0.999...

[Agent Smith]

We know that $0.999... = 0.\overline 9 = 1$

Does that mean ...

$0.\overline 9 \times 10^\infty = 1 \times 10^\infty = 10... = \infty$

?

I've seen infinite decimal expansions like $\sqrt 2 = 1.414...$ or $2/3 = 0.666...$
but haven't seen that with natural numbers. Is this, $2013...$ a valid mathematical entity? Isn't $2013... = \infty$?

 

[khansaheb]

with natural numbers. Is this, 2013...2013...2013... a valid mathematical entity? Isn't 2013...=∞2013... = \infty2013...=∞

This must be that hallucination that some people are talking about.........

 

[fresh_42]

Infinity cannot be treated like a number. If you want to have a calculus that deals with infinity as a separate admissible object, then it is the so-called non-standard analysis. In that case $10^\infty =\infty$ and $c \cdot \infty =\infty$ for any (finite) real number $c.$
$$0.999\ldots =0.\overline{9} =1$$is normally an abbreviation for
$$0.\overline{9} := 0.999\ldots := \sum_{k=1}^\infty 9\cdot (0.1)^k := 9\cdot \lim_{n \to \infty} \sum_{k=1}^n (0.1)^k =9\cdot \lim_{n \to \infty}\left(-1+\dfrac{1-(0.1)^{n+1}}{1-0.1}\right)=9\cdot \left(-1+\dfrac{10}{9}\right)=9\cdot \dfrac{1}{9}=1$$where $:=$ reads "is defined as".

It is not clear to me what $2013\ldots$ means. $2013.000\ldots = 2013$ and $2013000\ldots$ is not a defined entity.

 

[Agent Smith]

This must be that hallucination that some people are talking about.........

Haha. So $2013... = \infty$?

Infinity cannot be treated like a number. If you want to have a calculus that deals with infinity as a separate admissible object, then it is the so-called non-standard analysis. In that case $10^\infty =\infty$ and $c \cdot \infty =\infty$ for any (finite) real number $c.$
$$0.999\ldots =0.\overline{9} =1$$is normally an abbreviation for
$$0.\overline{9} := 0.999\ldots := \sum_{k=1}^\infty 9\cdot (0.1)^k := 9\cdot \lim_{n \to \infty} \sum_{k=1}^n (0.1)^k =9\cdot \lim_{n \to \infty}\left(-1+\dfrac{1-(0.1)^{n+1}}{1-0.1}\right)=9\cdot \left(-1+\dfrac{10}{9}\right)=9\cdot \dfrac{1}{9}=1$$where $:=$ reads "is defined as".

It is not clear to me what $2013\ldots$ means. $2013.000\ldots = 2013$ and $2013000\ldots$ is not a defined entity.

Gracias. Also is $2013... = \infty$?

Is $...2013 = \infty$?

 

[blamocur]

Is this, 2013...2013...2013... a valid mathematical entity?

It's not.

 

[Agent Smith]

It's not.

Because ... that would mean I know the leading digits of $\infty$?

 

[fresh_42]

Haha. So $2013... = \infty$?

Gracias. Also is $2013... = \infty$?

Is $...2013 = \infty$?

Not if the dots at the beginning represent zeros. Then it is simply $2013.$

 

[Agent Smith]

It's not.

Thank you.

So $3.14159...$ is ok
but
$2013...$ and $...2013$ are not! For the reason that it implies I know something that I possibly can't know, how infinity begins and ends?

 

[Agent Smith]

Not if the dots at the beginning represent zeros. Then it is simply $2013.$

No, the dots are not 0's

 

[Agent Smith]

I ask because for an irrational number like $\pi = 3.14159...$, I could say that if $s = \text{the numerical string that is the decimal expansion of } \pi$, then $\pi = 3 + \frac{s}{999...} = 3 + \frac{14159...}{999...}$. Notice the $999...$ and $14159...$ (infinite repetend).

 

[fresh_42]

No, the dots are not 0's

Well, that is the problem! What are those dots? In $0.999\ldots$ they represent repeating nines, in $3.141592653589793238462 \ldots$ they represent "any numbers", in $2013.000\ldots$ they represent zeros, and at the beginning, they are not defined except for p-adic numbers, and at the end without a decimal separation, they aren't defined either.

So what are the dots is the central point here. You cannot use them outside of context or convention without explicitly saying what you mean. That is why I used the symbol "$:=$" for "is defined as". I explicitly said what I meant!

 

[Agent Smith]

Well, that is the problem! What are those dots? In $0.999\ldots$ they represent repeating nines, in $3.141592653589793238462 \ldots$ they represent "any numbers", in $2013.000\ldots$ they represent zeros, and at the beginning, they are not defined except for p-adic numbers, and at the end without a decimal separation, they aren't defined either.

So what are the dots is the central point here. You cannot use them outside of context or convention without explicitly saying what you mean. That is why I used the symbol "$:=$" for "is defined as". I explicitly said what I meant!

