# .999 and 1, convergence

1. May 14, 2010

### srfriggen

1. The problem statement, all variables and given/known data

Using the geometric series formula, a/1-r , where l r l < 1, it is easy to see that:

1/10 + 1/10(1/10)^2 + 1/10(1/10)^3... = 1.

applying the same formula to say, 2/3 = .666... does not give the answer 7, but rather 20/27.

My question is, what is the intuition behind .999... = 1?

It is easy to prove, and seemingly obvious (my girlfriend said, "well sure, even if it is infinite you would always have a 9 at the end so all them would round up like a domino effect from the end". I said, "sure, but then why doesn't that work for .666?" and I'm stumped (obviously her intuition about rounding up isn't what matters. Can someone tell me what does?

2. Relevant equations

3. The attempt at a solution
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. May 14, 2010

### Cyosis

Luckily it doesn't. I suggest you grab a ruler and measure 0.67 cm then measure 7 cm to see the difference between the two. It is also not equal to 20/27. In fact the method you've tried to use (albeit incorrectly) to prove that 0.9..=1 works for 0.6.. as well. The answer you should get is 0.666.=2/3.

This is also wrong and would in fact be equal to 1/9.

There is no end, the 9s keep going on forever. You can represent such a decimal as 0.9+0.09+0.009+.... and link it to the geometric series to see it converges to one.

Last edited: May 14, 2010
3. May 14, 2010

### srfriggen

I don't have a ruler near me lol, but I must have used the formula incorrectly then?

my a = 2/3 and my r = 1/10...

a/1-r = (2/3)/(9/10) = (2/3)*(10/9) = 20/27

4. May 14, 2010

### D H

Staff Emeritus
That is the case (i.e., you have used the formula incorrectly). Why did you pick those values?

5. May 14, 2010

### Cyosis

Nevertheless I am sure you know that something that is slightly more than half a centimeter is smaller than 7 cm. So I am at a loss why you think that 0.6666..=7.

You can't set a=2/3, 0.6..=2/3 and therefore your suggested sum would be 2/3+2/30+2/300+...=2/3, which is obviously false.

Try to write 0.66... like I wrote 0.99... in post #4 (last line).

6. May 14, 2010

### srfriggen

I picked the values from the sequence of partial sums...

.6+.06+.006... = 2/3(1/10)^0 + 2/3(1/10)^1 + 2/3(1/10)^2...

When I picked those values for .999... I used .9 = 1/10 = .09 = (1/10)^2 = .009 = (1/10)^3...

I'm seeing that my values must be incorrect. Can you explain how then to prove that 2/3 = .666... by using the geometric convergence series formula?

7. May 14, 2010

### Cyosis

You claim that 0.6=2/3, which is false (note that the 6 is not repeating here). That's why you get the wrong answer. You have made the same error in trying to prove that 0.9..=1.

8. May 14, 2010

### srfriggen

to quote myself, "I'm seeing that my values must be incorrect. Can you explain how then to prove that 2/3 = .666... by using the geometric convergence series formula?"

9. May 14, 2010

### srfriggen

I'm sorry, I mean how 2/3 = 7

10. May 14, 2010

### srfriggen

Sorry, I meant to say, can you show me how to prove 2/3 = .666... = 7 using the geo conv series formula, a/1-r

11. May 14, 2010

### Cyosis

No I am not going to do your work for you. I have told you where you went wrong multiple times. It is up to you to do something with that information. Your idea that 0.66...=0.6+0.06+0.006 is correct, but you then claim that 0.6=2/3 *1/10, which is wrong. If you fix your claim of 0.6=2/3 with the correct fraction you're done.

12. May 14, 2010

### Cyosis

Honestly are you pulling our leg here? How can 0.6, which is something smaller than 1, be equal to 7, which is something 7 times as large as 1?

To answer your question. I cannot, no one can, because it is not true.

13. May 14, 2010

### srfriggen

I don't like asking for direct help and know this forum is certainly not for that reason, so I appreciate your answer and thank you for pointing me in the right direction (that my fraction was incorrect). This isn't a homework question, it is just something I am stuck on.

I'll try some different methods...

14. May 14, 2010

### srfriggen

Listen, I realize my mistake now. My question was flawed from the start. I was trying to actually show that 6.666 = 7.

Can you at least let me know if THAT is true so I can start working on it?

(I know questions can be frustrating sometimes, but please keep in mind that some of us are very novice when it comes to math. I only learned what a series was a week ago).

15. May 14, 2010

### D H

Staff Emeritus
Of course 6.666... is not 7.

16. May 14, 2010

### Cyosis

srfriggen. It is very important that you read the replies to your thread thoroughly. As I have said on numerous occasions, your proof of 0.999=1, in post #1 is also wrong. Yet you use that proof as the basis to prove the other 'equalities' in this thread. As a result all the other things you want to proof will be wrong too.

As for.

I really suggest you grab that ruler. You are now claiming, using a different analogy, that six whole pizzas plus two thirds of a pizza equals 7 whole pizzas. That's not true either.

You don't have to worry about this. However it is not the series that go wrong, but very simple arithmetic. If you fix your arithmetic you will obtain the correct answers. I refer once again to the last line in post #11.

Last edited: May 14, 2010
17. May 14, 2010

### Mentallic

If 0.999...=1 then 6+0.999...=6.999...=6+1=7

Obviously 6.666...= 7 is false.

18. May 14, 2010

### srfriggen

19. May 14, 2010

### Cyosis

You can definitely use geometric series to obtain the proof you desire. However what you're saying is wrong.

Using the geometric series:

$$\sum_{n=0}^\infty \left(\frac{1}{10}\right)^n=\frac{1}{1-\frac{1}{10}}=\frac{10}{9}=1.11..>1$$

20. May 14, 2010

### D H

Staff Emeritus
That sum is 0.111..., not 0.999...