.999... = 1

[ser stiefel]

Doc, don't get sucked into Prowler's nonsense. Some people argue for the sake of achieving a better understanding, but that is not his quest. I don't even believe it is a quest to insist that his belief is correct while everyone else is wrong. Rather, he starts with the assumption that everyone else is wrong and makes his belief the opposite simply to have a point to argue.

Thank you Prowler for making my day and vehemently resuming this discussion where it was left months ago. I hope "0.999..." day we can all look back on this and laugh. I know I will!

lol... your use of 1 interchangeably with 0.9999... in these posts has had me chuckling several times. Nicely done.

[a.sub]

this is hopeless, im going to go do something more productive, if someone with a BS in math give u a page proof and all prowler
does is shrug it off without actually realizing what he is saying (or even reading it) then this is clearly a fruitless effort.

adios

[Army of GOD]

Haha, yea, Doc, that was lame. This is supposed to be a debate about how liberals are bad or something.

But that was a good read. Very intellectual and I could barely follow it, but it was interesting (no, I'm not a geek!).

[Snorri1234]

The problem Prowler is having is in thinking that infinity+1 is larger than infinity.

Or, that infinity-1 is smaller than infinity.

Yes, that's also a flawed assumption.

[Phatscotty]

.999 does not = 1

[the.killing.44]

.999 does not = 1

No, but .999… does.

[Phatscotty]

.999 does not = 1

No, but .999… does.

that .sooo....... minimizes the concept of 1

In order to define .999...., you must first understand what 1 is

[john9blue]

I don't know which is worse: Prowler's trolling or the fact that our opinion of his intelligence is so low that we think he is serious.

[Snorri1234]

I don't know which is worse: Prowler's trolling or the fact that our opinion of his intelligence is so low that we think he is serious.

The problem is that there are actually people who really believe it, so we can't be sure Prowler doesn't believe it.

[max is gr8]

The proof I believe is by doing the following

Code: Select all

999.999999999999999999999999999999999999999999999999999999 = 1000x
000.999999999999999999999999999999999999999999999999999999 = x
----------------------------------------------------------
999.000000000000000000000000000000000000000000000000000000 = 999x

999
--- = x
999

999
--- = 1
999

[jonesthecurl]

One has to chuckle, doesn't .999999999...?

[TheProwler]

Another quick thought (which a.sub reminded me of with his summation example):
It can be shown (see any Algebra 2 textbook) that a repeating decimal can be written as a rational number. In fact, there is a cute little trick you can use to write the rational representation quickly. Assume that the sequence of number that repeats contains N digits (0.1111.... has 1 repeating digit, 0.212121212... has 2 repeating digits, 0.314314314314... has 3 repeating digits, and so on). Take those N digits and put them in the numerator of your fraction. The denominator is exactly given as a number containing N 9s. The three above examples can all be written as 1/9, 21/99, 314/999 respectively.

To prove this, you can use the same proofs that have been presented elsewhere in this thread, or you can use the definition for the sum of an infinite geometric series. Recall that the sum is an approximation for a large number of terms in the series, but it is exact when you include infinite terms.

The upshot is that 0.999... can be written (exactly) as 9/9 or 99/99 or 999/999, etc... All of these are exactly equal to 1.

What you have shown in your first paragraph is not a proof that 9/9 = 0.999... It only demonstrates that when you are using repeating decimals (of non-zeroes), you are only writing a good estimate. 21/99 is exact. 0.212121... is not.

But it looks like the start of a good proof. Why? Because you are noticing a pattern. And patterns are great, right? They are often the catalyst needed to get a good proof formulated.

