Rational powers of irrational numbers

[BWV]

√2 is irrational but √2² is rational

Is there any way to know if given some irrational number α, if αⁿ is rational for some n?

Or can it be proven that ∏ⁿ or eⁿ are irrational for all n?

 

[PeroK]

√2 is irrational but √2² is rational

Is there any way to know if given some irrational number α, if αⁿ is rational for some n?

Or can it be proven that ∏ⁿ or eⁿ are irrational for all n?

##\pi## and ##e## are transcendental numbers:

http://mathworld.wolfram.com/TranscendentalNumber.html

 

[fresh_42]

√2 is irrational but √2² is rational

Is there any way to know if given some irrational number α, if αⁿ is rational for some n?

The answer probably depends on how ##\alpha ## is given. What is it? E.g. in your example, you see it immediately. In general you have to investigate the roots (solutions) of ##ax^n + b = 0## with ##a,b \in \mathbb{Z}## This equation can be solved with complex numbers, so you only have to compare its solutions with your given number.

Or can it be proven that ∏ⁿ or eⁿ are irrational for all n?

Yes.
##e## and ##\pi## are not only irrational, they are transcendental, which means there is no rational equation ##x^n+a_1x^{n-1}+\ldots +a_n=0## which has one of them as a solution. So if a power would be rational, then we would have such an equation, which is impossible.

 

[BWV]

Ok thanks, so the real question would be knowing if some irrational number picked at random is trancendental or not.

Put another way, i know that the odds of randomly choosing a rational number from (0,1) on the real line is zero. What are the odds of choosing a trancendental number?

 

[fresh_42]

Ok thanks, so the real question would be knowing if some irrational number picked at random is trancendental or not.

This would be sufficient, but not necessary. ##1+\sqrt{2}## is not transcendental, but irrational. However, none of its powers is in ##\mathbb{Q}##.

 

[PeroK]

Ok thanks, so the real question would be knowing if some irrational number picked at random

... which is not that easy to do, as a matter of fact!

 

[TeethWhitener]

What are the odds of choosing a trancendental number?

I think you mean to say that the rational numbers form a set of measure zero within the reals. This is because the rationals are countably infinite, and any countable set in the reals has measure zero.

The algebraic numbers are countably infinite, which means they form a set of measure zero within the reals. Their complement is the transcendental numbers, which necessarily form a set of measure 1 (for the set of transcendental numbers in ##[0,1]##).

(Note that countable means measure zero, but uncountable does not necessarily mean measure not zero: consider the Cantor set for a counterexample).

 

[fresh_42]

Put another way, i know that the odds of randomly choosing a rational number from (0,1) on the real line is zero. What are the odds of choosing a trancendental number?

You can't.

The entire debate still lacks the definition of ##\alpha##. What is it? How is it presented? In which alphabet? Will a Turing machine come to a hold? You cannot pick a real number. You have to describe it somehow, and then the rules for such a description will influence the answer.

 

[PeroK]

Put another way, i know that the odds of randomly choosing a rational number from (0,1) on the real line is zero. What are the odds of choosing a trancendental number?

In terms of choosing a number at random - with each number having the same probability of being chosen - this is only of possible if the set of numbers you are choosing from is finite.

There are lots of places online where someone says: choose a number at random from (0, 1). Then they draw various conclusions, including that things happen that have a zero probability.

However, it is simply not possible to choose a number at random from (0,1). At best, you can have a countable subset to choose from. And then, not every number has an equal chance of being chosen.

 

[BWV]

In terms of choosing a number at random - with each number having the same probability of being chosen - this is only of possible if the set of numbers you are choosing from is finite.

There are lots of places online where someone says: choose a number at random from (0, 1). Then they draw various conclusions, including that things happen that have a zero probability.

However, it is simply not possible to choose a number at random from (0,1). At best, you can have a countable subset to choose from. And then, not every number has an equal chance of being chosen.

But isn't that like saying you can't talk about a square because no such thing exists? I have seen these problems - particularly picking a rational from (0,1) in formal measure theoretic discussions of probability.

 

[PeroK]

But isn't that like saying you can't talk about a square because no such thing exists? I have seen these problems - particularly picking a rational from (0,1) in formal measure theoretic discussions of probability.

You can talk about an uncountable set and handle it mathematically. But, you cannot choose one of its members at random, where all members have a chance of being chosen.

You can, however, have a random variable distributed uniformly on [0, 1]. But, that's different from "picking" a number at random from [0, 1].

If you don't believe this, then please provide a mechanism that would pick a number in [0, 1], where every number could be picked.

 

[BWV]

I don't claim to have any expertise in the measure theoretic foundations of probability, but does not the Lebesgue measure, which extendes the intuitive classical definition of (0,1) being limited to rationals to all possible subsets of the interval, accomplish this?

 

[fresh_42]

I don't claim to have any expertise in the measure theoretic foundations of probability, but does not the Lebesgue measure, which extendes the intuitive classical definition of (0,1) being limited to rationals to all possible subsets of the interval, accomplish this?

