# .99~ = 1 (revisted)

Diane_
Homework Helper
ram1024 said:
an infinite number of decimal sequential digits of 9 even if it WERE to simply EXIST, it is a unique number and quite simply NOT equal to 1. it is certainly NOT rational whereas 1 is
Ram, you keep saying things like this, but you never offer anything resembling proof. I presume that's because you consider it self-evident, but it isn't. You need to ask yourself why you believe that - I think the answer is that the two numbers don't look the same, so the back of your mind says that they aren't the same. That's not how mathematics works. There are certain things that must be taken as postulates in order for a mathematical system to get started - once you've postulated those things, though, the rest of the system must flow strictly from logic. If you choose to postulate that .9_ and 1 are, as you put it, "unique" numbers, that is certainly valid. You must be aware, though, that doing so will put you into a system of mathematics which is in disagreement with what the rest of the world has been doing for the past several centuries.

So, as specifically as you can, without reference to numbers which are not "known" (like the number between .9_ and 1) and without reference to intuition - why are .9_ and 1 two different numbers?

but within the confines of the problem it can be concluded that given INFINITE number of half distances you could never reach your destination.
No; it can be concluded that when considering successive half-distances that reaching the desintation will not be included in the consideration.
same thing different words. it does not change the fact that the destination will never be reached :P

As you have said it would take an infinite number of half distances to get there and that is similar to reaching 1, but there are an infinite number of 9s in
no i said an infinite number of half-distances will NOT reach a destination :P

ram1024: would you please state clearly what DEFINITION of "real number" you are using? I know several definitions that could be used- they all lead immediately to the conclusion that 0.9999... is exactly equal to 1.
A number inclusive in the set that is comprised of numbers less than infinity and greater than negative-infinity

You need to modify your concept of infinity. Zeno did not know about convergent infinite sequences, with that knowledge there is no paradox.
there is no paradox because Zeno is absolutely correct. you cannot reach the destination EVER within the confines of his equation.

You really should not put so much faith in a single quote, why not make an effort to learn more about the construction of the Real number line before being so argumentative.
you should really open your mind to the possibility that you and many other very very smart people could be wrong as well. i'm not argumentative, i'm just having a rational calm discussion with other people who (seem to) have interest in the same subject

Whoa, where have I been?

Just to explain things to the native forum users: this is a continuation of a thread from the Warcraft 3 Off-Topic forum which was closed because 0.999 = 1 threads are banned there. Sorry to bother you with all this. :/

What happened is that there was a particularly vicious round of 0.999 = 1 related posts, and then ram1024 made a thread asking everyone to come to him with their questions, and in which he 'explained' to us that 0.999 does not equal 1. The thread, or rather, flames went on for a page or two before the thread was closed, so apparently ram1024 has decided to bring them here.

I guess he wasn't counting on anyone/everyone here actually posting to disagree with him, since he's convinced tha he's right. Actually, I'd just like to the say thanks ram1024... I've been posting on www.scienceforums.net[/url] with my biology/chemistry relates posts, but this is good too. :) (use this url if you want ot help me out by giving me referrals :) [PLAIN]http://www.scienceforums.net/forums/index.php?referrerid=764 [Broken])

Okay, I'll try to join in here by dissecting ram1024's post (making references to another post that I made on the other forum which, I hope, he read before they were deleted, otherwise I'll just have to type them out again):

same thing different words. it does not change the fact that the destination will never be reached :P
This being in reference to the two quotes above it. I already proved before that infinity does not act similarly to other real numbers, in that you cannot divide and multiply by it, (as would be suggested by your earlier claim that infinity/infinity = 1), because then you would end up with statements like 1 = 0 x infinity, which lead to, as I showed before, contradictions like 2 = 3. So while it is true that the series which halves with each term will not reach 1, because there is only a finite number of terms, can you confidently say the same of infinity?

no i said an infinite number of half-distances will NOT reach a destination :P
Same reason as above, and the same as your first point; by taking one point out of context, that being someone else's conclusion, you can't state your own unverified conclusion to 'prove' that they are wrong.

A number inclusive in the set that is comprised of numbers less than infinity and greater than negative-infinity
Here's another definition... "One of the infinitely divisible range of values between positive and negative infinity" taken from here: http://dict.die.net/real number/
That's not the same as less then positive infinity and more than negative infinity. Here's one from www.dictionary.com: "any rational or irrational number". It doesn't specify limits. The one before does, and those are infinity, but they're not really necessary because there is NOTHING beyond infinity or negative infinity.

However, not all rational and irrational number behave the same. 1/0 is undefined, for example. Wait... I forgot. You said on the other forum that zero is not a number, right? Then these definitions are incorrect, as is yours... you defined a real number as one being less than infinity and more than negative infinity, but you forgot that you don't think zero is a number and it lies between those two!

there is no paradox because Zeno is absolutely correct. you cannot reach the destination EVER within the confines of his equation.
The confines of HIS equation, which was based without the knowledge of these series. As I said before, you are assuming that when you multiply by any number, including infinity, the series never reaches its destination, which as I said before is based on the assumption that infinty acts like a real number (even though your own definition denies that it is one).

you should really open your mind to the possibility that you and many other very very smart people could be wrong as well. i'm not argumentative, i'm just having a rational calm discussion with other people who (seem to) have interest in the same subject
Yes, we are having a calm discussion, but you are not being rational... you have not desproved a single point we have made and yet still claim to be correct without proof. The quotes you have selected are countered by repeats of your own unproven assumptions, and when someone points out that your single (outdated and fallacious) source may be wrong you tell them that it is the rest fo the modern mathematical community which may be wrong. Remember, Zeno clearly didn't know what infinity was, because they probably hadn't invented it then.

