Thread: Is 0.999... equal to one?

Originally Posted by astromark

I would have thought you might jump on that number.. and the completely wrong expression of it..

888... is a infinitival stream of 8's. adding a 1 or 2 to the end of that ... does not make any sense.

and that would be because the ... gives indication of a infinite term.

Exactly right.

Hooray, I finally found a use for this summation series.

1/1 = 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ...

1/1 = 0.999999999999999999999999 ...

Originally Posted by Jeffrey DreamKing

And I'd like to point out that ( 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ...) will not equal 1, but will always be some fraction short of one, and just go on forever.

And much credit goes to grapes, who helped me find that

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

Originally Posted by grapes

Algebraicly, it's easy to show that no matter what a and r are (except for r=1)



But, in particular, if r is less than 1, then (and ) becomes zero when n goes to infinity, so the formula becomes pretty simple:




For that one, on the right hand side, a = 1/50 and r = 1/50, so it is equal to

(1/50)/(1 - 1/50) = (1/50)/(49/50) = 1/49, as you've found.

Using the formula in the other direction, we can start with 1/7 and see that it equals (1/8)/(1 - 1/8), so

1/7 = 1/8 + (1/8)^2 + (1/8)^3 + (1/8)^4 + ...

which, BTW, repeats after 6 places.

from

I noticed a pattern in 7^n, expressed as a summation series.

Originally Posted by astromark

So I pinch this bit...

So is that supposed to be a reference to passing a joint ? Funny, I almost didn't get that. I just kept thinking of things that you pinch, things that you pinch... women, money, I don't know...

Originally Posted by astromark

Mark @ 39' 55" 52.92 South.

And I suppose this is where I can find a good crop ?

I may have just proved something, but I still don't believe it.

Is 1.0000... equal to 1 ?

yes, because it is zero's forever.

Does 0.999... equal 1 ?

No, because it is nine's forever.

Originally Posted by Jeffrey DreamKing

No, because it is nine's forever.

But why does that mean it doesn't equal 1?

(There are so many ways to show, with valid math and number theory, that it does equal 1. That the 9 are "forever" is a big part of it.)

Originally Posted by Jeffrey DreamKing

And much credit goes to grapes, who helped me find that

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

To answer your other question, try this. Take that formula, and let n be 9.

Then, next step, multiply everything on both sides by 9.

Originally Posted by Bananas

But why does that mean it doesn't equal 1?

There is only one answer to, ' What repeating decimal is equal to one ? '

1.000... is equal to 1 .

It's like a game of musical chairs with only one chair. 1.000... sat down and there is only one winner. One absolute winner. The seat is taken.

The problem I see with such statements as "0.333... is equal to 1/3" , is that 0.333... is only an approximation.

One is indivisible by 3.

Three is just not a common divisor of 10 like 2 and 5 are.

It's just IN-DIVISIBLE. It can only be expressed as a/b , algebraically.

.999... + .111... = 1.111...

.999... + .111... - .111... = 1.111... - .111...

.999... + 0.000... = 1.000...

Originally Posted by Jeffrey DreamKing

There is only one answer to, ' What repeating decimal is equal to one ? '

1.000... is equal to 1 .

It's like a game of musical chairs with only one chair. 1.000... sat down and there is only one winner. One absolute winner. The seat is taken.

The problem I see with such statements as "0.333... is equal to 1/3" , is that 0.333... is only an approximation.

One is indivisible by 3.

Three is just not a common divisor of 10 like 2 and 5 are.

It's just IN-DIVISIBLE. It can only be expressed as a/b , algebraically.

That's all largely just a truism. The idea that 0.333... is only an "approximation" of 3 comes down to an opinion. Which is fine, as long as anyone reading this can see that that's an opinion at odds with common mathematical understanding.

In modern, current, accepted math: 1/3 = 0.333... (Not nearly, not approximately, but equals: 0.333... is the exact base ten decimal representation of 1/3).

(And in any case, the "1/3 = 0.333... so 3 x 1/3 = 3 x 0.333... thus 1 = 0.999..." proof is just one of several.)

The common thread in all the objections to 1 = 0.999... (and 1/3 = 0.333...) seems to come down to discomfort with the infinitely repeating decimal*, which in turn seems to stem, from thinking about it as a process, which if looked at at any time "must" mean the number is "unfinished". Once a person can fully comprehend that the "..." in a repeating decimal includes all of the decimals (all infinite of them), the issue can go away.

(* the mathematical theory of limits (also a current, accepted mathematical "thing") has that covered.)

Originally Posted by Jeffrey DreamKing

.999... + .111... = 1.111...

.999... + .111... - .111... = 1.111... - .111...

.999... + 0.000... = 1.000...

Yes. (So you agree that 0.999... = 1 ?)

However I'd show that (essentially ther same thing) as:

0.999... + 0.111... = 1.111...
and
1.000... + 0.111... = 1.111...
so
0.999... + 0.111... = 1.000... + 0.111...
Therefore
0.999... = 1.0