0.9999.....to infinity is equal to 1

Discussion in 'General Chat' started by 996 911 Turbo, Jul 22, 2006.

1. holy shit, so because they proved it using simple arthimetic, it's not enough?

you must be so bad at math. what you've just said is that i cant say half of four is 2, because 4/2 = 2. you literally said the exact same thing. the proof is in the math, nothing else, you chud.

2. Start here:

x = .9999999 repeating

Multiply by ten:

10x = 9.99999999 repeating

Subtract x from the left side, and subtract .9999999 repeating from the right side, since they are equal:

9x = 9

Solving:

x = 1

So we know that since x is equal to both 1 and .9999999 repeating, they are themselves equal.

3. That proof has nothing to do with eqauting fractions to decimals whatsoever. You should be good to go.

4. I didn't look at the mathematical explanations myself, I figured they would all be there. I agree in that the 1/3 + 1/3 + 1/3 = 1 thing isn't really a sufficient answer, all it does is sort of loophole around the question with things people already take advantage, i.e. 1/3 = .3333... It just raises more questions about why 1/3 = .333... and believe me it does.

The real reason is something called an infinite series. Unless you took calculus, then you probably don't know what it is. Series' can do 2 things. They can converge to a number, or diverge and go off to infinity(please correct me if I'm wrong, its been a while since calc II). I don't remember the specific series' for 1/3 and 1, but I'll try to look them up. The reason why 1/3 = .3333... and .999999... = 1 is because the number that these infinite series ocnverge to, is 1/3 and 1 respectively. The series for .3333... essentailly adds .3 + .03 + .003 to "n" number of decimal places. Then the limit of "n" is taken to infinity, and when this happens, the series "converges" to 1/3.

Thats a lot more complex, and its pretty much described only in laymens terms(considering thats most of what I remember of it). Thats why the 1/3 + 1/3 + 1/3 = 1 is a much more common and easy to understand explanation. Its just simpler.

5. Thank you, that was a good explanation.

6. Also a good explanation, and i know you posted it before. thanks

7. You are, to my knowledge completely correct.

Another way to look at this exact same idea without a knowlege of calculus is to simply do the long division by hand. 1/3 will obviously leave a 0 in the "ones" position, so we move on to the tenths position. We drop down another 0, as we have 10 thenths, and we see that three goes into this number three times. Three times three is nine, leaving us with a remainder of 1. We move on to the hundredths and that one tenth we have left is obviously equal 10 100ths. Three goes into this three times, leaving us with a remainder of one. And so on forever. Simply doing this it becomes clear that the patter repeats itself exactly forever. It is the same as saying the sum of 3/(10^n) from n=1 to n=infinity is one third and converges on 0.333 repeating, but using tools learned in grade school.

8. I would say definitely NO, though it's a very interesting concept.

Yes, the farther out you go (meaning 999/1000, 9999/10000, 99999/100000) the closer you get to 1 and the number is infinite. Edit: meaning the number of digits is infinite.

But the same could be said for .333 to infinity. The further out you carry it, the closer you are to the value of 1.

While in both cases you (could) argue that it infinitely becomes closer to 1, thus, being valued at 1. But what you have to consider is that it also infinitely reduces the amount that it increases by a factor of 1/10.

Example

.9 = 9/10

.99 = 9/10 + 9/100

.999 = 9/10 + 9/100 + 9/1000

.9999 = 9/10 + 9/100 + 9/1000 + 9/10000 ... and so on

Note that, while it for eternity increases in value, the value of the increase itself becomes smaller each time (for eternity). Thus, negating the concept that .999... = 1

9. Obviously not.

10. Read Neoptolemus' post on convergence, it answers exactly what you're saying. If it both increased forever in value and the increment of addition increased, it would not be convergent. But this decreases by a factor of 10^n, which is a decrease great enough to be convergent. By the ratio test, using 9/10^n as our a(n), then we have (9/100)/(9/10)

11. "But this decreases by a factor of 10^n, which is a decrease great enough to be convergent"

I'm not familiar with "convergent" as a mathematical term, but I'm assuming it means it will converge, come together, simply put, it will = 1?

If I'm interpreting what you're saying right then I have to disagree. So long as the increase steadily reduces by a factor of 1/10, the value will never reach 1.

12. Did you read my post? It explains convergence pretty well.

13. Your assesment of what I mean is exactly correct and your intiution is half correct, but has lead you ultimately to the wrong answer. So long as the series is finite, as you point out, it will never reach 1. BUT it is not finite. The series is infinite and therefore it EXACTLY equals one. I posted it before, but here it is again with an explaination:

Lim [1/10^n] = 0
n->infinity

This means that as n approaches infinity, the equation in the brackets approaches 0. This equation is the leftover you're thinking of. The little bit that is leftover when you subtract 0.999 from 1. As 0.9 part gets larger, the leftover bit gets smaller. So long as there is an end to the 9's, then there is a real little bit of a leftover, no matter how large the series of nines is. But it repeates INFINITELY. It does not end. Therefore, we are at the limit of infinity for the leftover bit, which is one.

I hope this made some sort of sense.

14. I havent read the last 6 pages of this thread, but I have a question:

When, in everyday life, is the number 0.999999 repeated, infinity, etc, is ever used?

15. there were several links posted, including one from wikipedia with a ton of proofs.

16. you're wrong, links are a couple pages back.

17. So, you just make these threads so that you can come in and tell people they are wrong, when they inevitably post the wrong thing? I think its fun, too, but its one thing to tell someone they are wrong, and another entirely to come in after the issue has been addressed and just say "You're wrong!!!!!" before he gets another chance to say anything. Seriously, you don't really add much to the conversation of your own knowledge, I think you just get satisfaction from telling someone they are wrong.

18. what's the point typing it again when there are like 10 proofs on the wiki page.

and yes, I vastly enjoy people making fools of themselves.

EDIT: and I hadn't seen that it had been addressed because i didn't f5 before quoting him.

19. Whats the point in typing anything at all when its already taken care of? You just wanted to get a word in.

20. "and I hadn't seen that it had been addressed because i didn't f5 before quoting him."

21. One can make very logical arguments either way on this subject. So you learned something in class, that .999... to infinity = 1. That's not exactly an intuitive conclusion. If you wanted to bring up the subject for an interesting discussion, that's cool. But now it seems you only wanted to say, "you're wrong! I made a fool of you!" That's pretty pathetic.

22. you can make a argument that can seem logical for the =/= side, but it's incorrect.

23. It APPROACHES 1.

24. it EQUALS 1. a number can't approach anything.

25. 0.9999.... =/= 1

It's #\$%#ing obvious.