# .99999 repeating = 1? Whats your opinion?
**URL:** https://boards.straightdope.com/t/99999-repeating-1-whats-your-opinion/85321
**Category:** Factual Questions
**Created:** [October 4, 2001, 3:03am UTC](https://boards.straightdope.com/t/99999-repeating-1-whats-your-opinion/85321 "2001-10-04T03:03:53Z")
**Posts on this page:** 15
**Page:** 1
### Author:  [@EagleEye](https://boards.straightdope.com/u/EagleEye)
#### Post date: [October 4, 2001, 3:03am UTC](https://boards.straightdope.com/t/99999-repeating-1-whats-your-opinion/85321/1 "2001-10-04T03:03:53Z")
Do you think that .9 repeating equals one? What is your theory. I’ll give you mine tommorrow.
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### Author:  [@Shayna](https://boards.straightdope.com/u/Shayna)
#### Post date: [October 4, 2001, 3:10am UTC](https://boards.straightdope.com/t/99999-repeating-1-whats-your-opinion/85321/2 "2001-10-04T03:10:37Z")
Lots of people had theories [in this thread](http://boards.straightdope.com/sdmb/showthread.php?threadid=82064) (most recently posted to on 8/21/01). It seems the consensus of opinion is that .99999 does, in fact, equal 1.
* * *
I really appreciate your consideration in avoiding stepping on my penis - **Spiny Norman**
Jeg elsker dig, Thomas
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### Author:  [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)
#### Post date: [October 4, 2001, 3:12am UTC](https://boards.straightdope.com/t/99999-repeating-1-whats-your-opinion/85321/3 "2001-10-04T03:12:59Z")
This isn’t a matter of opinion. .999… does equal 1 in the real number system.
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### Author:  [@friedo](https://boards.straightdope.com/u/friedo)
#### Post date: [October 4, 2001, 3:13am UTC](https://boards.straightdope.com/t/99999-repeating-1-whats-your-opinion/85321/4 "2001-10-04T03:13:51Z")
There’s nothing to opine. By definition 0.999… equals one.
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### Author:  [@Wumpus](https://boards.straightdope.com/u/Wumpus)
#### Post date: [October 4, 2001, 3:27am UTC](https://boards.straightdope.com/t/99999-repeating-1-whats-your-opinion/85321/5 "2001-10-04T03:27:38Z")
Nooooooo! Mods, kill it before it breeds.
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### Author:  [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)
#### Post date: [October 4, 2001, 3:59am UTC](https://boards.straightdope.com/t/99999-repeating-1-whats-your-opinion/85321/6 "2001-10-04T03:59:13Z")
> [@](#):
>
> \*Originally posted by friedo \*
> \*\*There’s nothing to opine. By definition 0.999… equals one. \*\*
It’s not a definition, just a consequence of the axiom of completeness and the definition of decimal expansions. This may seem like a minor point, but definitions are supposed to be non-creative (i.e., you can’t prove anything with a definition that you can’t prove without it), and a definition like that could be creative.
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### Author:  [@ski](https://boards.straightdope.com/u/ski)
#### Post date: [October 4, 2001, 4:59am UTC](https://boards.straightdope.com/t/99999-repeating-1-whats-your-opinion/85321/7 "2001-10-04T04:59:12Z")
Yes. Look at it a different way, in fractions. 1/9 = .11111… 9 x 1/9 = 1. .1111… x 9 = .99999…
Therefore, .99999… = 1.
Sound good?
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### Author:  [@The\_Peyote\_Coyote](https://boards.straightdope.com/u/The_Peyote_Coyote)
#### Post date: [October 4, 2001, 5:19am UTC](https://boards.straightdope.com/t/99999-repeating-1-whats-your-opinion/85321/8 "2001-10-04T05:19:16Z")
No. .9999999… does not equal 1. Say you can extend .9999… out to k decimal places. I can then pick the number k + 1 (after all, the set integers is infinite). Multiply .99999… by k + 1 and you will get a different answer from 1 \* (k+1). If you assume that .99999… = 1 than (k + 1) \* .9999999… must = (k + 1) \* 1.
Furthermore, the set of real numbers, to which .9999… and 1 both belong, is a type of mathematical structure called a field. One of a field’s properties is that it has a multiplicative identity, a number y such that y \* x = x for all x’s belonging to the field. In the case of the real numbers the multiplicative identity is 1, and by the definition of a field, the multiplicative identity is UNIQUE. Therefore, .99999… cannot be 1 as that would mean the real numbers would have two multiplicative identities which is impossible by definition.
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### Author:  [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)
#### Post date: [October 4, 2001, 5:41am UTC](https://boards.straightdope.com/t/99999-repeating-1-whats-your-opinion/85321/9 "2001-10-04T05:41:07Z")
> [@](#):
>
> \*Originally posted by The Peyote Coyote \*
> **No. .9999999… does not equal 1. Say you can extend .9999… out to k decimal places. I can then pick the number k + 1 (after all, the set integers is infinite). Multiply .99999… by k + 1 and you will get a different answer from 1 \* (k+1). If you assume that .99999… = 1 than (k + 1) \* .9999999… must = (k + 1) \* 1.**
A series is not a sum of an infinite number of terms. A series is the limit of the sequence of partial sums. .999… and 1.000… are both shorthand for two series with the same value. I refer the reader to Walter Rudin’s _Principles of Mathematical Analysis_.
