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 Post subject: Re: Exponential functionPosted: Sun, 27 May 2012 08:13:13 UTC
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glebovg wrote:
But why is it unknown whether is irrational?

Because that's equivalent to knowing if is rational...

Why is that equivalent???

BTW, is irrational because is transcendental.

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 Post subject: Re: Exponential functionPosted: Sun, 27 May 2012 08:19:16 UTC
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glebovg wrote:
But why is it unknown whether is irrational?

In general, deciding whether a specific number is irrational or transcendental or ... is a very difficult thing, and indeed for most numbers it is probably impossible (heuristically, proofs are finite strings from a countable collection of letters, so only countably many numbers can be proven to be irrational/transcendental/...).

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 Post subject: Re: Exponential functionPosted: Sun, 27 May 2012 09:04:45 UTC
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glebovg wrote:
But why is it unknown whether is irrational?

Because that's equivalent to knowing if is rational...a very difficult thing to decide in general. Understanding which numbers we can easily express are or are not algebraic is a VERY difficult prospect in general.

I do not understand. e is irrational, so cannot be rational, because like you said if .

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 Post subject: Re: Exponential functionPosted: Sun, 27 May 2012 09:08:13 UTC
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outermeasure wrote:
glebovg wrote:
But why is it unknown whether is irrational?

In general, deciding whether a specific number is irrational or transcendental or ... is a very difficult thing, and indeed for most numbers it is probably impossible (heuristically, proofs are finite strings from a countable collection of letters, so only countably many numbers can be proven to be irrational/transcendental/...).

So it all goes back to Godel's theorem?

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 Post subject: Re: Exponential functionPosted: Sun, 27 May 2012 09:11:29 UTC
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glebovg wrote:
outermeasure wrote:
glebovg wrote:
But why is it unknown whether is irrational?

In general, deciding whether a specific number is irrational or transcendental or ... is a very difficult thing, and indeed for most numbers it is probably impossible (heuristically, proofs are finite strings from a countable collection of letters, so only countably many numbers can be proven to be irrational/transcendental/...).

So it all goes back to Godel's theorem?

No. It doesn't depend on Godel's theorem. Just basic counting.

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 Post subject: Re: Exponential functionPosted: Sun, 27 May 2012 09:22:02 UTC
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outermeasure wrote:
glebovg wrote:
outermeasure wrote:
glebovg wrote:
But why is it unknown whether is irrational?

In general, deciding whether a specific number is irrational or transcendental or ... is a very difficult thing, and indeed for most numbers it is probably impossible (heuristically, proofs are finite strings from a countable collection of letters, so only countably many numbers can be proven to be irrational/transcendental/...).

So it all goes back to Godel's theorem?

No. It doesn't depend on Godel's theorem. Just basic counting.

I presume when you say countable you mean the cardinality of the set of numbers which can be proven to be irrational or transcendental is , but the Lindemann–Weierstrass theorem, for instance, states that is transcendental if is a non-zero algebraic number. This set forms an infinite subset of the reals, and therefore is not countable.

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 Post subject: Re: Exponential functionPosted: Sun, 27 May 2012 10:38:28 UTC
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outermeasure wrote:
glebovg wrote:
But why is it unknown whether is irrational?

Because that's equivalent to knowing if is rational...

Why is that equivalent???

BTW, is irrational because is transcendental.

If , then e would be a root of , but e is transcendental. That is exactly what I thought.

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 Post subject: Re: Exponential functionPosted: Sun, 27 May 2012 16:55:43 UTC
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glebovg wrote:
outermeasure wrote:
glebovg wrote:
outermeasure wrote:
glebovg wrote:
But why is it unknown whether is irrational?

In general, deciding whether a specific number is irrational or transcendental or ... is a very difficult thing, and indeed for most numbers it is probably impossible (heuristically, proofs are finite strings from a countable collection of letters, so only countably many numbers can be proven to be irrational/transcendental/...).