So let's look at another pattern (try to keep an open mind now people):

0.9 + 0.9 = 1.8
0.99 + 0.99 = 1.98
0.999 + 0.999 = 1.998
0.9999 + 0.9999 = 1.9998
0.99999 + 0.99999 = 1.99998
0.999999 + 0.999999 = 1.999998
0.9999999 + 0.9999999 = 1.9999998
0.99999999 + 0.99999999 = 1.99999998
0.999999999 + 0.999999999 = 1.999999998
0.9999999999 + 0.9999999999 = 1.9999999998
0.99999999999 + 0.99999999999 = 1.99999999998
...
0.99<784 287 368 nines>99 + 0.99<784 287 368 nines>99 = 1.99<784 287 368 nines>98

You can see the pattern, right?

Just stop for a second. Just because this isn't a mathematical text book being quoted doesn't mean you shouldn't keep an open mind. Just take a look at the pattern.

Ask yourself.

When does it stop looking like this:

0.99999999999999999999999999999999999999999999
+0.99999999999999999999999999999999999999999999
=1.99999999999999999999999999999999999999999998

and start looking like this:

0.999...
+0.999...
=2



If you are not taking a second look at this, you are not keeping an open mind.

If you are not questioning why an obvious pattern suddenly disappears as soon as we introduce the concept of infinity, you are not the thinker you thought you were. I think.

I'd also point out that I (and others) can cite published mathematical texts that assume, prove, and even require that 0.999...=1. Prowler has essentially said, "No it's not, and no one else understands why but me." For his claims to be valid, one of two things has to happen. Either there is a group of respected mathematicians that would agree with his claims (and I would like to see documented proof of this) or he is the first to prove his claims, in which case he deserves to win the Field's Medal!

Hahaha!! Sorry for laughing. I'm actually laughing at your species, not you in particular.

Why do you think we are even capable of fully understanding the concept or the reality of infinity? Much less prove anything to do with infinity.

Us humans are always overestimating ourselves.

The point of all this is about the size of infinity. There are in fact multiple values of infinity (aleph null, aleph 1, aleph 2, ...), however, you don't increase the size of a set from one level to the next by simply adding a finite number of elements. Nor can you do it even by squaring the number of elements. It might be best to think of the relations between them as being on the order of the factorial operation. This isn't strictly valid, but it's convenient to think about the relative sizes in this way.

"There are in fact multiple values of infinity"

"relative sizes" (of infinity)

Okay. Time for some of you to be honest. Raise your hands if you were quite sure that there was only one "value" of infinity. Don't be shy. We have over 20 pages of people who were quite confident they had a solid understanding of infinity. I think there has to be at least one or two of you that swallowed a little air when they read some of these theories.

Did you get the feeling that maybe you were over-simplifying things in your mind?

When reading of these theories, did anyone think "Wow, even the most brilliant mathematical theorists end up talking like first-year philosophy students when they run into some abstract ideas!"? Or was it only me?

The problem Prowler is having is in thinking that infinity+1 is larger than infinity.

Or, that infinity-1 is smaller than infinity.

Yes, that's also a flawed assumption.

Hahaha!!!

Yes snorri, you are quite confident. But be true to yourself. Be honest with yourself.

The point is that you can stand up, look for the popular opinion, and get behind that opinion. And you will be confident that you are right. And you will be cocky about yourself. And you will be the Master of Infinity.

But you are talking about something that has perplexed man for ages. You are over-simplifying the entire concept. You are fooling yourself if you think you really have a good grasp of infinity.


Seriously, why would the pattern all of a sudden end at infinity?

[Koganosi]

This problem aint that hard.

1/3 = 0,3333333.......

1/9 = 0,1111111.......

3 * 1/3 = 1

But if you look at the sentence above 1/3 = 0,3333... Then it also is

3 * 1/3 = 0,9999999....

So that makes 0,999999.... = 1.

You can do the same story for 1/9.

Urs

Koganosi

[THORNHEART]

could it be that mans mind cant quite grasp infinity?

could we be failible?

could there be a greater power that has estabilished all this

could it be that somethings we will never quite understand because we were not created with the ability nor purpose to understand.

