0.999... equals 1

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http://en.wikipedia.org/wiki/0.999…

Thread created as referenced here

 
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Good. Let me just reply to the latest post here.

It doesn’t have an end. If it did it would be 8.

A number either has an end or it doesn’t. If it has an end and it ends with 8, it is finite. If it is infinite, it would have an infinite amount of 9s as decimals.

It is infinite.

Do you know the definition of what is infinite and what is not? You’re ending the number of decimals with an 8, you cannot state that 0.9999…8 is infinite, you cannot imply there’s an infinite amount of 9s in between the first four 9s and the 8. You’re ending the number with an 8, no matter how many 9s there are. Therefore, it is not infinite.

 
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Originally posted by Darkruler2005:

Good. Let me just reply to the latest post here.

It doesn’t have an end. If it did it would be 8.

A number either has an end or it doesn’t. If it has an end and it ends with 8, it is finite. If it is infinite, it would have an infinite amount of 9s as decimals.

It is infinite.

Do you know the definition of what is infinite and what is not? You’re ending the number of decimals with an 8, you cannot state that 0.9999…8 is infinite, you cannot imply there’s an infinite amount of 9s in between the first four 9s and the 8. You’re ending the number with an 8, no matter how many 9s there are. Therefore, it is not infinite.

But isn’t 0.999… Stating 0.999…9? I don’t understand. That’s ending it with 9.

Are you saying there is no number which is less than 0.999… and next to it?

It’s next to 1, but what’s on the other side?

 
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But isn’t 0.999… Stating 0.999…9? I don’t understand. That’s ending it with 9.

No, that’s not the same thing. The difference between 0.9999… and 0.9999…9 is that one is infinite and the other is finite. You’re ending the latter with a 9, no matter if the rest are 9s too. The amount of decimals is finite. 0.9999… implies the amount of 9s is infinite.

Are you saying there is no number which is less than 0.999… and next to it?

There isn’t, since 0.9999… is infinite. Any number higher or lower than this has an infinite amount of real numbers in between itself and 0.9999…

It’s next to 1, but what’s on the other side?

Nothing directly, that is why 0.9999… is set equal to 1.

EDIT: I know my first post about this stated I logically assumed 1 cannot equal to 0.9999…, but now that I understand the concept, I can argue for it.

 
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Nothing directly, that is why 0.9999… is set equal to 1.

So there is no number next to any number except recurring 9s and the whole number after?

No, that’s not the same thing. The difference between 0.9999… and 0.9999…9 is that one is infinite and the other is finite. You’re ending the latter with a 9, no matter if the rest are 9s too. The amount of decimals is finite. 0.9999… implies the amount of 9s is infinite.

I don’t see how it implies it’s finite. The … is there. The dots symbolise never ending. So it doesn’t actuyally end but if it did it would be there. Maybe it’s an unreal number?

 
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So there is no number next to any number except recurring 9s and the whole number after?

Not directly, because in between almost every two real numbers, you can find an infinite amount of other real numbers. Infinity in between two numbers is rather huge to say the two are next to each other.

I don’t see how it implies it’s finite. The … is there. The dots symbolise never ending. So it doesn’t actuyally end but if it did it would be there. Maybe it’s an unreal number?

You putting the dots there doesn’t automatically imply a real infinity in between the first four 9s and the last 9. And again, a number either has an infinite amount of decimals or it doesn’t. If it does have an infinite amount of decimals, it doesn’t end, if it has a finite amount of decimals, it does end. 0.9999…9 doesn’t actually exist, you’re implying infinity while also ending it with a 9.

 
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Not directly, because in between almost every two real numbers, you can find an infinite amount of other real numbers. Infinity in between two numbers is rather huge to say the two are next to each other.

Surely one number has to be next to it. :S Are we saying numbers aren’t linear now?

0.9999…9 doesn’t actually exist, you’re implying infinity while also ending it with a 9.

So an unreal number.

 
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Surely one number has to be next to it. :S Are we saying numbers aren’t linear now?

You have to understand that if we take 0.9999999999 and 0.9999999998, we can find an infinite amount of real numbers in between the two. You can always add a 9 behind the 8, or two 9s, or three. In fact, you can add an infinite amount of 9s behind the 8, still making the latter number lower than the former. Therefore, I can state you can always find one number that’s “more” next to any number you give me than another.

So an unreal number.

Not really. The problem lies here in the concept of infinity. You’re allowing an infinite number to end, which is impossible. It contradicts itself if it were to happen. “Infinity” means it doesn’t end, you’re letting infinity end.

