.999.... (Repeating) is equal to one?

Kexruct and I have been arguing for hours about this. (Read title.) Does any one have evidence they would like to present on this topic?

It's not equal, but it'd damn close enough that in most cases, the math would be easier done if you just made it one.

... .999 repeating is a useless number. Why wouldn't you just use 1...?

for all intents and purposes, yes. The differences are so minute its not worth taking the time to calculate.

I think it's one of those weird math things. They're not actually equal, but the numbers work that way. Wikipedia uses this as an example:

1/9= 0.111...

9 times 1/9 = 9 times 0.111...

9 times 1/9 = 1.

Therefore, 1 = 0.999...

A common proof used is that 1/3 + 1/3 + 1/3 is one, and 1/3 can be represented as .333 repeating. Add them together, and you get .999... Therefore, .999.... most also equal one.

@DDR, midget, Wulf: No, the difference is actually zero. .999... == 1. Exactly.

The 1/3 = .333... thing is a good proof of this. Another is:

• 1 - .99 = .01
• 1 - .999 = .001
  • And so on
• Therefore, 1 - .999... = .000... (that is, an infinite string of zeros)
• An infinite string of zeros is equal to zero.
• Therefore since the difference between .999... and 1 is 0, .999... = 1

I vaguely recall figuring out a proof involving ∑ and 0.9 + 0.09 + 0.009 + etc, but I can't remember the exact details now.

edited 9th Sep '11 3:58:22 PM by Blueeyedrat

It is exactly equal to 1. Take any calculus class and it should be obvious.

@Black Humor- Doesn't make sense. They're functionally the same because there's no end to infinity, but if that end did come, there'd be a last nine and a one.

It's one of those things where Mathematicians do weird things and expect everybody else to accept it.

I figured it was due to the 1/3rd and .333-repeating thing, and multiplying them both by 3.

So yes, basically equal, although '1' is a lot easier to write than '.9999....'

• 0.999...
• multiply by ten: 10*0.999... = 9.999...
• 9.999... minus 0.999... is 9
• thus, 9 = (10 - 1)*0.999... = 9*0.999...
• dividing that by 9, we get 1 = 0.999...

Oh great it's this thing again.

Sure the difference at a large enough chain of digits is insignificant, but .999 repeating is not equal to 1. (I mean seriously, Blizzard Entertainment did this exact thing as an April Fools Joke about 7 years ago.)

@OP: It's a fairly common debate that crops up everywhere. Don't expect any kind of general agreement on which side is right.

On the bright side, at least your friend isn't arguing that division by zero should be possible. I don't think we need the message of "everymathaxiomever.exe has stopped working" popping up on this wiki.

edited 9th Sep '11 4:20:48 PM by TotemicHero

No, mathematicians are pretty unified on this one.

If anybody is not convinced, there are far too many proofs right on Wikipedia. Note the deeper implications about real numbers.

edited 9th Sep '11 4:58:23 PM by Tzetze

It's exactly one, not nearly, not almost but exactly. The "thirds" thing is the easiest to understand proof, I think, and issues regarding it are to do with limitations in numerical representation.

It's easy to understand that 3 x 1/3 = 1. That is, three thirds are one whole. But representing that decimally is the issue, because numbers do not allow that. Thinking about it, they're actually pretty limited in that regard. Thus, the best we have is 0.3333 repeating.

When 0.333 is taken as representative of a third, three times that number has to equal three thirds, or one whole. So, 0.999 is three thirds, or, one.

That said, 0.999 does not necessarily have to equal 1, because it can also be the closest number to 1 in its own right. That is, 0.333 need not be a decimal representation of 1/3, it exists as a number on its own anyway. As does 0.666, as does 0.999 or any other fraction.

Conclusion: If 0.333 is simply a decimalisation of the fraction "one third" then 0.999 as a decimalisation of "three thirds" does indeed equal exactly one. But when 0.999 is simply a number in itself, then it means no more than that. Everything depends on the context.

Oh this again? Look, it's equal. Learn a bit of algebra. It's just a quirk of how our number system isn't a great way to represent stuff because there's more than one way to represent the same value.

Saw this gem of a joke on Wikipedia.

Q: How many mathematicians does it take to screw in a lightbulb?

A: 0.999999....

Yep. Cross-multiplication is all you need to think of this. (this would be easier if I could use multiple lines for the equations effectively)

3/3 = 1/1. Cross multiply, and you get 1 = 3 x 1/3. Divide both sides of the equation by three and decimalise the result on the right hand side and you get 1/3 = 0.333 repeating, and from that you simply multiply both sides of the equation by three to give you 3/3 = 0.999 repeating.

Thus, 1 = 0.999 repeating.

Note that it is fallacious to assume there can ever be a "last 9". When we're talking about infinity, the notion of a final number simply goes against logic - by definition, infinity does not have an end.

edited 9th Sep '11 6:01:49 PM by CaissasDeathAngel

The difference between 1 and .999... is infinitely small, as in the opposite of infinitely large. And the opposite of infinitely large is zero.

No, mathematicians are pretty unified on this one.

QFT. To a mathematician, this is like arguing whether 1 + 1 = 2* In the integers

OH this again...

Every fraction has a decimal equivalent, and by equivalent, they're equal. In decimal system, certain numbers have more than one decimal forms.

1/1 = 1.0000... = 0.999...

1/2 = 0.5000... = 0.499...

1/3 = 0.333...

edited 9th Sep '11 7:07:19 PM by abstractematics

No amount of math trumps logic.

As soon as you put that decimal that 0.999..., it's not 1.

Man, I got enough of this debate during my time on gamefaqs' CE board.