Does .999~ = 1?

[Han Solo]

0.9 repeating is a fixed value. It is not a limit nor does it get closer to 1 over time nor is it infinitely close to or far away from any integer. It is a fixed value.

Agree.

We will not be able to plot a graph where the 0.99999999.... reaches the 1 line on a y-axis.

Han Solo

[yittz]

So if .999~ = 1, that would mean that 1/infinity = 0, but that makes no sense.

My first thought in regards to this question was that .999~ is a concept, thus, different from 1 (a real number).

I guess my issue is that I can't quite mentally grasp the idea of ANY recurring decimal. I mean, I can easily imagine having 1/3 of something, but the idea of having .333~ of something really seems like something else.

I am late to the debate, but I have read all that anyone has posted so far.

I like the elegant proof of 1/3*3=1. I do not see a problem with any of that logic. I am convinced.

I am not convinced by the 10 x .999...argument. I am no mathemetician, but it seems to me that 10 x .999... would = 9.999...0. I realize the 9's go on out to infinity, but if .0000...1 is possible, why not .999...0?

let us start with something everyone can agree with: .999...+.000...1=1.

From there let us compare results from operations that substitute these numbers for the values we are supposing they equal.

1-0=0, but .999...-.000...1=.999...8. Those two results are not the same. unless .999...8=1 also. That doesn't hold upt to the testy of conceiveing of a value between two numbers. If it did, we could lather, rinse and repeat infinitely, and at some point along the way we would discover that 1=.999...=.999...8=.999...7 and so on until eventually 1=0. That's quite a slippery slope. If you must use the infinite to prove your case you cannot supress it in attempts to disprove.

Let's try it on a number line. I know it is a little elementary sounding, but imagine a segment where one end represents 0 and the other end represents 1 and all conceivable numbers that fall between have their place alongthe path. .9 represents the place that is 9/10 the distance from 0 to 1. .99 represents the place that is 9/10 the distance from .9 to 1. .999 represents the place that is 9/10ths the distance from .99 to 1 and so on. If every time we add a 9 to the end of the number we are only traveling 9/10 the distance to 1 how can we ever say we reach 1? Until add 10/10 of the difference the is no 1 to carry and change those infinite rascally 9's to 0's and no 1 to the left of the decimal.

The same applies to the water glass mentioned earlier. To say it is not full so we add a little more is horribly imprecise for a math discussionI think. If we arbitrarily "add a little more" until it is full of course it will be full eventually. However, adding a little more is not what the numbers ar showing. It would more accurately represent the number if the 9/10 full glass had an amount of water equal to 9/10 the difference between a full glass and the current volume of the water already in the glass. Then we would have 99/100 of a glass of water. Again, if we constantly close the gap by 90% we never reach the destination.

But 1/3*3=1.

I propose that problem is not in values but in notation, and that the decimal system's rigid base 10 structure was not designed with these problems in mind. While fractions may be clumsier, they are far more precise and mean exacly what they say rather than approximately what they say.

I am no mathemetician. I am not saying anyone is wrong or right. I only want to be shown why these two things that seem to me to be undeniably true are in opposition to one another so I can sleep tonight.

Missouri.slim

0.9 repeating is a fixed value. It is not a limit nor does it get closer to 1 over time nor is it infinitely close to or far away from any integer. It is a fixed value.

Omg, I pressed the back button and deleted all my post.

In summary, infinity is a concept, you can't do arithmetic with infinity using +-x/. So it makes no sense 1/infinity. Nor do I get how you came to that conclusion based on 0.999~=1.

0.9999~x10=9.9999~, not 9.9999...0 (1/3x10=10/3=3+1/3, the 1/3 you multiplied is the same as the 1/3 you get at the end if you minus 3, ditto with 0.999~). Nor does 0.000...1 = 0, nor is it an infinite series, so the following arguements aren't valid.

If you repeat the step infinitely, then it is the same as reaching the line or filling the glass.

Not sure what K2Grey mean by fixed value. What I assume you mean is 0.9 repeating does not equate to the infinite series with sum of 9/10^n?

[Meh]

I think explaining the concept behind this equality is useless. You either believe in all the concepts of higher level math, or you don't.

I will say that higher level math is the basis for just about every modern technology in existence. You can deny the concept behind them all you want. But scientists and mathmaticians have used these "BS" concepts for ages on practical things that are tangible and do work.

Want an example? How about the computer you're reading this post on?

[missouri.slim]

Omg, I pressed the back button and deleted all my post.

In summary, infinity is a concept, you can't do arithmetic with infinity using +-x/. So it makes no sense 1/infinity. Nor do I get how you came to that conclusion based on 0.999~=1.

0.9999~x10=9.9999~, not 9.9999...0 (1/3x10=10/3=3+1/3, the 1/3 you multiplied is the same as the 1/3 you get at the end if you minus 3, ditto with 0.999~). Nor does 0.000...1 = 0, nor is it an infinite series, so the following arguements aren't valid.

