Question on Infinite

[Pyrotex]

Thanks CraigD!

Just found a book called "The Infinite Book - A short guide to the boundless, timeless and endless". You are right, it's very interesting indeed.

I learned about infinity while in the 10th grade. I found a book: "One, Two Three... Infinity" by George Gamov.

 

It may be difficult to find now, but that was one wonderful, mind-expanding book.

[Pyrotex]

If the limit of infinitely small is infinity, then does zero ever exist?

Absolutely!!

 

Zero is the limit of 1/x as x goes to infinity.

[Qfwfq]

However, unlike infinity, zero does not exist only as a limit, it is a number.

[Pyrotex]

However, unlike infinity, zero does not exist only as a limit, it is a number.

"I am a free concept! I am not a number!"

 

--Zero, from the TV series, "The Prisoner"

[Qfwfq]

Zero is my prisoner and I decide whether or not he is a number!!!!!!

 

[Pyrotex]

Zero is my prisoner and I decide whether or not he is a number!!!!!!

ehhh, you can have him.

 

He doesn't amount to much.

 

:)

[sanctus]

ehhh, you can have him.

 

He doesn't amount to much.

 

:hihi:

have you considered a world without the concept of zero? I think that amnesty international should step in if Qfwfq really holds nothing as prisoner :)

[Jet2]

:confused:If infinitely small can come to an end, would it be absolute zero?

[Tormod]

I think absolute zero would amount to nothing.

[Qfwfq]

If infinitely small can come to an end,

If it could come to an end, it wouldn't be infinitely small! :hihi:

[ughaibu]

What about subtracting 0.9 recurring from 1?

[sanctus]

ughaibu, I guess that you actually get 0.0000..... i.e. not 0, hence you get the infinitely small but not 0.

[CraigD]

What about subtracting 0.9 recurring from 1?

ughaibu, I guess that you actually get 0.0000..... i.e. not 0, hence you get the infinitely small but not 0.

The mathematically correct answer to ugh’s question is that you get zero, not an infinitesimal, because [math]0.\overline{9} = 1[/math]. Proof:

[math]\mbox{Given}\, A = 0.\overline{9}[/math]

[math]10A – A = 9.\overline{9} - 0.\overline{9}[/math]

[math]9A = 9[/math]

[math]\mbox{Therefore}\, A = 1[/math]

 

The critical idea here is that repeating decimals are concrete, fixed rational numbers, not algorithms for approximating such numbers, such as:

1. Set A to 0

2. Set B to 9

3. Set B to B divided by 10

4. Set A to A plus B

5. Repeat steps 3 through 5

[Turtle]

The mathematically correct answer to ugh’s question is that you get zero, not an infinitesimal, because 0.9… = 1. Proof:

Given A = 0.9…

10A – A = 9.9… - 0.9…

9A = 9

Therefore A = 1

 

The critical idea here is that repeating decimals are concrete, fixed rational numbers, ...

 

Nicely shown Craig. Many texts use a straight line over the repeating part of the decimal to show that it is in fact a repeating decimal; does Latex have that symbol capability? :confused:

 

PS (barring an available bar, I'll use ellipsis marks to indicate a repeating decimal) following on Craig's admonition that repeating decimals are in fact fixed rational numbers, I conjured this. it is easy to show that 1/3 = .333... and that 3*.333...=.999..., and since 3*1/3 = 1 then .999...=1. :turtle:

[CraigD]

does Latex have that symbol capability?

Indeed it does – the overline function. I’ve edited my last post to use it.

 

At the moment, hypography’s nice new math tag isn’t working, making it necessary to use the older, uglier latex tag. Perhaps I’m unconsciously avoiding using it ‘til the pretty renderer is fixed :confused:

[Turtle]

Indeed it does – the overline function. I’ve edited my last post to use it.

 

At the moment, hypography’s nice new math tag isn’t working, making it necessary to use the older, uglier latex tag. Perhaps I’m unconsciously avoiding using it ‘til the pretty renderer is fixed

 

Nice! I see the edit. Now if we want to restore an infinitude to this oft misconstrued and misrepresented proof you so succinctly demonstrated and explained, we can move to using all the other bases*. :confused:

 

* repeating digits under long division

[von Faulkenstein]

What about negative infinity and positive infinity?:confused: