.999~ = or != 1

[AnimeJet]

So does .9 infinitely equal one or not? Discuss. Unless this has already been discussed.. then tell me what the general concensus is on these boards ._.

EDIT: Now 100% more poll!

[HemlockGrey]

...how can it possibly equal one?

[SyntaxVorlon]

.9~/3 = .3~
.3~= 1/3
1/3*3=1
.9~=1

[Mad]

.111~ = 1/9
.222~ = 2/9
.333~ = 3/9
...
.888~ = 8/9

All of the above are exact and can easily be proven by doing the division 1/9, 2/9, etc.

Therefore: .999~ = 9/9 = 1

[Crazedwraith]

Yes o.9 reaccuring does indeed equal one.
.9~ = x
.99999999999=x
9.9999999999= 10x
9x = 9
x=1

that make it clear?

[aerius]

So does .9 infinitely equal one or not? Discuss. Unless this has already been discussed.. then tell me what the general concensus is on these boards ._.

It equals 7, but I haven't taken any math courses in a couple years so I could be wrong.

[Darth Wong]

It equals 1 when it is taken to a reasonable number of significant figures.

[Kuroneko]

So does .9 infinitely equal one or not? Discuss. Unless this has already been discussed.. then tell me what the general concensus is on these boards ._.

Of course it does. The decimal representation of a number is simply a series in a particular form: 0.999... = 9/10 + 9/100 + 9/1000 + ... in the exact same way that 295.1 = 2*100 + 9*10 + 5 + 1/10 + 0/100 + ... . This particular series has a sequence partial sums of (.9, .99, .999, ...) that is monotonically increasing and has an upper bound, so it must converge to the lowest upper bound of (.9, .99, .999, ...), which is 1. Therefore, 0.999... = 1.

[HemlockGrey]

Could someone explain this slightly better? I mean, won't it always be just short of 1?

[Kuroneko]

Could someone explain this slightly better? I mean, won't it always be just short of 1?

No. It's a number--it doesn't change. It isn't any of (0.9, 0.99, 0.999, ...); it is the lowest real number greater than or equal to every number found in that sequence. That number is 1.

[CrimsonRaine]

Could someone explain this slightly better? I mean, won't it always be just short of 1?

No. It's a number--it doesn't change. It isn't any of (0.9, 0.99, 0.999, ...); it is the lowest real number greater than or equal to every number found in that sequence. That number is 1.

Oh, Christ, I'm still lost. Even though I can see it can be rounded to 1, since it is so close to that number, it still doesn't equal one. Does it?

I'm an English major. I know words, not numbers. Heeeelp!

Crimson Raine

[Mad]

Could someone explain this slightly better? I mean, won't it always be just short of 1?

For any easy explanation, SyntaxVorlon's is probably the easiest to understand:

1/3 * 3 = 3/3 = 1
.333~ * 3 = .999~

1/3 = .333~, therefore 1 = .999~

As for details, Kureneko's explanation is correct. It follows the same concept that is used very often in calculus: if you have an infinite number of terms to add, with each term smaller than the previous term, then they will all add up to a specific number.

In this case, we're adding up .9 + .09 + .009 + .0009 + ... . If this stretches out to infinity, they will add up to be 1. (The same way the area under a curve can be determined by slicing the problem up into an infinite number of columns.)

You could also look at it by taking the sum of an infinite series, as that'd work, as well.

I guess the easiest way would require a picture of the infinite series being added together, along with the equation and limits and everything.

[Captain Cyran]

Could someone explain this slightly better? I mean, won't it always be just short of 1?

No. It's a number--it doesn't change. It isn't any of (0.9, 0.99, 0.999, ...); it is the lowest real number greater than or equal to every number found in that sequence. That number is 1.

Oh, Christ, I'm still lost. Even though I can see it can be rounded to 1, since it is so close to that number, it still doesn't equal one. Does it?

I'm an English major. I know words, not numbers. Heeeelp!

Crimson Raine

Just do what I do. Nod your head, smile, and file it away in that useless information part of your brain for when you want to impress people.

[Darth Wong]

Technically, we would say that it is a term which approaches 1 as its terms approach infinity.

[Kuroneko]

Oh, Christ, I'm still lost. Even though I can see it can be rounded to 1, since it is so close to that number, it still doesn't equal one. Does it?

Why wouldn't it? Let's call it X = 0.999... .

Pick any number Y between 0 and 1. There is always some number in (0.9, 0.99, 0.999, ...) that's greater than it. For example, suppose I pick Y = 0.931. Then, 0.99 > Y. If I pick Y = 0.9934, then 0.999 > Y. Et cetera.

If X < 1, then there must be a number between them (for example, Y = (1+X)/2). But then by the above demonstration, there is some terminating 0.999...99 > Y. But that cannot be, as X should be greater than any of (0.9, 0.99, 0.999, ...)!


The above doesn't actually strictly prove the result mathematically, but it should work as demonstration.

[Captain Cyran]

Oh, Christ, I'm still lost. Even though I can see it can be rounded to 1, since it is so close to that number, it still doesn't equal one. Does it?

Why wouldn't it? Let's call it X = 0.999... .

Pick any number Y between 0 and 1. There is always some number in (0.9, 0.99, 0.999, ...) that's greater than it. For example, suppose I pick Y = 0.931. Then, 0.99 > Y. If I pick Y = 0.9934, then 0.999 > Y. Et cetera.

If X < 1, then there must be a number between them (for example, Y = (1+X)/2). But then by the above demonstration, there is some terminating 0.999...99 > Y. But that cannot be, as X should be greater than any of (0.9, 0.99, 0.999, ...)!


The above doesn't actually strictly prove the result mathematically, but it should work as demonstration.

*blink, blink*

Okay I think I kind of understood that. I'm so glad I'm gonna go for an English Major...

[Jeremy]

That's about all I have to say on this matter.

[AnimeJet]

Well it seems that there is a general agreement, that .999~ does in fact, equal.. one! Odd, most places dissagree and rip at each other until the topic reaches a limit.. this place scares me.

[Kuroneko]

... if you have an infinite number of terms to add, with each term smaller than the previous term, then they will all add up to a specific number.

Which isn't strictly true--it is so if and only if the sequence of partial sums is Cauchy. As a well-known counterexample, the Harmonic series 1/1+1/2+1/3+1/4+... adds smaller and smaller terms, and yet fails to converge to any real number.

[Master of Ossus]

Since the difference between this and one is 1/infinity, it does in fact equal one.

[Andrew J.]

This makes so much more sense now that I've studies sequences in math...

[Rye]

Guhi hate this...personally i've always thought of it in terms of thirds, the whole 0.3333....equalling a third, times by 3 = 1.

Damn decimals...

[kojikun]

Absolutely, no. Its infinitesimally close to but not equal to one. Effectively, and for any reasonable purposes, yes.

[Mad]

Absolutely, no. Its infinitesimally close to but not equal to one. Effectively, and for any reasonable purposes, yes.

You'd be right except for the fact that the fundamental concepts behind calculus beg to differ, as explained earlier in this thread...

[kojikun]

You'd be right except for the fact that the fundamental concepts behind calculus beg to differ, as explained earlier in this thread...

shit, right, yeah, sorry. Forgetting calculus = naughty me.