# .999_=1, so does 1=2?

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 .999_ is less than 1 by an infinitely small difference, but a difference nonetheless.

No, it isn't smaller than 1. Several proofs have been posted already.

The misconception usually seems to stem from the idea that the decimal system has only one representation for a number. There's no question that 1.0 is the *preferred* representation, but 0.9 recurring is the same number with a different representation.

Another way to look at it: there is no "infinitely small" real number that you could add to 0.9 recurring to get one. If you have any small number greater than zero, then there is a smaller number between that small number and zero as well. The real numbers are continuous.

This is also known as the intermediate value theorem.

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Probably one of the better threads imo, people are thinking more in this thread than any other one currently.

That is a parlor trick equation.

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 Could the difference between 1 and .9_ be written as .1x10^-?

Probably yes, but why bother if 0 is much shorter.

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 No, it isn't smaller than 1. Several proofs have been posted already. See also 0.9 Recurring, in Wikipedia. The misconception usually seems to stem from the idea that the decimal system has only one representation for a number. There's no question that 1.0 is the *preferred* representation, but 0.9 recurring is the same number with a different representation. Another way to look at it: there is no "infinitely small" real number that you could add to 0.9 recurring to get one. If you have any small number greater than zero, then there is a smaller number between that small number and zero as well. The real numbers are continuous. This is also known as the intermediate value theorem.

Well goddamn. ##### Share on other sites

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 Well goddamn. Once you get into infinite-dimensional integrals, things get a bit more hairy though.

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 I understand the logic, but by my calculations 3/2 is 2, not 1. Because you would round up removing the decimal places. I believe your teachers logic is flawed as well as his math.

Integer Division. There is no rounding.

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_

.9 does equal 1. He explained it this way.

1/9 = .1 repeating...

2/9 = .2 repeating

3/9 = .3 repeating

4/9 = .4 repeating

5/9 = .5 repeating

etc etc....

8/9 = .8 repeating

9/9 = 1.

Whether that is the actual proof ##### Share on other sites

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If there are an infinite number of points possible between two points...how can someone walk over an infinite number of points? (one of Xeno's paradoxes)

hint: "infinite number?"

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why are there people saying that .999... APPROACHES 1... it's not like .9(repeating) is a function... it's a number...

besides, if 1 == .999... , then there would be some kind of difference, or a number you could add to .999... to bring it to one. no one can describe any actual number to fit here, so they are equivalent.

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 If there are an infinite number of points possible between two points...how can someone walk over an infinite number of points? (one of Xeno's paradoxes) hint: "infinite number?"

because all points are an element of the interval you have defined.

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 If there are an infinite number of points possible between two points...how can someone walk over an infinite number of points? (one of Xeno's paradoxes) hint: "infinite number?"

There's a fundamental wrong assumption in that paradox, assuming that a mathematical infinity maps onto reality.

Reality seems to be discrete, i.e. if you break it down far enough you end up with the "smallest possible size" - see Planck Length.

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 Quoted for fail.

how can u even try to quote others for fail when your OP was so terribly terribly wrong?

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 There's a fundamental wrong assumption in that paradox, assuming that a mathematical infinity maps onto reality. Reality seems to be discrete, i.e. if you break it down far enough you end up with the "smallest possible size" - see Planck Length.

I would argue that the fundamentally wrong assumption is that an infinite sum of lengths must result in infinite length. If this were true, it would be impossible to have asymptotic behavior, which in reality is readily apparent in simple mathematical functions. The geometric series describes Zeno's motion paradox. In case anyone wants to argue that the sum of the series only converges to 1 and never reaches it, therefore Zeno was right, should note that even if you don't believe the series reaches 1, it does get bigger than zero - which is where Zeno thought it was stuck. It would seem obvious that if you can travel 99.999999% of the distance you can travel that last bit too ##### Share on other sites

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The problem here is that many people don't understand what infinity means, mathematically. Which isn't surprising; it's a tough concept, especially seeing as infinity (arguably) doesn't really exist in the world that we experience. But mathematically, it is very real and very well understood.

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 The problem here is that many people don't understand what infinity means, mathematically. Which isn't surprising; it's a tough concept, especially seeing as infinity (arguably) doesn't really exist in the world that we experience. But mathematically, it is very real and very well understood.

Aleph null, or aleph one infinity?

;-)

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 Aleph null, or aleph one infinity? ;-)

Unfortunately, I never really went that far in set theory... I was a physics major so most of my math was based on calculus and a little probability. But I had a good dose of (introductory) number theory along the way.

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 Unfortunately, I never really went that far in set theory... I was a physics major so most of my math was based on calculus and a little probability. But I had a good dose of (introductory) number theory along the way.

Aleph null infinity is "countable" - like the set of integers. They are infinite, but there is only a finite number of integers in any interval.

Aleph one infinity is not countable - like the set of real numbers. Any interval of the real numbers has an infinite number of real numbers.

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Aha. Yeah, I know countable and uncountable... didn't know that terminology though.

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