# Does .999999999 (repeating) = 1?

I know they are different numbers, but consider this 1/3= .33(rep). 3 times 1/3 = 1 so .333(r) times 3 = .9999(r)

So 1=.99(r)?

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• Anonymous

Yes, they really do equal the same thing. This question has already been answered here literally thousands of times.

0.999...., with the nines REPEATING INFINITELY, is indeed EQUAL to 1. It's not "about" 1, or "really, really close" to 1. It's EQUAL to 1. This is mathematical fact, not a matter of opinion. The people who say otherwise are simply wrong. They usually make the stupid assumption that infinity is an actual integer and that there's a "0.01" you can add to the end. But there is no "end" here.

There are several ways of proving this. I think the simplest way is this:

1/3 + 1/3 + 1/3 = 1

0.333... + 0.333... + 0.333... = 1

0.999... = 1

Again the ellipsis (...) here implies that the decimals are infinitely repeating.

If 0.999... were NOT equal to 1, then this would beg the question of what positive number you need to add to 0.999... to GET one. There isn't one. You can't put an infinite amount of zeroes between "0.00..." and "...001". That's not a number.

Source(s): BS & MS in mathematics. I'd like to see the credentials of the people who claim otherwise.
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• its an easy way out to not be confused as hell. .3 times 3 is .9 repeating. but 3/3 is one. just consider 1/3 as .3, 2/3 as .6 and 3/3 as 1

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• particular, 0.9(repeated) = a million. As for this project: "yet then on the different hand, between .999999999999999999... and a million theres constantly that .000000000000000001 so... i'm so perplexed by using this finished concept!" Why are you so particular a a million happens someplace? Which place fee does the a million happen at?

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• YES. Assuming the 9's repeat forever, 0.99999... does equal 1. They have to repeat forever though. If they ever terminate, then it will clearly not be equal to one.

Proof:

0.99999999...

=

0.9 + 0.09 + 0.009 + 0.0009 + ......

=

[n=0 n=∞]Σ(0.9*0.1^n)

This is an infinite series with a=0.9, r=0.1. Thus its sum is equal to a/(1-r), a standard result.

=

0.9/(1-0.1)

=

0/9/0.9

=

1

NOTE : this question was posed recently, and a couple of people gave my answer a Thumbs Down. IT IS CORRECT. Please do not do so.

Source(s): Degree in Mathematics
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• Yes.

more proof:

x=0.999....

10x=9.9999

10x-x=9

9x=9 => x = 1

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• No, theoretically 1 is only =1

however, practically, you can assume it to be infinitesimally close to 1, and so ~= 1.

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• yes it does.

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