# How Is This Possible

1. Apr 13, 2005

### krypto

Everyone knows that 3/3 is equal to 1....
but 1/3 is equal to 0.3333333 recurring....
this means that 2/3 is 0.6666666 recurring.....
and therefore 3/3 is 0.99999999 recurring...
So therefore 3/3 does not equal 1! :surprised
Id like to try and see you prove all this wrong!

2. Apr 13, 2005

### CRGreathouse

Your first mistake is on line 5, where you assume there is a contradiction. There's nothing wrong with 0.9999... = 1.

3. Apr 13, 2005

### snoble

This is a classic dilema. In fact what you've shown that 1 actually has two decimal representations. ie 0.9999...=1. The reals are actually just Cauchy sequences of rational numbers modded out by the equivilancy that two sequences are equivilant if the interpolated sequence is still Cauchy. So the claim is the sequences $$(\sum_{j=1}^k 9\times 10^-j )_{k=1}^\infty$$ is equivilant to the sequence $$(1)_{k=1}^\infry$$. It is not hard to show the interpolated sequence is Cauchy. And clearly the first one is 0.999... and the second is 1.

4. Apr 13, 2005

5. Apr 13, 2005

### krypto

shhh lol 0.999 recurring isnt 1, just look at it simply

6. Apr 13, 2005

### master_coda

I assume "like at it simply" really means "look at it without bothering to learn how decimal notation works".

7. Apr 13, 2005

### Hurkyl

Staff Emeritus
Do not make duplicate posts.

8. Apr 13, 2005

### CRGreathouse

You mean that the two don't look the same? Well, sure. 1+1=2, but the string "1+1" isn't the same as the string "2". They just happen to have the same numeric value.

9. Apr 14, 2005

### eNathan

This has been discussed so many times here. Hmn, do you know what the mathematical keyword called "difference" means? It means subtraction. So tell me, what is the difference from 1.0 and 0.999...~ ?

Hmn, you have two options. Either 0 or .000...~1
The second answer is wrong. .999... = 1
This concept does not violate common sense at all. Its a mathematical thing. Were you aware that 5/10 is the same as 1/2? Hmn, does that violate your logic too?

Settled

Last edited: Apr 14, 2005
10. Apr 14, 2005

### shyboy

strictly speaking 1/3 is not equal 0.33333...
what do we have here is an infinite series 0.3, 0.33, 0.333, 0.33333....
this series converges to 1/3.
At the same time there is an infinite number of another series, i.e.
0.4,0.2, 0.34,0.32,0.334,0.332... which converges to the same number.

So 0.9999... is actualy a representation of series 0.9, 0.99, 0.999.... which converges
to the 1. So, roughly speaking, 1=0.99999... if we keep in mind that on the left side we have not the number, but a set of numbers.

In the same way we can write: 1=1.1, 0.9, 1.01, 0.91, 1.001,0.991... But

11. Apr 14, 2005

### uart

Man I get sick of these threads but here goes anyway. Shyboy an infinite series does NOT approach it's limit point it EQUALS it. A finite series of length n approaches the limit as n approaches infinity but a convergant infinite series is (let me repeat) EQUAL to the limit.

12. Apr 14, 2005

### shyboy

you may be right, but why all of these mathematicians use the term convergence?

13. Apr 14, 2005

### master_coda

Why wouldn't they use the word convergence? Convergence isn't a reserved word that you're only allowed to apply to sequences.

14. Apr 14, 2005

### arildno

shyboy:
We say that the sequence of finite partial sums converges to some limit.
An infinite series is typically defined as that limit (i.e, another name for it).

Last edited: Apr 14, 2005
15. Apr 14, 2005

### jcsd

Right, infinite decimal expansions are defined (I can't stress this too much as it really is only a matter of how soemone choose to define something), as the value that the sequence of partial sums of the related decimal series converges to.

To be fair infinite sequenecs are sequences of numbers not numbers themselves, but if the are convergent thye do have a partocualr number associated with them (i.e. the value they converge to).

16. Apr 14, 2005

### snoble

It's all a matter of definition really. As I said before a standard way to build the reals is just the completion of the rationals under the standard norm. If you think of the rationals this way then every number is just a congruency class of Cauchy sequences. Even non-infinite decimal representations. Even numbers in radical form. Perceived this way there is no $$\pi$$ beyond the Cauchy sequences that represents $$\pi$$. Or rather I should say "Perceived this way there is no $$\pi$$ beyond the Cauchy sequences that are represented by the symbol $$\pi$$."

17. Apr 14, 2005

### shyboy

I guess that a congruency class of Cauchy sequences is not what you can find in a high school texbook, but the question itself is within high-school range.

18. Apr 14, 2005

### Hurkyl

Staff Emeritus
Another important thing to keep in mind is that $0.\bar{9}$ is a number. Words like "convergence" or "member" don't apply to numbers.

We might build numbers, or represent numbers, with objects for which convergence and membership do mean something, but those words only apply to those objects, not to $0.\bar{9}$ as a number.

19. Apr 14, 2005

### moose

since when is 1/3=.333333..........its the same as
.99999.....=1, which i dont believe is true....
I don't care what kinda proofs might even show that it is, it just shows that something contradicts in those proofs.....

20. Apr 14, 2005

### Hurkyl

Staff Emeritus
You can disbelieve it all you want. Why you would come to a math forum if you reject mathematics?

Last edited: Apr 14, 2005