This is irrelevant here. You said, given a real number ##\alpha##. In order to decide, whether it has rational powers or not, we need a certain number, not any number. So how is it given? We cannot do calculations such as a proof on transcendence on an any number. Theoretically the odds to hit a transcendental number is one as @TeethWhitener already explained above, but that doesn't mean that you couldn't hit e.g. ##2##. So here we are again: how is it given? Without a definition of that number, it cannot be said, whether it's rational, algebraic or not. In this case, you can only do abstract considerations as in post #7.

 

[PeroK]

I don't claim to have any expertise in the measure theoretic foundations of probability, but does not the Lebesgue measure, which extendes the intuitive classical definition of (0,1) being limited to rationals to all possible subsets of the interval, accomplish this?

No. You're looking for a process that allows you to choose a number. And, in fact, there's a related issue that it's not possible to identify a typical real number. How do you identify the one you've chosen?

The trap you can fall into here is to confuse a mathematical process with a physical one. If you stick to measure theory, then fine. But, to illustrate the point, if you said something as innocuous as "write the number down on a piece of paper", then you have fallen into this trap. Because:

a) There's no way to pick from an uncountable set
b) There's no way to identify more than a countable set of things.

This does not undermine Real Analysis or Measure Theory, but it does mean you must be careful not to assume mathematical processes translate into things you can do.

That's where the websites that say: "you pick a number in [0,1] and I'll pick a number in [0, 1] and the probability they are the same is 0" are wrong. They are assuming a physical process that cannot be carried out.

 

[mathwonk]

the odds of choosing a number from (0,1) which is not transcendental are also zero, and for the same reason as the odds of choosing a rational number are zero. I.e. there are the same number of non transcendental numberts as rational, namely one for every positive integer. Hence if we number them off, we can cover the first one with a small interval of length 1/10, and the second one with a smaller intyervl of length 1/100, and the 3rd one by an interval of length 1/1000, and so on. Added up these intervals have total length 1/9 by the formula for summing a geometric series. So the odds of choosing one are at most 1/9.

But if we start with an interval of length 1/100, and then 1/1000, and so on, the sum is only 1/90, so the odds are at most 1/90. In the same way the odds are also at most 1/900, 1/9000, and so on, so we say the odds are zero.

 

[PeroK]

the odds of choosing a number from (0,1) which is not transcendental are also zero, and for the same reason as the odds of choosing a rational number are zero.

I assume you mean choosing a number from (0, 1) where all numbers have the same chance of being chosen (uniform distribution).

Except, that it's simply not possible to do that.

In any case, I suggest you choose a number from (0, 1) and if it's transcendental you win; if it's not, I win.

I'll leave you to do the choosing and tell me whether you chose a transendental number or not.

Do you accept my bet?

 

[TeethWhitener]

The entire debate still lacks the definition of αα\alpha. What is it? How is it presented? In which alphabet? Will a Turing machine come to a hold? You cannot pick a real number. You have to describe it somehow, and then the rules for such a description will influence the answer.

This is a tricky area of mathematical logic. Computable numbers (that is, numbers that can be calculated to arbitrary precision with a Turing machine) are countable. But in terms of describing a number using a model of logic or set theory, it's not so clear. This paper:
https://arxiv.org/abs/1105.4597
for example, argues that there exist (non-standard) models of ZFC set theory where every real number is definable.

 

[stevendaryl]

If you don't believe this, then please provide a mechanism that would pick a number in [0, 1], where every number could be picked.

Sure. At time t=11:00, flip a coin. If it's heads, write down 0. If it's tails, write down 1. At time t=11:30, flip again. Then at time t=11:45. Then at time 11:52:30. Etc. If you time it right, you'll have flipped an infinite number of coins by 12:00, and you will have generated the binary expansion of a random real in [itex][0,1][/itex].

 

[PeroK]

Sure. At time t=11:00, flip a coin. If it's heads, write down 0. If it's tails, write down 1. At time t=11:30, flip again. Then at time t=11:45. Then at time 11:52:30. Etc. If you time it right, you'll have flipped an infinite number of coins by 12:00, and you will have generated the binary expansion of a random real in [itex][0,1][/itex].

It would be quicker, surely, simply to toss an infinite number of coins at 11:00.

 

[stevendaryl]

It would be quicker, surely, simply to toss an infinite number of coins at 11:00.

Don't be ridiculous. Nobody can toss coins infinitely fast. But there's no obvious upper bound on how quickly you can toss them (if you ignore relativity, anyway).

 

[PeroK]

Don't be ridiculous. Nobody can toss coins infinitely fast. But there's no obvious upper bound on how quickly you can toss them (if you ignore relativity, anyway).

Interestingly, almost my first contribution on PF was one of @office shredder's maths puzzles on "Gambler's Hell". It had more or less been solved, but I came up with a solution where the gambler puts in a coin for every real number - between 0 and 1 seconds. Now that's fast!

 

[jbriggs444]

Don't be ridiculous. Nobody can toss coins infinitely fast. But there's no obvious upper bound on how quickly you can toss them (if you ignore relativity, anyway).