If you repeat the step infinitely, then it is the same as reaching the line or filling the glass.

Not sure what K2Grey mean by fixed value. What I assume you mean is 0.9 repeating does not equate to the infinite series with sum of 9/10^n?

I reassert that the problem is in notation. As I am sure the members of an Asian Entertainment forum who are all posting in English can understand, meaning can be lost in translation.

The basis of my objection is that numbers are instructions. Each of those 9s means something—that you have closed the gap before the all the digits to the left of its place and 1 by exactly 9/10. It doesn’t matter how many of them there are – each 9 by definition falls short of reaching one. If your instructions are always to stop 10% short of finishing filling the glass then repeat ad infinitum how can you fill the glass without ignoring the instructions? If you think the glass is full tell that to the poor sap with the pipette who is stuck adding tiny drops to it for eternity. Just because I can’t define how short of the top of the glass you have come does not mean that you have filled it.

I believe K2grey meant that .999~=9/9 in response to my objections to filling the cup and walking down the number line. K2Grey’s objection I believe is the action of filling the cup and walking down the line indicates changes in value while .999~ is a value that will never change while my narrative sounded like I was describing a function instead. I agree that it is a fixed value. The walking and filling are merely illustrations of how we come to the value.

The problem is the decimal system because there is no way of truly representing 1/9, 1/3 or any other value that is not easily divisible except for a clumsy shorthand that had to be invented as an afterthought.

I believe that 1/3*3=1. I never questioned that. I will even accept your other assorted proofs that.999~=1. My confusion was created by the earlier proof in post 39 that appears to have been thrown out for procedural reasons in post 51 because it defines a concept rather than a number and therefore cannot be used with mathematical operators. I think this is what was causing Skylee’s confusion as well

Fractional notation is accurate. It means what it says. We should learn to use fractions again.

I think explaining the concept behind this equality is useless. You either believe in all the concepts of higher level math, or you don't.

I will say that higher level math is the basis for just about every modern technology in existence. You can deny the concept behind them all you want. But scientists and mathmaticians have used these "BS" concepts for ages on practical things that are tangible and do work.

Want an example? How about the computer you're reading this post on?

I am not saying the concepts are BS. I just want to understand.


Missouri.slim

[Meh]

I am not saying the concepts are BS. I just want to understand.

It seems the examples here should be more than convincing enough. But if you're still skeptical...

Have you at least learned Pre-Cal in school? Or at least a good grasp of high school Algebra/Geometry? If so, to understand simply read a Calculus book. No need to do problems. And no need to go in depth. But just reading and understanding the first few chapters will allow you to free your mind from the "restrictions" you've boxed yourself with.

[Nefertari]

It seems the examples here should be more than convincing enough. But if you're still skeptical...

Have you at least learned Pre-Cal in school? Or at least a good grasp of high school Algebra/Geometry? If so, to understand simply read a Calculus book. No need to do problems. And no need to go in depth. But just reading and understanding the first few chapters will allow you to free your mind from the "restrictions" you've boxed yourself with.

Hey Meh, just wondering why would you want to call those concepts "BS" in the first place? If what I'm thinking is actually what you meant. Unless BS has another kind of meaning. i mean, from what you wrote, you obviously respect those concepts, right? Just don't write anything that could let anyone misunderstand you.

Actually, reading Missouri.slim's post, i think he did learn Pre-Cal. Reading everyone's posts here, I think it's safe to say everyone in this thread learned some basic Caculus, or we wouldn't be attracted to this thread, agree? We're here coz we love to discuss Maths. And from what I gather, most people here also love Physics?

I agree with you that explaining the concept behind this equality is useless. This just makes for something to tick your mind on when you have too much time or grey brain cells. It's not practical, just something that you might want to play with.

Is there any practicality in memorizing all the decimals of pi, except to flaunt your memory? Someone did that anyway. This thread is here because we want something interesting to discuss after work, looking for something to destress.

Don't take this TOO seriously guys, unless you're a mathematician and proving or disproving it could get you the Nobel prize.

[Nefertari]

I believe K2grey meant that .999~=9/9 in response to my objections to filling the cup and walking down the number line. K2Grey’s objection I believe is the action of filling the cup and walking down the line indicates changes in value while .999~ is a value that will never change while my narrative sounded like I was describing a function instead. I agree that it is a fixed value. The walking and filling are merely illustrations of how we come to the value.

Missouri.slim

Even with those functions, we will NEVER come to the value because infinity has no restrictions.

Just like the words FOREVER and NEVER

[Nefertari]

So if .999~ = 1, that would mean that 1/infinity = 0, but that makes no sense.

My first thought in regards to this question was that .999~ is a concept, thus, different from 1 (a real number).

skylee, is that a typo you made there? don't you mean

if .99999~ = 1, that would mean that 1/infinity = 1

coz 1/1 = 1

isn't that what you mean?
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Hey guys, actually pondering more...... it is never quite possible to divide exactly a cake into 3 parts. We only imagine the divided cake to be equal. But right down to the last molecule, there must be a part of a cake that is bigger or smaller than the other parts. Isn't that right?

