + 5 * (-10)^1 + 0 * (-10)^0 = "150"
950 is trickier, since you want 1000-50, but the 1000s column is negative so you need to go one along:
950 in base -10 = 1 * (-10)^4 + 9 * (-10)^3 + 0 * (-10)^2 + 5 * (-10)^1 + 0 * (-10)^0 = "19050"
So the system works, and needs 10 symbols so at the very least it's now |b| symbols not just b symbols
As for fractional bases, consider what "base 0.5" actually means. Each position going to the left is halved, each position going to the right is doubled. So, it would basically be binary, reflected around the units column. The number of symbols needed is still 2. You could pull the same trick and prove that base 0.1 would be mirror-image base 10, using 10 symbols.
This just shows that the "rule" that base-X needs X symbols is a simplification that's not true once we look at alternative bases. The number of symbols needed depends on the relative size of each power of the base, in both directions.
As for bases that aren't some neat multiple or divisor, they work, but you'll probably get an infinite expansion when trying to encode integers; just like base 10 cannot neatly encode 1/3, base 0.71 would not be able to neatly encode ANY integers. However we can use logic to work out how many symbols is needed.
Take base 1/7 for example, that can use 7 symbols and is just mirror-image base-7. But base 7/2 or 2/7 is different. none of the places to either left or right of the decimal are whole numbers, so it can only express integers as decimal expansions.
With base 7/2 = 3.5, you'd need 4 symbols, because you'd need to express either 0,1,2,3 times any power of the base. you don't need more than that because 3.5 * 4 would move it over a column, then you could move the remainder down. So yeah, because it's not an integer multiple between powers of the base, you'd have remainders to redistribute any time you add 1 to a column and it carries over.
As a rule, basically anything base greater than 1 up to 2 would need 2 symbols, then anything greater than to up to 3 would need 3 symbols etc. The inverse would hold true, that any fractional base less than 1 down to 1/2 needs 2 symbols, then 3 symbols below 1/2 down to 1/3 and so on.
Qooper on 13/11/2023 at 15:08
Quote Posted by Cipheron
for example - base sqrt(2). If you try and express normal integers in base-sqrt(2) then you can just note every even column is just the powers of 2, so you realize you can do it with exactly 2 symbols: by leaving all the in-between values as 0 and filling in the normal binary digits every 2nd column. Job complete.
Interesting find! What about base e, can we find a way to make it work? My example of using 0, 1 and 2 for base e obviously won't work because you wouldn't even be able to add 2 + 2. Of course using only 0 and e works, but that's trivial and has the downside of not being able to express any integers expect 0.
Cipheron on 13/11/2023 at 16:53
Quote Posted by Qooper
Interesting find! What about base e, can we find a way to make it work? My example of using 0, 1 and 2 for base e obviously won't work because you wouldn't even be able to add 2 + 2. Of course using only 0 and e works, but that's trivial and has the downside of not being able to express any integers expect 0.
Heh I edited that out because the area is just too big and I didn't want it to ramble too much from the main point about fractional bases. Sqrt-bases are kind of cheating, because they have that property of hitting every whole number power of the squared base.
but keep in mind what all these tricks mean. Negative bases work. Inverse bases work, and square roots of bases work. So you can construct a base of "negative one over square root of 10" and get it to work.
As for base e, you're right, that you can express 2 by having a 2 in the 1's column, but then if you add 2, you now have 4, which is some value "k" minus e. You then have a 1 in the e^1 column, then a 1 in the e^0 column, then you have some fractional part which you have to compute and subtract powers of 1/e from. You can do that using standard long division. However you're going to have to estimate it as it's an infinite expansion.
The issue is just as e has no clean representation in base 10, integers greater than e have no clean representation in base-e
demagogue on 15/11/2023 at 16:25
Posting a meme basically comparing Ramanujan's summation business vs. the however many other ways you can arbitrarily factor a geometric sum to probably come up with a lot of different answers.
I guess the punchline is that there's a lot more behind the former than just the cheap proof I gave above, but on the other hand, it's no less arbitrary if you're looking at it as a one special regime out of a lot of others you could choose from. (All this keeping in mind the points already made before that strictly speaking you're not supposed to do algebra with divergent series to begin with.)
Well it's a cheap meme. I don't really have to explain it much.
Inline Image:
https://i.ibb.co/48FDt41/398939051-1298901677475209-8202301721206038281-n.jpg
Anarchic Fox on 4/12/2023 at 01:59
Ramanajan summation would cause far less havoc among enthusiasts if it used a colon instead of an equals sign. Next week I'll ask some of my fellow grad students about Ramanajan summation. I vaguely remember them expressing frustration with its misinterpretation.
(Yes, I'm in graduate school again, this time in math. I belong in academia. This is my way back in.)
Cipheron: the p-adic numbers sure are mind-bending, aren't they! There's an analogy to be made with Laurent series in complex analysis.
Anarchic Fox on 5/12/2023 at 01:36
I asked three classmates. The consensus was that it's bad notation. Two said the equals sign should not be used, one said the summation symbol should not be used.
Cipheron on 14/12/2023 at 21:58
Thought about where to post this, maybe it should go in the math thread
(
https://twitter.com/realstewpeters/status/1733236101555327317)
Stew Peters (the dollar store Alex Jones) posts a video by a guy who's proved 1x1=3, says we should listen to that guy because he's blown everyone out of the water proving everything we're taught is a lie.