## 0.999... = 1

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While looking into using html for academic docs, I typed this one up (based on Thomas Park's PubCSS formatting of ACM SIG Proceedings articles).

As a matter of representing numbers, wouldn't most be fine with 9/9 = 9 × (1/9) = 9 × (0.111...) ?

Anyway, please feel free to point out any errors in the attached note.

(For some reason, internal links in the pdf don't seem to work, not sure why; they're find in the original html.)

EDIT: Incidentally noticed that one of the reference links had the wrong url. Fixed.
• 220
Are you asking if others agree with this mathematical tautology?
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, typically when "0.999... = 1" is brought up, you'll find a flurry of objections, you just watch. :)

Either way, I think the attached proof is valid, but others are invited to point out errors.

Another argument, more or less following similar thinking, is whether a number could be found between 0.999... and 1.000... (like the mean).
If no such number can be found, then we might reasonably say they're one and the same.
• 6.1k

Looks fine to me!

I think there's a common source of confusion related to the one you pointed out, 1 isn't in the infinite series, but the infinite series converges to its supremum, which is 1. People confuse maximum and supremum a lot.
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Another argument, more or less following similar thinking, is whether a number could be found between 0.999... and 1.000... (like the mean).
If no such number can be found, then we might reasonably say they're one and the same.

This argument isn’t actually valid, because it could arguably be the case (if not for other, valid proofs that 0.999... = 1) that 0.999... is the very last number before 1, so there is nothing between them even though they’re not (this hypothetical person would argue) the exact same.
• 220

I think the main issue is whether someone is familiar with the notation conventions within mathematics.
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the very last number before 1

Both the rationals and the reals are densely ordered.
For any two different numbers, there's a third between them.

I suppose they might say that 0.999... isn't a number.
• 220

Indeed, if that argument was taken to be valid in mathematics, I could also argue that since there is no natural number between 9 and 10 to be found, that means that 9=10.

Besides a notation like 1.000 isn't a mathematical notation if it doesn't differ from 1. If there is no difference the preferred and obliged notation is 1 not 1.000.

1.000 is a valid notation in applied sciences like physics and chemistry as in applied sciences there is something like numbers of significance. 1.000 in physics or chemistry refers to any number in the collection [0.9995,1,0005>, while in mathematics 1.000 just means 1.
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, yeah, reading through the note takes familiarity with the mathematics and notation.
And it's a fairly large area; might scare some away.
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This argument isn’t actually valid, because it could arguably be the case (if not for other, valid proofs that 0.999... = 1) that 0.999... is the very last number before 1, so there is nothing between them and even though they’re not (this hypothetical person would argue) the exact same.

It would be valid when restricted to the real numbers as originally intended, I think. There does not exist an x between the limit of that sequence and 1; it would simultaneously have to be greater than every sequence element of 0.9 0.99, 0.999... and less than 1, but since that sequence is monotonic, the "least" such strictly positive real number strictly between the sequence limit and 1 doesn't exist as you can always find one less than it. It's another way of reading the epsilon N proof.
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since there is no natural number between 9 and 10 to be found, that means that 9=10

The naturals aren't densely ordered like the rationals and the reals. ;)
• 667
It comes down to how the language is used .9999... equals 1. This is not true. In calculus one is only allowed to say that a series converges to a limit. In this case the series is .999... and the limit is 1. That is, incorrectly speaking, .999...becomes one after an infinity of 9s. But you can't have 'an infinity of' 9s because infinity is not a number. So no, .999... is not 1. It only converges to 1. That is all the link you posted is showing. They are replacing 'tends towards the limit 1' with 'equals'.
You can say The limit = 1 but not .999...= 1
• 4.6k
The total sum of an infinite series is defined as the limit of the series of partial sums. The sum of the infinite series
{0.9, 0.09, 0.009, ... }
is thus the limit of the series
{0.9, 0.9 + 0.09, 0.9 + 0.09 + 0.009, ...}
or in other words the limit of the series
{0.9, 0.99, 0.999, ... }
which you can clearly see converges to 1.

“0.999...” is of course also equal to the sum of the infinite series
{0.9, 0.09, 0.009, ... }
because that’s just what decimal notation means. And since the limit of the series of partial sums of that infinite series is 1, that means the total sum of that infinite series represented by 0.999... is also 1, so 0.999... = 1.
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depends on how accurate the system needs to be. NASA does have room for error but it has extremely less room for error than chevrolet does. 0.999 might be good for some systems but not good enough for other systems.