[IcyFox]

[FONT=Verdana]So if .999~ = 1, that would mean that 1/infinity = 0, but that makes no sense.

Yes, 1/INF = 0.

If you want to be more precise, the limit of n to INF for 1/n = 0.

0.9 repeating is a fixed value. It is not a limit nor does it get closer to 1 over time nor is it infinitely close to or far away from any integer. It is a fixed value.

Yes it is a fixed value. Because 0.999... = 1

If you want to argue, consider this : If 0.999 is fixed at some value other than one, then where?

We will not be able to plot a graph where the 0.99999999.... reaches the 1 line on a y-axis.

If we could, what are you gonna set as the x-axis and y-axis, dude?

I think explaining the concept behind this equality is useless. You either believe in all the concepts of higher level math, or you don't.

I will say that higher level math is the basis for just about every modern technology in existence. You can deny the concept behind them all you want. But scientists and mathmaticians have used these "BS" concepts for ages on practical things that are tangible and do work.

Want an example? How about the computer you're reading this post on?

I think you're wasting your time. Sorry.

if .99999~ = 1, that would mean that 1/infinity = 1

coz 1/1 = 1

isn't that what you mean?

That would also mean that 1 = INF. Wow good to know.

[IcyFox]

I am not convinced by the 10 x .999...argument. I am no mathemetician, but it seems to me that 10 x .999... would = 9.999...0. I realize the 9's go on out to infinity, but if .0000...1 is possible, why not .999...0?

let us start with something everyone can agree with: .999...+.000...1=1.

From there let us compare results from operations that substitute these numbers for the values we are supposing they equal.

1-0=0, but .999...-.000...1=.999...8. Those two results are not the same. unless .999...8=1 also. That doesn't hold upt to the testy of conceiveing of a value between two numbers. If it did, we could lather, rinse and repeat infinitely, and at some point along the way we would discover that 1=.999...=.999...8=.999...7 and so on until eventually 1=0. That's quite a slippery slope. If you must use the infinite to prove your case you cannot supress it in attempts to disprove.

First, 0.000...1 = 0, if it exists.

So 0.999... - 0.000...1 = 0.999... = 1


And 0.999...0 is out of the question because the 1st condition of this topic is

0.999... with recurring 9's all the way...


Also, the math stated in post #39 is sound. See post #65 for an explanation of infinity and infinitesimal.

[missouri.slim]

It seems the examples here should be more than convincing enough. But if you're still skeptical...

Have you at least learned Pre-Cal in school? Or at least a good grasp of high school Algebra/Geometry? If so, to understand simply read a Calculus book. No need to do problems. And no need to go in depth. But just reading and understanding the first few chapters will allow you to free your mind from the "restrictions" you've boxed yourself with.

I don't think you will win me over by questioning my education. I was unaware I needed to post my curriculum vitae to join in on the thread. I will keep it in mind next time.

Like I said, I am not a mathemetician, and I assume you are. I did not say you were wrong-- I actually agreed that .9~=1. We agree on the what, we just have trouble with the why. I merely asked if each of your infinite places to the right of the decimal brings you 90% closer by definition of the number nine you will always fall 10% short of reaching one, just like Sysiphus never got that rock to the top of the hill. Even if there are infinite nines to the right as I know there are, they are still nines each closing the gap by 90%. I am merely trying to find help reconciling two things I know to be true but seem in conflict with one another. And yes, I realize that is a whole lotta nines.

I guess a calc book and a fifth of scotch is what I need. And a fifth of scotch=25.6 oz=25.5999~oz=.2 gallons=.1999~ gallons. See, I have been paying attention. I get it, every bottle with a terminating decimal has a twin with trailing 9's.

Missouri.Slim

[missouri.slim]

First, 0.000...1 = 0, if it exists.

So 0.999... - 0.000...1 = 0.999... = 1


And 0.999...0 is out of the question because the 1st condition of this topic is

0.999... with recurring 9's all the way...


Also, the math stated in post #39 is sound. See post #65 for an explanation of infinity and infinitesimal.

I think that Wolfram Mathworld has already been introduced as an authority in this thread. They (at least to my feeble non mathematical mind) seem to have a different definition of inifinitesimal:

An infinitesimal is some quantity that is explicitly nonzero and yet smaller in absolute value than any real quantity.

To me that reads as less than anything (smaller in absolute value than any real quantity) but more than nothing (explicitly nonzero).

Kind of like back in June of 1997 when Evander Holyfield's ear was infinitely close to Mike Tyson's mouth and yet Evander Holyfield's ear was not Mike Tyson's mouth.

I will go find a calculus book as Meh suggests and start working on a hangover and stop bothering you for now.

It has been fun,
Missouri.slim