## Does 0.9999(repeating) = 1?

Yes 705 54.95%
No 578 45.05%
Voters: 1283. You may not vote on this poll

### Does 0.9999(repeating) = 1?

First Riot Post

Crowcide

Senior Member

Quote:
Originally Posted by Eledhan
TL/DR - .999... does NOT equal 1, no matter how close to infinity you get. Speaking of infinity, it's not a real number, which means .999... isn't real either, but just a number to show a concept that isn't easily converted into a decimal number system.
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This has nothing to do with being close. You are arguing about a bunch of definitions you don't even know or understand.

What do you think a real number is? What do you think repeating decimals are? What do you think infinity is? Because your understanding of them is contrary to how math defines them.

.999~ = 1 because of what we define .999~ to be, a geometric series with a common ratio of 1/10 and an r value of 9. .999~ is a real number directly from the definition of the reals, it's the supremum of this set {.9,.99,.999,.9999,...}.

Eledhan

Senior Member

Quote:
Originally Posted by Kodoku
The only infinity needed for 0.99... is the infinity associated with sequences and limits. If you're telling me sequences and limits are no good, then you're saying that the set of real numbers can't be constructed. The construction of the reals, after all, is done by completing the set of rationals, i.e. adding (in a completely non-trivial way) the limits of cauchy sequences of rationals.
What I was trying to say is that if you use basic mathematical operations with infinite numbers, you get impossible results. I vaguely remember reals, rationals, integers, and the like...I'm not a math major or anything like that. However, these concepts are still not totally foreign to me.

If what you are saying is that portions of rational numbers are not possible without infinity concepts, then I see your point. However, the algebraic proof I was attempting to refute cannot work without rational numbers. Whenever you use an "X = Y" statement, for the rest of that proof, X can never equal anything other than Y. What I believe the proof is trying to do is say that "X = Y, and Y = Z, therefore, X = Z". I would go with this more than I would go with the way it was formatted.

If the definitions of X and Y were the same, then I would agree with this whole thing. But as far as I know, the definitions of .999~ and 1 are NOT the same... Am I wrong? My understanding of a need for .999~ was because 1 is not evenly divisible by 3. It has to do with different number systems, not high-level mathematics. Decimal number systems are only one system that is used. It's like a language...there's no TRUE translation for 1/3 in the decimal number system, since the number isn't able to be represented by decimals for its true value without using infinity.

It's kind of like trying to translate different languages...there's no TRUE translation for many words in ancient greek, latin, hebrew, aramaic, arabic, etc. into modern english because those languages had different meanings and inferences than we have today in our language.

That being said, I assume that for all intents and purposes, when someone says, ".999~ is the same as 1" it's no different than someone saying, "agape love is the same as unconditional love", even though the meanings are always going to have some infinitesimal difference, but the overwhelming majority of the meanings are so close that it's pointless to discuss the difference between the two.

Quote:
No such assumption is made. All that's assumed is that 9x/9 is x, and 9/9 is 1.

When you get to 9x = 9, the point is that by dividing both sides by 9, equality will be preserved. So 9x/9 is equal to 9/9.

Well the left side is just x, and the right side is just 1, ergo x = 1

Your example doesn't contradict anything. It just shows that if you start with x = 2, you end up with x = 2, as one would hope.
I think the definitions of the numbers is the key...if it's what I think it is, the problem with people's opinions is that they're trying to represent the same number in two different numeric systems. Is this the point everyone has been trying to make?

Kaolla

Senior Member

Quote:
Originally Posted by Kodoku
Zeno was an ancient philosopher who attempted to support Parmenides' claim that the existence of motion is contradictory. If you accept his argument, then it follows that motion is impossible. As it turns out, it's quite easy to resolve this particular argument because he implicitly assumes that if you sum an infinite number of finite numbers, you'll end up with something infinite. This is false. Calculus proves it's false.
wrong, because motion in reality is not dependent on calculating half-distances. and even if you DID have to, and you wanted to move 10 ft away, there always exists a distance 20ft to where moving HALF that distance will get you 10ft away in one calculated step. so really it's a THOUGHT exercise which CORRECTLY illustrates that infinite calculations DO NOT reach the correct answer, they merely approximate.