Everything can be quantified including the personality of a person or people. I'm not sure everything should be quantified though. There is an ancient book that talks about that in regards to whether a nation should have a census. I'm not actually against modern censuses.
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0.999

0.999... is not equal to 0.999

0.999... minus 0.999 equals 0.001
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true but how does that negate what i said. Are you familiar with scientific notation. Something to consider is Pie is a never ending set of integers after the decimal. For NASA the number of those integers would need to be much greater than for Chevrolet engineers. Perhaps i wasn't clear enough the first time.
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I did read what you said, my point is that this isn’t an issue of accuracy at all. Nobody is worrying about 0.999 here at all. They’re only talking about 0.999.... Which is an exactly accurate number, if the number you want is 1, because they’re the same number.
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.999 and 1 are not the same thing. Is that what the OP was saying? That would be incorrect. In some systems you could claim it is close enough while in other systems they would not be close enough. Everything including personalities can have systems analysis and design principles applied to them.
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typically when "0.999... = 1" is brought up, you'll find a flurry of objections, you just watch. :)

Either way, I think the attached proof is valid, but others are invited to point out errors.

Another argument, more or less following similar thinking, is whether a number could be found between 0.999... and 1.000... (like the mean).
If no such number can be found, then we might reasonably say they're one and the same.

___________________________

Perhaps if the OP was written with more clarity you could prove otherwise but 0.999 is different from 0.999999 and also different from 1.0 or 1. Would you like to rewrite the OP so that my above statement is invalid?
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9/9 = 9 × (1/9) = 9 × (0.111...)

1/9 doesn't equal exactly 0.111 nor 0.111111. But for some systems it would be close enough.
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I'm confused what sort of system would this sort of concept be applicable too. I would agree that for some systems 0.111 would be close enough and for NASA (they have less room for error), 0.1111111111111111111111111111111111111111111 might be required.
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Census doesn't have much to do with it, .

A NASA engineer may invoke a handful of mathematical theorems and formulae out of physics, involving π, differentiation and integration (which are calculus related to limits), only to find that x + 7m was what they were looking for. For example, there are all kinds of rules of differentiation, many of which are, or can be, proven by limits. Also, no manner of discussion and voting can somehow make √2 mysteriously become 1.414.

Years ago I vaguely recall having done error analysis in physics experiments. Tedious. Maybe that's more along the lines of what you're thinking of?

What's this ancient (science/mathematics) book you're referring to anyways?

for some systems 0.111 would be close enough and for NASA (they have less room for error), 0.1111111111111111111111111111111111111111111 might be required

A sensible person would just use 1/9. Loss-less. Unless or until they needed to write it out differently anyway.
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But you can't have 'an infinity of' 9s because infinity is not a number. So no, .999... is not 1. It only converges to 1.

Looks like the confuzzlement mentioned earlier.
There are infinitudes of numbers. Therefore there aren't numbers? Hmm...
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lets not worry about that ancient book because i was replying to someone else.

Yes 1/9 is fine. Like i said it depends on the system as to how many integers after the decimal will be used for a variable or component of an equation.

Yes limits as for example in calculus make things alot more simple if a person took high school calculus. I'm familiar with the greek method of exhaustion as well.

I don't have a problem with what you are saying however am i correct that you could summarize what your OP is stating with: "some systems require more precision than others in terms of how many integers are used after the decimal"? I wouldn't be surprised if you have a less clear but at the same time more professional way of saying what i just said. Most of the people on here including me are amateur arm chair quarterback mathematicians.
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The point I’m making is that that “...” is an important difference.

0.111 does not equal 1/9. It’s close, and you’re saying it might be close enough for some purposes, but for others you might need more 1s. But so long as you have finitely many 1s, it won’t equal exactly 1/9.

But 0.111... (with that “...”, that’s very important) equals EXACTLY 1/9, by definition. It has infinitely many 1s. That’s what the “...” means: “keep repeating the preceding pattern forever.”

0.111 x 9 = 0.999, which is not 1.

0.111... x 9 = 1/9 x 9 = 0.999... = 1, exactly.
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• 1.4k
The point I’m making is that that “...” is an important difference.

0.111 does not equal 1/9. It’s close, and you’re saying it might be close enough for some purposes, but for others you might need more 1s. But so long as you have finitely many 1s, it won’t equal exactly 1/9.

But 0.111... (with that “...”, that’s very important) equals EXACTLY 1/9, by definition. It has infinitely many 1s. That’s what the “...” means: “keep repeating the preceding pattern forever.”

0.111 x 9 = 0.999, which is not 1.

0.111... x 9 = 1/9 x 9 = 0.999... = 1, exactly.

i agree. I'm used to the line that goes over the 0.9999.... in which case you wouldn't need the .... . ______________________________________ i don't type alot of math equations so i didn't equate .... with that line that goes over the integers that come after the decimal. You win. No sarcasm intended.
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lol. this is the bullshit we do on here. this is so we drink less alcohol.
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Ppppphhhhorrrrrest explained to me what you were talking about.
• 880
So does this mean ...999.999... = 0?
• 220
The naturals aren't densely ordered like the rationals and the reals.

Depends on how one looks at it, the collection of any of those three are all infinite. I don't see a difference in the applied logic, no matter how densely ordered the collection of number is, the logic applied remains the same.
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