So it all goes back to Godel's theorem?

No. It doesn't depend on Godel's theorem. Just basic counting.

I presume when you say countable you mean the cardinality of the set of numbers which can be proven to be irrational or transcendental is , but the Lindemann–Weierstrass theorem, for instance, states that is transcendental if is a non-zero algebraic number. This set forms an infinite subset of the reals, and therefore is not countable.

Huh? is countable, so is countable, so is a countable subset of the set of transcendentals which (is computable and) can be proven to be transcendental.

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 Post subject: Re: Exponential functionPosted: Sun, 27 May 2012 17:18:58 UTC
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outermeasure wrote:
glebovg wrote:
But why is it unknown whether is irrational?

Because that's equivalent to knowing if is rational...

Why is that equivalent???

BTW, is irrational because is transcendental.

oops, Sorry, I kept unwrapping the same definition over and over in my memory and that time the wrong quantity came out. Thanks for the catch.

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 Post subject: Re: Exponential functionPosted: Sun, 27 May 2012 17:20:01 UTC
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glebovg wrote:
outermeasure wrote:
glebovg wrote:
outermeasure wrote:
glebovg wrote:
But why is it unknown whether is irrational?

In general, deciding whether a specific number is irrational or transcendental or ... is a very difficult thing, and indeed for most numbers it is probably impossible (heuristically, proofs are finite strings from a countable collection of letters, so only countably many numbers can be proven to be irrational/transcendental/...).

So it all goes back to Godel's theorem?

No. It doesn't depend on Godel's theorem. Just basic counting.

I presume when you say countable you mean the cardinality of the set of numbers which can be proven to be irrational or transcendental is , but the Lindemann–Weierstrass theorem, for instance, states that is transcendental if is a non-zero algebraic number. This set forms an infinite subset of the reals, and therefore is not countable.

What are you talking about? is an infinite subset of the reals.

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 Post subject: Re: Exponential functionPosted: Sun, 27 May 2012 23:22:31 UTC
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Quote:
What are you talking about? is an infinite subset of the reals.

I meant an uncountable infinite subset. Sorry.

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 Post subject: Re: Exponential functionPosted: Sun, 27 May 2012 23:38:04 UTC
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Quote:
Huh? is countable, so is countable, so is a countable subset of the set of transcendentals which (is computable and) can be proven to be transcendental.

I do not understand why use chose , but I get the point. For some reason I thought algebraic numbers were uncountable.

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 Post subject: Re: Exponential functionPosted: Sun, 27 May 2012 23:44:34 UTC
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glebovg wrote:
Quote:
Huh? is countable, so is countable, so is a countable subset of the set of transcendentals which (is computable and) can be proven to be transcendental.

I do not understand why use chose , but I get the point. For some reason I thought algebraic numbers were uncountable.

Northcott's Theorem establishes their countability.

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 Post subject: Re: Exponential functionPosted: Sun, 27 May 2012 23:55:14 UTC
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glebovg wrote:
glebovg wrote:
But why is it unknown whether is irrational?

Because that's equivalent to knowing if is rational...a very difficult thing to decide in general. Understanding which numbers we can easily express are or are not algebraic is a VERY difficult prospect in general.

I do not understand. e is irrational, so cannot be rational, because like you said if .

Shadow, could you reply to my previous post. I still do not understand. Thanks.

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 Post subject: Re: Exponential functionPosted: Mon, 28 May 2012 00:03:03 UTC
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glebovg wrote:
glebovg wrote:
glebovg wrote:
But why is it unknown whether is irrational?

Because that's equivalent to knowing if is rational...a very difficult thing to decide in general. Understanding which numbers we can easily express are or are not algebraic is a VERY difficult prospect in general.

I do not understand. e is irrational, so cannot be rational, because like you said if .

Shadow, could you reply to my previous post. I still do not understand. Thanks.

You should ignore that post, telling that is no easier than the original question.

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