Quote:
Originally Posted by Kodoku
Though it's not hard to see it intuitively: It should be clear that the sum
0.1 + 0.01 + 0.001 +... does not sum to infinity. In fact, it sums to exactly 0.11... = 1/9.
it does not, the long division of 1 divided by 9 proves that it does not

Eledhan

Senior Member

Quote:
Originally Posted by Crowcide
This has nothing to do with being close. You are arguing about a bunch of definitions you don't even know or understand.

What do you think a real number is? What do you think repeating decimals are? What do you think infinity is? Because your understanding of them is contrary to how math defines them.

.999~ = 1 because of what we define .999~ to be, a geometric series with a common ratio of 1/10 and an r value of 9. .999~ is a real number directly from the definition of the reals, it's the supremum of this set {.9,.99,.999,.9999,...}.
So what you're saying is that the very definition of .999~ is the same as 1.000~, right?

If this is true, then I guess that's the argument people should be making...not that the number represented by .999~ is the same value as the number represented by 1.000~.

For all intents and purposes, I see how there's no reason to suggest that .999~ is any different than 1.000~. Where I get irritated is when people try to say they are different, and then say they are the same. They can't be both different AND the same simultaneously.

Crowcide

Senior Member

Quote:
Originally Posted by Eledhan
If the definitions of X and Y were the same, then I would agree with this whole thing. But as far as I know, the definitions of .999~ and 1 are NOT the same... Am I wrong? My understanding of a need for .999~ was because 1 is not evenly divisible by 3. It has to do with different number systems, not high-level mathematics. Decimal number systems are only one system that is used. It's like a language...there's no TRUE translation for 1/3 in the decimal number system, since the number isn't able to be represented by decimals for its true value without using infinity.
Is 2/2 1? Is 2/2 the same definition as 1?

What is .9999~ defined to be? The mathematical definition is a geometric series with a common ratio of 1/10 and an r value of 9. That series equals 1. .9999~ is defined to be 1 in the same way 2/2 is through equivalency.

Crowcide

Senior Member

Quote:
Originally Posted by Eledhan
So what you're saying is that the very definition of .999~ is the same as 1.000~, right?

If this is true, then I guess that's the argument people should be making...not that the number represented by .999~ is the same value as the number represented by 1.000~.

For all intents and purposes, I see how there's no reason to suggest that .999~ is any different than 1.000~. Where I get irritated is when people try to say they are different, and then say they are the same. They can't be both different AND the same simultaneously.
I'm guessing there aren't very mathematicians in this thread, hence the non mathematical arguments.

1.000~ and .9999~ are the same.

1.000~ is defined to be a 1 + geometric series with a common ratio of 1/10 and an r value of 0, which equals 1.

.9999~ is defined to be a geometric series with a common ratio of 1/10 and an r value of 9, which also equals 1.

Not lookign the same doesn't mean they are different, does 2/2 look like 1?

Colonel J

Member

612 people, and counting, are retarded.

Kodoku

Senior Member

Quote:
What I was trying to say is that if you use basic mathematical operations with infinite numbers, you get impossible results.
Operations with infinite numbers do indeed cause problems - if you extend the ordinary numbers (reals) to include infinity, you break a lot. The point is that 0.99... does not require infinitely large numbers at all.

Quote:
If what you are saying is that portions of rational numbers are not possible without infinity concepts, then I see your point.
What I was saying is that the same concepts used to define 0.99... are used to define the set of real numbers. 0.99... is well defined if and only if real numbers in general are well defined.

Quote:
However, the algebraic proof I was attempting to refute cannot work without rational numbers. Whenever you use an "X = Y" statement, for the rest of that proof, X can never equal anything other than Y. What I believe the proof is trying to do is say that "X = Y, and Y = Z, therefore, X = Z". I would go with this more than I would go with the way it was formatted.
A statement "x = y" means that the object on the left is the same as the object on the right. The initial step states that x is being used to denote the same object denoted by 0.99...

What the proof shows, in essence, is that x = 0.99... implies x = 1. From this it follows that 0.99... = 1, by the transitivity of equality. The only implicit assumption here is that 0.99... is a real number (whatever real number it may be), as that's required to use ordinary operations on it.

