# Insights Why Do People Say That 1 and .999 Are Equal? - Comments

1. Oct 17, 2015

### Multiple_Authors

2. Oct 17, 2015

### HallsofIvy

While there are several arguments here that people might accept as proofs, they all, as stated, are non-rigorous, involving some basic, unstated assumptions. For example, in argument two, it is assumed that multiplying x= 0.999... by 10 gives 10x= 9.999... and then that subtracting x give 9x= 0.999... again. Both of those assume the usual arithmetic properties are true for 0.999... That is true but exactly the sort of thing people who object to "0.999...= 1" would object to anyway. In argument four, it is accepted that 0.333...= 1/3. Why would a person who objects to "0.999...= 1" accept that?

It is not too difficult to give a rigorous proof using "geometric series". It is easily shown that the sum $\sum_{i= 0}^n ar^i$ is $$\frac{a(1- r^n}{1- r}$$. To do that, let $S_n= \sum_{i= 0}^n ar^i= a+ ar+ ar^2+ \cdot\cdot\cdot+ ar^n)$. Since that is a finite sum, the usual properties of arithmetic hold and we can write $S_n- a= ar+ ar^2+ \cdot\cdot\cdot+ ar^n= r(a+ ar+ \cdot\cdot\cdot+ ar^{n-1})$. The quantity in the last parentheses is almost $S_n$ itself. It is only missing the last term, $ar^n$. Restore that by adding $ar^{n+1}$ to both sides: $S_n- a+ ar^n= r(a+ ar+ \cdot\cdot\cdot+ ar^{n-1})+ ar^{n+ 1}$. Taking that last term inside the parentheses we have $S_n- a+ ar^n= r(a+ ar+ \cdot\cdot\cdot+ ar^{n-1}+ ar^n)= rS_n$. We can write that as $rS_n- S_n= (r- 1)S)_n= ar^n- a= a(r^n- 1)$ and dividing both sides by r- 1, $S_n= \frac{a(r^n- 1)}{r- 1}$ which, for r< 1 can be written more as $S_n= \frac{a(1- r^n)}{1- r}$

The geometric series, $\sum_{i= 0}^\infty ar^i$, is, by definition, the limit of the "partial sums", $S_n= \sum_{i= 0}^n ar^i$, as n goes to infinity. As long as $|r|< 1$, $\lim_{n\to\infty} r^n= 0$ so that limit is easily calculated as $\sum_{i= 0}^\infty ar^i= \frac{a}{1- r}$.

Now, turning to the problem at hand. 0.999... is, by definition of the decimal numeration system, $0.9+ 0.09+ 0.009+ \cdot\cdot\cdot= 0.9+ 0.9(0.1)+ 0.9(0.01)+ \cdot\cdot\cdot= 0.9+ 0.9(.1)+ 0.9(.1^2)+ \cdot\cdot\cdot$, precisely a geometric series with a= 0.9 and r= 0.1. So the sum, and the value of 0.9999..., is $\frac{0.9}{1- 0.1}= \frac{0.9}{0.9}= 1$.

3. Oct 17, 2015

### Greg Bernhardt

4. Oct 18, 2015

### Algr

Proofs #2 & #3 are what convinced me that 1 and .999... are NOT equal. It's just insulting to assume that .333... = 1/3 in this context. And why assume that .999... and 9.999... have different numbers of nines when this would never happen for any finite number?

5. Oct 18, 2015

I alread covered this in blog a while ago, though it's in spanish: https://preguntaspococientificas.wordpress.com

6. Oct 18, 2015

### FactChecker

If they are different, how different are they? Suppose you say that they are different by more than 0.0001. Then 1 - 0.99999 is closer and the more 9's you add, the closer it gets. So you can not say there is any difference.

7. Oct 18, 2015

### SlowThinker

I'm not an expert on these things, but you can easily tell that a proof is wrong when it proves 0.9999...=1 on surreal numbers. Which most of the above do.
I'm not entirely sure about HallsofIvy's proof, it might be OK, even if I think it's swapping $\infty$ with $\omega$ somewhere along the way.

8. Oct 18, 2015

### FactChecker

You can easily prove it by contradiction, as I indicated. Once that is done, you know that they are equal. Then I would to be careful about saying that other proofs are wrong unless I completely understood the other proofs and can pinpoint the error.

9. Oct 18, 2015

### Staff: Mentor

How is this assumption insulting. Just by ordinary division you can show that the 3 digit keeps repeating.
To the right of the decimal point, both have the same number of 9 digits. Also I don't understand what you are saying about "finite numbers". Both .999... and 9.999... are finite numbers; i.e., strictly less than infinity.
I did a quick scan of all of the proofs. None of them mentioned surreal numbers. If you can point out where I might have missed this, I would appreciate it.

Last edited: Oct 18, 2015
10. Oct 18, 2015

### SlowThinker

My idea was: if the proof, applied over surreal numbers, proves that 0.9999...=1, then the proof is not correct. There must be some implicit assumption.
I would say that, by definition of the decimal numeration system, 0.9999...=$\sum_{n=1}^\omega{}9\ 0.1^n$ rather than $\sum_{n=1}^\infty{}9\ 0.1^n$. The difference is, obviously, 0 in real numbers.

11. Oct 18, 2015

### Staff: Mentor

Why would the proof need to invoke surreal numbers? The proofs are all concerned with two representations of the same real number.
And that's the set we're concerned with here.

12. Oct 18, 2015

### SlowThinker

If the proof can be used to prove a false statement, then the proof must be wrong.
0.9999...$\neq$1 in surreal numbers, so if the proof works with them too, then it's wrong.
I dare say that all of the "informal proofs" from the article immediately work with surreal numbers, so they must be wrong.

This quote from the original article,
seems to be the answer that people are looking for, together with "Real numbers are defined in such a way that 0.999...=1".

13. Nov 2, 2015

### Ernest S Walton

So does this mean that .99999..... = .99999...8 ?

14. Nov 2, 2015

### pwsnafu

No, because the right hand side is nonsense. You can't have a non-terminating run of nines and then terminate it.

15. Nov 2, 2015

### Ernest S Walton

Okay... but it seems like to make .999 = 1 you also have to terminate the run of nines. Anyway I know this has been discussed several times so I'll read up some more.

16. Nov 2, 2015

### Staff: Mentor

As already mentioned, the right side is meaningless for two reasons. On the left side the ellipsis (...) means that the pattern you see repeats endlessly. On the right side, it is not known how many 9s there are in front of the final 8 digit.

Don't confuse .999 with .999... They mean two very different things. The first is identical to 999/1000 which is smaller than 1, and the second is equal to 1.

17. Dec 8, 2015

### Nemika

*Both .999… and 9.999… are finite numbers; i.e., strictly less than infinity.*

How can you call 0.999...... or 9.999...... as finite?
If these are finite then infinity can also be considered finite!

18. Dec 8, 2015

### WWGD

Please look up the definition of a finite number. There is a difference between an infinite _decimal expansion_ and an infinite number.

19. Dec 8, 2015

### Staff: Mentor

WWGD is correct. Finite numbers are bounded; infinite numbers are unbounded.

20. Dec 9, 2015

### mathexam

They are clearly different but if you're approximating, they're almost equal.