Mathematics

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?

"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?"

Click to expand...

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:

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?

let me explain that better.

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.

hell, they're probably trancendental too.

IdleRich said:

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.

Click to expand...

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.

"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"

Click to expand...

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.
Decimal places would work in exactly the same way as in base ten, the first digit after the point would represent the number of (1/pi)s and the next digit would represent the number of (1/(pi squareds)) and so on.

"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"

Click to expand...

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.

Yes, the point is that the rationals are countably infinite - there are in fact "just as many" rationals as there are natural numbers (because a one-to-one mapping can be established between the rationals and the natural numbers), whereas the irrationals are uncountably infinite.

I'll tell you what does my head in a bit, and that's the fact that ZFC implies that there's a well-ordering of the reals...

How's the programme going, Josef? Have you solved Goldbach's conjecture yet?

Like it, like it.

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.

Click to expand...

Edit: shouldn't 'thou' be 'thine'?

this thread is hugely interesting but i don't really understand any of it, and it's scaring me off that astrophysics course i was considering applying for. God, i wish i understood this stuff more.

Is this an appropriate place to ponder what the ontological implications of mathematics are? - esp, does the existence of mathematical objects contradict materialism?
I'm only capable of discussing this in a rambling, confused way, but that doesn't usually stop me.....

Well, as I understand it, if you think of sets and numbers as objects, rather than formal constructions, then they have to be abstract objects, which by definition exist outside of space and time. Is this not a part of what is at stake in the dispute between nominalism and mathematical realism?
As you can see I ain't no philosopher.

Ditto with rambling and confused but aren't there implications for mathematics from ontology? I speak as someone who suffered "extra maths", i.e quite thick and with a bone to pick.

I had to take maths in a seperate year from all my other Highers to make sure I could pass it. Good times, good times.