If you could toss them infinitely fast, all in the same place, then surely it would be no problem to toss them at infinitely many nearby places, all at the same time.

Of course, that's still a very big if.

 

[SSequence]

Sure. At time t=11:00, flip a coin. If it's heads, write down 0. If it's tails, write down 1. At time t=11:30, flip again. Then at time t=11:45. Then at time 11:52:30. Etc. If you time it right, you'll have flipped an infinite number of coins by 12:00, and you will have generated the binary expansion of a random real in [itex][0,1][/itex].

I don't whether this was a fully serious post, but it is an interesting thought experiment at any rate.

When I first saw this, my first impression was it is not valid from intuitionistic point of view (so I will first try to see the experiment from this view). In fact, the "first act" of LEJ Brouwer (which you might have read) is essentially very relevant here.
The main issue is how does one record the results of coin flips. If you imagine writing it yourself, the problem quickly becomes that during a finite period of half an hour one can only write finite number of bits.

If one trusts the output of some kind of other mechanism that was supposedly able to store infinite number of bits (during this finite time) ... which we can presumably reuse later on, then the problem quickly becomes that we are then making an empirical assertion that is tied up with our underlying account of physical reality (and that's without bringing up the issue of randomness).

And this is why I mentioned intuitionsim in this post, because it tends to assume highly minimalist assumptions regarding "physical reality" to be able to formulate a coherent mathematical account (I think perhaps something to this effect is also mentioned in the essay (containing the first act) but I don't remember very well). However, if one challenges even those highly minimalist assumptions then it is another matter*** (and another discussion really).

At any rate, it seems to me, the main point that I discussed above will still hold (in a pov different from intuitionism) as long as one believes in mathematical meaning as "more basic" than a certain specific physical model of the world.

=============================================

Regarding defining probability of picking an element from A⊆ℝ, I would first consider the problem of defining probability for picking an element A⊆ℕ.

One way for us to consider the probability with some subset A (of N) is to consider increasingly larger subsets:
{0}, {0,1}, {0,1,2}, {0,1,2,3},...
and consider probability in the limit (defined precisely in a suitable way).

Below assume card(ℝ)=##\aleph_1##. One can think in a similar way as in the case of natural numbers. If one assumes a well-order of ℝ (with order-type ω₁) then I don't know of any result that explicitly forbids picking up increasingly larger subsets of ℝ (but I have read that such a description won't be "definable" ... I don't know what that means). After this one can define probability in the limit (defined precisely in a suitable way) I think. But the limit will be to ω₁ instead of ω.

I don't really believe much of anything I said in last paragraph (with any confidence) but that's besides the point.

What is interesting in the case of natural numbers is that probability in the limit will turn out to be 0 for finite sets. So for the case of ℝ, we would want to define both the well-order and selection process in such a way that the probability in the limit will turn out to be 0 for any countable subset of ℝ. Since much of this involves quite advanced notions (it seems to me), I don't have anything meaningful to add beyond this.

=============================================

*** In words of Georg Cristoph Lichtenberg:
"Everything happens in the world of the self. This self, within which everything unfolds, resembles in this regard the cosmos of physics, to which the self also belongs by which that cosmos appeared mentally in our representation. … So the circle is complete"

 

[Mark44]

Sure. At time t=11:00, flip a coin. If it's heads, write down 0. If it's tails, write down 1. At time t=11:30, flip again. Then at time t=11:45. Then at time 11:52:30. Etc. If you time it right, you'll have flipped an infinite number of coins by 12:00, and you will have generated the binary expansion of a random real in [itex][0,1][/itex].

f one trusts the output of some kind of other mechanism that was supposedly able to store infinite number of bits (during this finite time) ... which we can presumably reuse later on, then the problem quickly becomes that we are then making an empirical assertion that is tied up with our underlying account of physical reality (and that's without bringing up the issue of randomness).

@stevendaryl's response was not meant to be taken in a literal sense, with some sort of machine to store the bits, nor was the matter of being able to record the results over the period of an hour. @SSeqence, you're overthinking this.

 

[PeroK]

In words of Georg Cristoph Lichtenberg:
"Everything happens in the world of the self. This self, within which everything unfolds, resembles in this regard the cosmos of physics, to which the self also belongs by which that cosmos appeared mentally in our representation. … So the circle is complete.

Or, slightly more prosaically, if you tried to write down an infinite sequence then at some point you'd run out of paper.

 

[stevendaryl]

Or, slightly more prosaically, if you tried to write down an infinite sequence then at some point you'd run out of paper.

No, you just make each character half the size of the last.

 

[jbriggs444]

No, you just make each character half the size of the last.

Or make an infinitely precise mark on an rod constructed of unobtainium.

 

[Stephen Tashi]

I have seen these problems - particularly picking a rational from (0,1) in formal measure theoretic discussions of probability.

You might see such a discussion in an application of measure theoretic probability, but in the theory itself, there are no definitions or assumptions about the process of taking random samples. The intuitive notion that a random variable can have probable outcomes and then have an actual outcome is not formalized.