 
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If the ellipsis (three dots) implies an infinitely long repetition, you cannot add something to the end of it. It simply doesn’t make any sense. Infinity is not a real number (more specifically, there does not exist an element of the real set that is infinity) and you cannot add 1 [more decimal place] to it. By saying 0.999…8, you can only be saying that there is some finite repetition of nines, followed by an 8. Whether that repetition is 3 digits long or a googolplex long (a number so big it has been suggested that there is not enough “stuff” in the Universe to write its decimal value, it is still finite and makes the number < 1.

 
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The set of real numbers is said to be “dense”, whereas the set of integers, for example, is discrete. There is no integer x such that 1 < x < 2. But for any two non-equal reals a and b, there will always be an infinite amount of such xes.

 
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I can state you can always find one number that’s “more” next to any number you give me than another.

Scientists created a machine that can calculate the highest number possible… The last 100 digits of Graham’s Number (the scientists) are …940424826501819385156253579639961899396790549663800322234872396701848518643905910457562
7262464195387. It is said to be “higher than a googol or googolplex” and “power towers… are useless”. It’s always bothered me that we couldn’t find the largest number possible. I guess the highest number we can find just depends on how many new names we can make for that number place.

Oh, this is the link talking about Graham’s Number.

 
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y saying 0.999…8, you can only be saying that there is some finite repetition of nines, followed by an 8.

Why not? Just always add a 9 before the 8.

 
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Ion, I know about the Graham’s Number, though this discussion isn’t really about how high numbers can get, more about how many decimals we can add to a relatively low number.

Why not? Just always add a 9 before the 8.

It still wouldn’t be infinite. It’s infinite when it has no end. The 8 is the end in this case. Adding 9s doesn’t suddenly make it infinite.

 
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It still wouldn’t be infinite. It’s infinite when it has no end. The 8 is the end in this case. Adding 9s doesn’t suddenly make it infinite.

Adding an infinite number of 9’s doesn’t make it infinite? I’m confused.

 
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Originally posted by RMcD:

y saying 0.999…8, you can only be saying that there is some finite repetition of nines, followed by an 8.

Why not? Just always add a 9 before the 8.

It doesn’t make sense to insert digits into the middle of a number; and certainly not an infinite amount of times.

 
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RMcD, you have difficulty understanding the concepts. An infinite amount of decimals means that there is no last decimal. In this case, you could say it’s a 9, but that would not be true, because you could add another 9. That’s why there is no last decimal.

To already state that the last decimal is an 8, you’re implying there is no infinity. These two rule each other out. Adding an infinite amount of decimals in between the last 8 and the former 9s is impossible, since the last 8 implies finity.

 
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To already state that the last decimal is an 8, you’re implying there is no infinity.

I am?

Adding an infinite amount of decimals in between the last 8 and the former 9s is impossible, since the last 8 implies finity.

Why?

 
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I am?

Conceptually, you are. Can’t change the concepts.

Why?

All you need to know is that a finite amount of decimals implies an “end”. A number with an infinite amount of decimals does not have this end. You’ve already declared the end of your number with an 8. Adding an infinite amount of numbers in between does not change the end of the decimals.

 
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I am?

Rather, that the number in question is not infinite, but in essence, yes.

Why?

Because adding an 8 to the end would require an end, and infinity, by definition, is unending.

More fun mindfucks, courtesy of infinity. A circle of infinite radius can be represented as a line, and a sphere of infinite radius can be represented as a plane.

 
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Originally posted by Darkruler2005:

I am?

Conceptually, you are. Can’t change the concepts.

Why?

All you need to know is that a finite amount of decimals implies an “end”. A number with an infinite amount of decimals does not have this end. You’ve already declared the end of your number with an 8. Adding an infinite amount of numbers in between does not change the end of the decimals.

ok. I think I understand now.

 
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Isn’t there a fundamental difference between proportionality and actual numbers.

i.e 9/9 is a representation of one but in a different way then 1 = 1 unit.

 
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I thought the whole explanation there is rather vague. I’d rather just have them state “0.9999… = 1 kthxbye” than to give a self-fulfilling, mathematical explanation. 9/9 = 1, no doubt about it, and any 1 cannot be different from “another” 1. The point here is that 0.9999… can be set equal to 1, since there simply isn’t any real number in between that and 1. Name any two other numbers and you can find an infinite amount of real numbers in between them.

 
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9/9 does not =0.99999… any more than 4/4 = 0.9999… or 1 = 0.99999…..

An infinite number doesn’t end BY DEFINITION. By “ending” it with a number 0.9999…2 or 8 etc. you create a finite number. They are NOT the same.

 
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Yes; 1, 0.999… and 9/9 all refer to the same point on the continuum, they are just different representations of it.

 
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ooh, yay I love these.

1/3 = 0.3333….

0.3333... × 3 = 0.9999….

1/3 × 3 = 1

Therefore:

0.9999…. = 1