Sorry to all the native forum users about this thread. ram1024 made his thread personally asking us to come to him with questions, and then he tries ot give us these 'arguments' which, even though the subject is actually very trivial, is still quite offensive. I don't usually argue this much. :(

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NateTG
Homework Helper
ram1024 said:
infinitely repeating decimals can only be approximated as fractions because of the inclusion of infinity in their nature.
According to you $$\frac{1}{1} \neq 1.\bar{0}$$.

Do you think that 0.99999... is a real number?

If so, what is 0.99999... + 0.99999... equal to?

If the answer is 2, then you've got :
(0.99999...+0.99999...)=2
so
2(0.9999...)=2(1)
0.9999...=1

If the answer is 1.9999.... then you've got
0.9999...+0.9999...=1.9999...
(0.9999...+0.9999....)-0.999...=1.9999....-0.9999...
0.9999...+0.9999...-0.9999..=1
(1+1-1)(0.9999...)=1
(1)(0.99999...)=1
0.9999=1

1/3 = 0.33....
3*(1/3) = 3/3 = 1
3*(1/3) = 3*0.33... = 0.99...

Then... 1 = 0.99..

matt grime
Homework Helper
If we make a reasonable assumption about zeno's paradox, that both speeds are constant, then it just states that they will not meet before some fixed time (dependent on the ratio between the speeds). given that the observation that zeno's paradox does not stop you overtaking on the motorway, perhaps you ought to look at it and figure out where it's wrong for yourself, ram.

matt grime said:
If we make a reasonable assumption about zeno's paradox, that both speeds are constant, then it just states that they will not meet before some fixed time (dependent on the ratio between the speeds). given that the observation that zeno's paradox does not stop you overtaking on the motorway, perhaps you ought to look at it and figure out where it's wrong for yourself, ram.
Zeno's Observation is ABSOLUTELY correct. the conclusion he came to is incorrect because of the times he was living in this was a relatively unexplored field.

Tell me where i'm wrong in saying you will NEVER reach 1 in this series outlined in this problem

Object A travelling From Point B to Point C which is a measured distance of 10 meters, and moving at a speed of 10 meters per second.

Object A must first travel 1/2 the distance (5 meters) then 1/2 the remaining distance (2.5 meters) then 1/2 that remaining distance (1.25 meters) etc ad infinitum.

the series has a LIMIT of 10 meters, but the SUM of the series will NEVER be 10 meters. there will ALWAYS be some portion "half distance" left over to travel, infinitely small and beyond human comprehension.

SOMETHING cut into 2 parts will ALWAYS yield SOMETHING. 1/Infinity ≠ 0

Integral
Staff Emeritus
Gold Member
A number inclusive in the set that is comprised of numbers less than infinity and greater than negative-infinity
This is pretty funny. Clearly all that you know about the real number line can be written on a decimal point.
The world of mathematicans defines Infinity as an addition to the Real Number line. In doing so the field nature of the Real numbers is lost. In reality Infinity is not a number, but a definition, part of the definition is how Infinity behaves under each of the operations. It is by DEFINITION that

$$\frac 1 {\infty} = 0$$
no i said an infinite number of half-distances will NOT reach a destination
But Since that is a convergernt series, Math says that it DOES reach the end after an infinite number of steps.

Integral
Staff Emeritus
Gold Member
http://home.comcast.net/~rossgr1/Math/one.PDF

I have a sneaking suspicion that you simply do not have enough mathematical sophistication to understand formal proofs, so I doubt that the above link will be meaningful. If you can get yourself out of the lecturing mode into the learning mode you may actually be able to get a glimmer of how mathematic ans look at this problem. In this thread you are arguing with at least one Mathematics PhD and several graduate level math students. Given acess to this level of knowledge it is a shame that you cannot make some effort to learn something.

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Integral said:
This is pretty funny. Clearly all that you know about the real number line can be written on a decimal point.
The world of mathematicans defines Infinity as an addition to the Real Number line. In doing so the field nature of the Real numbers is lost. In reality Infinity is not a number, but a definition, part of the definition is how Infinity behaves under each of the operations. It is by DEFINITION that

$$\frac 1 {\infty} = 0$$
well that definition is CLEARLY wrong then. I don't know why you guys are using "an infinity" that has clear limits as such, but it's ridiculous.

it's a good thing you guys live in seclusion and never come out to see the real world, because in the real world, something is never nothing, not matter how you cut it.

find me the guy that invented your infinity cause i wish to verbally abuse him.

Hurkyl
Staff Emeritus