don_quixote
Trent End
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?
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?"
The most obvious thing that you're sacrificing is the fact that sums and products of 'rational numbers' are 'rational numbers'. Which means that whether or not things are rational kind of stops being of so much interest... as far as I can see the nice thing about rational numbers is less the 'terminating or recurring decimal' part than the 'quotients of two integers' thing.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.
Are you saying that numbers that are rational in base ten would have a recurring decimal expansion in base pi? If so you're wrong, they (except for 1,2 and 3) would be irrational in base pi and therefore would have a non-terminating and non-recurring expansion."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"
This does not in itself constitute a proof that there are "more" irrational numbers than rational ones (though there are in the sense that you mean) - you have to demonstrate that there can't be a one-to-one mapping, it's not enough to suggest one mapping that exhausts the rational numbers without exhausting the irrationals."ok every rational number in that base is irrational, right? now think of those numbers as written in a decimal base - this is a one to one relationship, so every rational number has an irrational partner by writing it in base pi.
further, every irrational number which cannot be written as a sum of products of pi is going to be irrational in base pi - so there's a huge amount of these, including those we've already written"
A graduate student at Trinity
Computed the square of infinity;
But writing the digits
Gave him the fidgets,
So he dropped Maths and took up Divinity.