Quote:
If the definitions of X and Y were the same, then I would agree with this whole thing. But as far as I know, the definitions of .999~ and 1 are NOT the same... Am I wrong?
The definitions don't need to be the same. The definitions of 2/2 and 1 are different, yet they denote the same object. You can define the same things in many ways.

Quote:
My understanding of a need for .999~ was because 1 is not evenly divisible by 3.
There's actually no need for .99... at all. In fact, to avoid confusion, some textbooks explicitly restrict decimal expansions to those not involving repeating 9s, since otherwise you can get two decimal expansions that denote the same number (e.g. 1.00... and 0.99...).

Quote:
It's like a language...there's no TRUE translation for 1/3 in the decimal number system, since the number isn't able to be represented by decimals for its true value without using infinity.
There is, in fact, a true translation in decimal form for 1/3. A decimal expansion is defined by assigning a digit to each slot in the decimal expansion. There is a slot for every natural number.

i.e. 1 corresponds to the first decimal place in the expansion.
n corresponds to the nth decimal place.

By assigning 3 to every natural number n, and correspondingly to every decimal place, you've defined a number in the decimal system.

Quote:
That being said, I assume that for all intents and purposes, when someone says, ".999~ is the same as 1" it's no different than someone saying, "agape love is the same as unconditional love", even though the meanings are always going to have some infinitesimal difference, but the overwhelming majority of the meanings are so close that it's pointless to discuss the difference between the two.
You can't do that in mathematics. If you say x = y when there is a difference, no matter how trivial or insignificant that difference is, you'll blow your mathematics up. You'll literally be able to derive anything, such as 1 = 2, or jam = square root of your mom.

Quote:
I think the definitions of the numbers is the key...if it's what I think it is, the problem with people's opinions is that they're trying to represent the same number in two different numeric systems. Is this the point everyone has been trying to make?
The proof shows that the two definitions (of 0.99... and 1) denote the same number. It's not just different systems though. After all, you can define 1 in decimal notation in another way: 1.00...
These are both infinite decimal expansions.

Quote:
612 people, and counting, are retarded.
612 + every single mathematician in the world.

Eledhan

Senior Member

Quote:
Originally Posted by Kaolla
wrong, because motion in reality is not dependent on calculating half-distances. and even if you DID have to, and you wanted to move 10 ft away, there always exists a distance 20ft to where moving HALF that distance will get you 10ft away in one calculated step. so really it's a THOUGHT exercise which CORRECTLY illustrates that infinite calculations DO NOT reach the correct answer, they merely approximate.

it does not, the long division of 1 divided by 9 proves that it does not
See, this is exactly what I was referring to...

If the definition of 1/9 is just that...1 step of the total of 9 needed to reach your goal, you would never be able to represent this 1 step in a decimal numeric system. Hence the abbreviation of it to .111~. If you add up all the .111~ 9 times, you'd never actually get to 1 using the decimal system. However, the representation .111~ is the closest possible representation to the value of 1/9, meaning that although it isn't a perfect translation, it is a translation that means close enough to the same thing to be treated as if it actually were the same thing.

Eledhan

Senior Member

Quote:
Originally Posted by Kodoku
You can't do that in mathematics. If you say x = y when there is a difference, no matter how trivial or insignificant that difference is, you'll blow your mathematics up. You'll literally be able to derive anything, such as 1 = 2, or jam = square root of your mom.
(I cut out the other stuff for brevity)

If you can't say that there is a difference between x and y if they are the same, then why bother with representing .999~ and 1.000~ at all if they are the same number? It seems to be an exercise in futility.

Why not just call it one or the other? Why was there ever a representation of .999~ if 1.000~ would suffice? I have always thought the reason was because there really IS a difference between the two...

However, based on what I've read here, I'm not so sure that it isn't any difference from my using "Eledhan" in exchange for my real name...I'm the same person, just referenced by a different name in different settings.

Quote:
The proof shows that the two definitions (of 0.99... and 1) denote the same number. It's not just different systems though. After all, you can define 1 in decimal notation in another way: 1.00...
These are both infinite decimal expansions.
Yes, both numbers represent one whole "numerical unit". In essence, if it were a pie, you could say you have a whole pie (1/1 or 1.000~), or three thirds of a pie (3/3 or .999~). They all mean the same thing...

Is this the point of the whole topic?