> [@](#):
>
> \*\*Furthermore, the set of real numbers, to which .9999… and 1 both belong, is a type of mathematical structure called a field. One of a field’s properties is that it has a multiplicative identity, a number y such that y \* x = x for all x’s belonging to the field. In the case of the real numbers the multiplicative identity is 1, and by the definition of a field, the multiplicative identity is UNIQUE. Therefore, .99999… cannot be 1 as that would mean the real numbers would have two multiplicative identities which is impossible by definition. \*\*
The multiplicative identity element of a field is unique, but it need not have a unique representation. .999… and 1.000… are two different representations for the multiplicative identity element of the real numbers.
In fact, as the identity of any group G can be written as g\*g[sup]-1[/sup] for any g[sym]Î[/sym]G, the identity never has a unique representation unless |G| = 1. The non-zero elements of a field form a group under multiplication, so the multiplicative identity of a field never has a unique representation.
If you want to argue that .999… is not equal to 1.000…, you need to argue against the property of completeness. Specifically, you need to show that the set of Dedekind cuts of rational numbers is not complete. You can’t, of course, but it might be fun to try.
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### Author:  [@Zenster](https://boards.straightdope.com/u/Zenster)
#### Post date: [October 4, 2001, 5:56am UTC](https://boards.straightdope.com/t/99999-repeating-1-whats-your-opinion/85321/10 "2001-10-04T05:56:18Z")
[list]I’d say that; \_\_\_\_\_\_
The value 0.999999 is equal to the number one minus the reciprocal of infinity.
Someone please demonstrate that this is not the case.
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### Author:  [@cornflakes](https://boards.straightdope.com/u/cornflakes)
#### Post date: [October 4, 2001, 6:28am UTC](https://boards.straightdope.com/t/99999-repeating-1-whats-your-opinion/85321/11 "2001-10-04T06:28:33Z")
> [@](#):
>
> \*Originally posted by Wumpus \*
> \*\*Nooooooo! Mods, kill it before it breeds. \*\*
gry
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### Author:  [@DarrenS](https://boards.straightdope.com/u/DarrenS)
#### Post date: [October 4, 2001, 7:29am UTC](https://boards.straightdope.com/t/99999-repeating-1-whats-your-opinion/85321/12 "2001-10-04T07:29:19Z")
cornflakes - lol.
(for those who don’t know, there are some threads that crop up again and again on these boards - .9999… = 1 is one of them, and ‘gry’ refers to another which I don’t dare mention for fear it sparks off yet another debate)
EagleEye - that .999… = 1 is not a matter of opinion - AFAIK, it is accepted by the mathematical community in the same way that sin[sup]2[/sup]x + cos[sup]2[/sup]x = 1 .
[Here](http://forum.swarthmore.edu/dr.math/faq/faq.0.9999.html) is a formal proof for .9 recurring = 1. I found it in Google by searching for “.9999 = 1” - amusingly, you can find various different proofs by changing the number of 9s in your search. It’s almost a proof by “reductio ad nauseum” 😉
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### Author:  [@Triskadecamus](https://boards.straightdope.com/u/Triskadecamus)
#### Post date: [October 4, 2001, 9:02am UTC](https://boards.straightdope.com/t/99999-repeating-1-whats-your-opinion/85321/13 "2001-10-04T09:02:27Z")
Infinity is not a discrete mathematical value. Hence such terms as infinity minus one, infinity plus one, and other products and sums of a similar form are indeterminate. Infinity really means, “I don’t know how much, but it seems like a whole lot.” When you find a use for the term where that definition is nonsensical, you have made an error in your use of the term.
## Tris
“Here Kitty, Kitty, Kitty.” ~ Erwin Schrodinger ~
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### Author:  [@glee](https://boards.straightdope.com/u/glee)
#### Post date: [October 4, 2001, 9:40am UTC](https://boards.straightdope.com/t/99999-repeating-1-whats-your-opinion/85321/14 "2001-10-04T09:40:49Z")
I realise this seems puzzling, but it’s not a matter of opinion.
‘Who is the greatest athlete of all time?’ is a matter of opinion.
As previous posters have said, mathematics is founded on axioms, which give rise to all sorts of consequences. **There is no opinion involved.**
By using the phrase repeating (forever) you have introduced infinity. Infinity, as Triskadecamus showed, is tricky!
Does this help?
1 - .9 recurring = 0.0 recurring = defined as 0.
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### Author:  [@bibliophage](https://boards.straightdope.com/u/bibliophage)
#### Post date: [October 4, 2001, 9:59am UTC](https://boards.straightdope.com/t/99999-repeating-1-whats-your-opinion/85321/15 "2001-10-04T09:59:29Z")
This has been done to death. I’ll close this thread. Read the earlier thread, linked to above.
bibliophage
moderator GQ