0.999999...
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But that's just another decimal representation of 1.
0.1 in base 3 is 0.33333... in base 10.
0.5 in base 10 is 0.11111... in base 3.
You can have some fun with this here.
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Which can be written as 1, right? '1' is the most intuitive and 'ordinary' way to express the concept of unity - to write it as 0.9999999... is to deliberately express it as a non-terminating number, which is unnatural and (for most purposes) pointless. So if an argument can be made that 1 can be artificially and unnaturally written as 0.9999....., I'd say a more powerful counterargument is to say that 0.99999.... can be naturally and easily as 1.
Originally Posted by woops
Every number has a non-terminating representation, in a given base. Some numbers also have terminating representations in that base, while some do not. I'm not sure I understand what the fuss is about.
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A bit like I think."Ad hoc" here seems to mean something a bit like Kolmogorov randomness.
I'm not sure. I don't think that "most" could be broken down into a way that would count as an analysis of the kind that Josef K was requesting."I would imagine that most wiggly lines I could draw by hand on the back of an envelope would turn out to be quite compressible - it would be quite a challenge to produce one that couldn't be broken down into a handful of frequencies."
Do you mean that terminating numbers can be written in a form in which they don't terminate? That might be true but I don't see how that means that there are no terminating numbers."I'm saying there are no terminating numbers"
Not if you use the number in question as the base presumably."But of course irrationals are irrational in any base."
I think that the only reason you would do it (off the top of my head) is if you wanted to make, say, pi rational but you would be sacrificing an awful lot in terms of simplicity for most things. I don't see why it shouldn't be possible though."Well I was thinking that, but I can't really see why you'd do that."
woaaaah using pi as a base? i'm not entirely sure you can do that. at least im really struggling to visualise it. how would you count in it for a start?
hold on, thinking about place value in an irrational base; well each place in a number in base pi for example would have infinite number of possible digits; right?
Well, I guess you would do it in the same way as any other base. When you're talking about base 10 you have a number of digits, the final one represents the the number of zeroth powers of 10 - which is 1 as anything to the power zero is one. The column next to that represents first powers of 10 ie tens, then you have ten to the power 2 ie hundreds and so on so any integer in base ten can be written:"woaaaah using pi as a base? i'm not entirely sure you can do that. at least im really struggling to visualise it. how would you count in it for a start?"
a x (10 to the power 0) + b x (10 to the power one) + c x (10 to the power 2) + d x (10 to the power 3) + .... and so on. Supposing that we are talking about a number between a thousand and ten thousand ie one where all of the letters after d in the sequence I have just described are 0 then the number would be written dcba in base 10.
(if it wasn't an integer then you would need minus powers as well ie 10 to the minus one is one divided by ten ie 0.1 and so on)
Similarly in base pi you would have a column for 1s, a column for pi's, a column for pi squareds and so on. Suppose you wanted to write, say, the number which is 21 in base 10 in base pi then you would need to work out how many for each column. pi cubed is going to be more than 21 so you don't need any in the "thousands column", pi squared is between nine and ten so you are going to need two in the hundreds column (because 21 is between 2 and 3 times pisquared), 21 minus 2(pi squared) is less than pi so you are going to have zero in the tens column, 21 minus 2(pi squared) is between 1 and 2 so you are going to have a 1 in the units column. So the number is going to be 201 point something. You continue this process, moving into negative powers, to work out all the digits although obviously it will be non-recurring and non-terminating because 21 is irrational in base pi.
I hope that makes sense. I've written it quite quickly as I'm busy at the moment. I hope there are no errors. When I get a chance I'll write it out more generally.
no i understood that, i'm just not sure about recurring sequences in an irrational base. obviously numbers formed from the pure base would not recur, but any other number, including most rationals would. and how would (the equivalent of) decimal places etc. even work? i'm seriously struggling with the concept in the context that we're talking about
oh hold on, yeah, i get it!!
wow, that's a really powerful example of why there are far more irrational numbers than rational - every number that isn't related to pi creates a brand new irrational number; do you see?
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