Does 0.9999(repeating) = 1?

Yes 705 54.95%
No 578 45.05%
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Does 0.9999(repeating) = 1?

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Kodoku

Senior Member

01-28-2013

Quote:
Originally Posted by Crowcide View Post
If we're not talking about the reals then .99~ = 1 might not be true. But we might as well be arguing .99~ = chicken unless someone defines the formal system they are talking about.
Except that there is no reasonable sense in which 0.99... = chicken, but there are reasonable senses in which 0.99... is not equal to 1. Furthermore, a formal system is often not constructed until many years of study of a mathematical object. Dramatic examples are groups (quotient groups were used over 50 years before they were defined) and fractals.


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Peacë

Senior Member

01-28-2013

Quote:
Originally Posted by Crowcide View Post
You made a positive claim, I asked for evidence. There is nothing nonsensical about that.

Your whole post is incoherent rambling mostly. The reals aren't about events or probability. These things have nothing to do with the construction of the reals, the completeness axiom, or why 1 = .999~.

This whole thread seems to have a theme, non mathematicians ramble some math and say why it doesn't seem like it should be, or how it doesn't jive with their intuition. This is not relevant, this is math. Offer a real formal proof or at least address the proofs you've been given.
To be fair, it was probably a ramble. I am kinda out of it lol.

I don't mean to be mean but I was kinda offended
a) I am a mathematician
b) Intuition is an extremely valuable tool, it helps us be critical, persistent, and able to spot holes in established theories.

Certainly, one should not rely on their intuition, otherwise we suppress valuable math like chaos theory (read Lorenz's first major* presentation(?) on chaos theory; lots of preambles on how it sounds crazy but it *might* be true).

I will not be writing a formal proof for two reasons
1) I am lazy
2) I'm not ready

I only wish to inspire doubt, so I will ask the following question (on a whim, I have not looked at this one thoroughly):
Assume we pick a number A (at random, which is of course beyond human capability)
What was the probability of us not picking that number? If the probability of us not picking that number was 1, was it actually possible for us to pick that number? If it was impossible for us to have picked that number, then how could we pick it in the first place? Is it a contradiction if the probability is 1?
(btw, this might be 75% of a formal contradiction proof, albeit phrased wrong and in question form)

Lastly, in parallel, but slightly unrelated to what was previously stated: If a number exists between two real numbers, is it a real number?

I am sorry, this might not be useful either, and yeah it's probably rambling too (and I spent way too long writing it lol, distractions distractions).

EDIT: *Found out appropriate word was stared out lol.


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Kodoku

Senior Member

01-28-2013

Quote:
I only wish to inspire doubt, so I will ask the following question (on a whim, I have not looked at this one thoroughly):
Assume we pick a number A (at random, which is of course beyond human capability)
What was the probability of us not picking that number? If the probability of us not picking that number was 1, was it actually possible for us to pick that number? If it was impossible for us to have picked that number, then how could we pick it in the first place? Is it a contradiction if the probability is 1?
(btw, this might be 75% of a formal contradiction proof, albeit phrased wrong and in question form)

Lastly, in parallel, but slightly unrelated to what was previously stated: If a number exists between two real numbers, is it a real number?
What you're describing has absolutely nothing to do with real numbers and 0.999... and everything to do with the nature of probability. Standard probability theory has the intuitively bizarre consequence that the probability of randomly selecting one object out of infinitely many equally likely options must be 0. That has nothing to do with real numbers, as it applies to any infinite probability space.

As for a number being real if it's between two real numbers, that's a loaded question. Ordering is generally defined on a set, which is, set-theoretically, a subset of that set. So by definition, anything ordered by the ordering on the reals is a real number, so being between two real numbers makes it a real number.

Yet clearly, there is a sense in which the answer is no: we can come up with numbers (we can even invent some) and an ordering on them, together with the usual real numbers, such that we have a number between two real numbers yet clearly not a real number.


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Kaolla

Senior Member

01-28-2013

Quote:
Originally Posted by Crowcide View Post
So Pi isn't a number. How about sqrt(2), is it a number?
write it out, if you start to get tired, chances are it's not


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Peacë

Senior Member

01-28-2013

Quote:
Originally Posted by Kodoku View Post
What you're describing has absolutely nothing to do with real numbers and 0.999... and everything to do with the nature of probability. Standard probability theory has the intuitively bizarre consequence that the probability of randomly selecting one object out of infinitely many equally likely options must be 0. That has nothing to do with real numbers, as it applies to any infinite probability space.

As for a number being real if it's between two real numbers, that's a loaded question. Ordering is generally defined on a set, which is, set-theoretically, a subset of that set. So by definition, anything ordered by the ordering on the reals is a real numbers, so being between two real numbers makes it a real number.

Yet clearly, there is a sense in which the answer is no: we can come up with numbers (we can even invent some) and an ordering on them, together with the usual real numbers, such that we have a number between two real numbers yet clearly not a real number.
Is it bizarre? Can we be certain this probability space is not a subset of the real numbers?


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Kodoku

Senior Member

01-28-2013

Quote:
Originally Posted by Peac View Post
Is it bizarre? Can we be certain this probability space is not a subset of the real numbers?
eh?

The result applies to proper supersets of the real numbers as well, such as the power set of the set of real numbers. And I still don't see how it relates to the issue at hand.


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Crowcide

Senior Member

01-28-2013

Quote:
Originally Posted by Kaolla View Post
write it out, if you start to get tired, chances are it's not
What about 1/3, is that a number?


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Crowcide

Senior Member

01-28-2013

Quote:
Originally Posted by Kodoku View Post
The result applies to proper supersets of the real numbers as well, such as the power set of the set of real numbers. And I still don't see how it relates to the issue at hand.
It doesn't; it's an attempt at argument by obfuscation


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Peacë

Senior Member

01-28-2013

Quote:
Originally Posted by Kodoku View Post
eh?

The result applies to proper supersets of the real numbers as well, such as the power set of the set of real numbers. And I still don't see how it relates to the issue at hand.
The issue is that probability space is a subset of the real numbers.
Quote:
Standard probability theory has the intuitively bizarre consequence that the probability of randomly selecting one object out of infinitely many equally likely options must be 0.
The only way you can get to this conclusion is if you define 0.000....1 as 0. Which follows the same logic as defining 0.999..... as 1. What I'm trying to show is that you don't need to do this, that doing this is incorrect, and that not doing this offers a logical number space, instead of one that gives "bizarre" properties.

check


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Kodoku

Senior Member

01-28-2013

Quote:
The issue is that probability space is a subset of the real numbers.
You can have supersets of the real numbers (or entirely different spaces, for that matter) as probability spaces.

Quote:
The only way you can get to this conclusion is if you define 0.000....1 as 0. Which follows the same logic as defining 0.999..... as 1. What I'm trying to show is that you don't need to do this, that doing this is incorrect, and that not doing this offers a logical number space, instead of one that gives "bizarre" properties.

check
No, it follows from the fact that if you accept countable additivity, and have equally likely options, then the infinite sum P(A1) + P(A2) + ... will be infinite for any non-zero P(A)s. The only exception is to have infinitesimal probabilities, but there are independent reasons to avoid such probabilities. Aside from the difficulty of understanding what an infinitesimal probability even means, if you start with an infinitesimal prior probability for some hypothesis, then no amount of evidence will raise the posterior probability beyond an infinitesimal. That's a fatal problem.