The dots represent nonzero digits and continue on forever in the direction the ellipsis appears.
So $2013...$ is a number whose initial digits are known, but the dots indicate that their place value is uncertain (there are digits that have lower place values as indicated by $...$, but the lowest place value is ones i.e. the number is an integer.

For $...2013$ we know 2 is in the thousands place and 1 is in the tens place and so on, but there are preceding digits in higher place values and they go on forever in the direction indicated by the dots.

It looks as though, as defined above, for $2013...$, the place value of 2 is $10^\infty$ i.e. $2013... = \infty$. As for $...2013$, we're looking at some unknown digit which sits on a place value $\infty^\infty$

 

[Steven G]

We know that $0.999... = 0.\overline 9 = 1$

Does that mean ...

$0.\overline 9 \times 10^\infty = 1 \times 10^\infty = 10... = \infty$

?

I've seen infinite decimal expansions like $\sqrt 2 = 1.414...$ or $2/3 = 0.666...$
but haven't seen that with natural numbers. Is this, $2013...$ a valid mathematical entity? Isn't $2013... = \infty$?

Any positive number times infinity is infinity. $\displaystyle \overline 9$ is a positive number.

 

[fresh_42]

There is another possibility, points in between:
$$2013000\ldots 000:=20,130, \ldots,000, 000 =2.013\cdot 10^{16}$$but you have to write the RHS because the information $16$ is essential.

 

[fresh_42]

For $...2013$ we know 2 is in the thousands place and 1 is in the tens place and so on, but there are preceding digits in higher place values and they go on forever in the direction indicated by the dots.

This is only valid if we speak of p-adic numbers. Those are a bit like Parias in mathematics:

p-adic number - Wikipedia

en.wikipedia.org

Counting to p-adic Calculus: All Number Systems That We Have

I want to consider the numbers by their mathematical meaning. I will describe the mathematics behind our number systems as simple as possible.

www.physicsforums.com

However, they are only the p-adic version of the decimal dots on the right in usual real numbers.

 

[Agent Smith]

There is another possibility, points in between:
$$2013000\ldots 000:=20,130, \ldots,000, 000 =2.013\cdot 10^{16}$$but you have to write the RHS because the information $16$ is essential.

Si, in this case the dots have nothing to do with $\infty$ and represent only missing information.

 

[Agent Smith]

This is only valid if we speak of p-adic numbers. Those are a bit like Parias in mathematics:

p-adic number - Wikipedia

en.wikipedia.org

Counting to p-adic Calculus: All Number Systems That We Have

I want to consider the numbers by their mathematical meaning. I will describe the mathematics behind our number systems as simple as possible.

www.physicsforums.com

However, they are only the p-adic version of the decimal dots on the right in usual real numbers.

Gracias.

 

[fresh_42]

Gracias.

De nada.

 

[Agent Smith]

I ask because for an irrational number like $\pi = 3.14159...$, I could say that if $s = \text{the numerical string that is the decimal expansion of } \pi$, then $\pi = 3 + \frac{s}{999...} = 3 + \frac{14159...}{999...}$. Notice the $999...$ and $14159...$ (infinite repetend).

@fresh_42 , the above would be incorrect? unless there's an infinity that begins $14159...$ and another infinity that begins $999...$.


$3 + \frac{\infty_1}{\infty_2} = \pi$. Note, both $\infty_1$ and $\infty_2$ are natural numbers.

 

[fresh_42]

@fresh_42 , the above would be incorrect? unless there's an infinity that begins $14159...$ and another infinity that begins $999...$.


$3 + \frac{\infty_1}{\infty_2} = \pi$. Note, both $\infty_1$ and $\infty_2$ are natural numbers.

You really cannot deal with infinities as if they were numbers. And dots require a definition, or in many cases a convention we all agree on. I don't understand the sequence of nines in the denominator. If we use the decimal system to represent $\pi$ then
$$\pi = 3.141592653589793238462\ldots =3+\dfrac{1}{10}+\dfrac{4}{100}+\dfrac{1}{1000}+\dfrac{5}{10000}+\dfrac{9}{100000}+\ldots$$You cannot even write $3+\dfrac{14159265\ldots}{1000000000\ldots}$ - at least not if you want to be understood - let alone nines in the denominator. Note that I wrote dots although the numerator is a sequence of changing digits and the denominator is a regular sequence that only adds a zero from term to term.

Dots are always dependent on their context. That's why you cannot take $0.999\ldots = 1,$ rip off the context and replace ones by dot nines. If you want to write unusual dots in $\pi ,$ I suggest



Infinity is a symbol you should try to avoid. It makes sense in sums $\displaystyle{\sum_{k=0}^\infty a_k} := \lim_{n \to \infty} \sum_{k=0}^n a_k$ as an abbreviation and sometimes as the result of a limit like $\lim_{n \to \infty} n=\infty ,$ but even that is already an abbreviation for a complex logical statement, not something infinitely large. It is rather something growing beyond all finite bounds!