If this really is such an easy and simple question as many have said then how come there is so much controversy....infinity means just that there is no closure no final equation. Man cannot grasp that something might not have an expliantion because we live in a a finite world at least on earth everything has a solution and final ending...but that is not so in eternity and also in the infinity of space. Hahaha for all our great accomplishments man really is a pathetic creature and very small in true knowledge

[john9blue]

21/99 is exact. 0.212121... is not.

0.212121... is exact.

[MeDeFe]

21/99 is exact. 0.212121... is not.

0.212121... is exact.

As is 0.21...

I wouldn't give Prowler any more digits than strictly necessary, he'll start talking about "patterns" again.

[jonesthecurl]

Let's look at it as a variation on the "Achilles and the tortoise" conundrum.
Joe Swift is racing Speedy Fred.
Speedy Fred gets a 10-yard start.
Fred runs at 90% of Joe's speed.
By the time Joe reaches the point at which Fred started, Fred is 9 yards in front.
By the time Joe reaches there, Fred is 9/10's of that distance in front.
Logically, joe can never catch Fred, he can only get .999999.... of the way.
This is of course, nonsense, and it turns out he does pass him.
Because, as it happens, .999... is equal to 1. It's just a different way of expressing it.
If it weren't, overtaking anything would be physically impossible.

[Doc_Brown]

*double post*

[Doc_Brown]

What you have shown in your first paragraph is not a proof that 9/9 = 0.999... It only demonstrates that when you are using repeating decimals (of non-zeroes), you are only writing a good estimate. 21/99 is exact. 0.212121... is not.

So let's look at another pattern (try to keep an open mind now people):
0.9 + 0.9 = 1.8
0.99 + 0.99 = 1.98
0.999 + 0.999 = 1.998
...
When does it stop looking like this:
0.99999999999999999999999999999999999999999999
+0.99999999999999999999999999999999999999999999
=1.99999999999999999999999999999999999999999998

and start looking like this:
0.999...
+0.999...
=2

This is actually an excellent question. I think it's one of the first really valid paradoxes you've presented (it certainly is more valid than claiming that 0.333... is less than 0.333...!). But the answer is exactly what you hinted at. The pattern ends at infinity. Let's express the sum of the two numbers in mathematical terms:
S_N = 2 * sum(n=1 to N, 9*10^(-n))
Hopefully my notation makes sense. I'm expressing 0.999...9 as a finite sum containing N decimal digits. We can simplify the sum to:
S_N = 18 * sum(n = 1 to N, 10^(-n))
Using the formulas for finite sums, this simplifies to:
S_N = 18 * (1/9) * (1 - 10^(-N)) = 2 * (1 - 10^(-N)).
That is an exact value for any N. Notice that for any N contained in the natural numbers, S_N < 2. Yet S_N converges to 2. The sum of the infinite series version is also exact and is equal to the limit of S_N as N->infinity, that is, 2.

I think the only argument you can make with this is to claim that sums of infinite series are not exact. Yet to do so would completely destroy most of the equations we use in calculus!

Why do you think we are even capable of fully understanding the concept or the reality of infinity? Much less prove anything to do with infinity.
Us humans are always overestimating ourselves.
But you are talking about something that has perplexed man for ages. You are over-simplifying the entire concept. You are fooling yourself if you think you really have a good grasp of infinity.

Why do you suppose that just because you can't understand the concept no one else can? I don't claim to fully understand infinity, but I am completely comfortable working with it! You'd be surprised how often it comes into play in higher mathematics. Complex analysis, for example, is riddled with infinities. To integrate around poles in the complex plane, it is often necessary to draw a contour around the pole and then move that contour out to infinity. It turns out that path integrals along contours at infinity can be much easier to evaluate at times!

But I'm also a physicist, and the concept of infinity is one of the easier things to grasp! What does it mean for a material to have a negative refractive index? What causes light to have an infinite phase velocity? How can an electron exist on one side of an infinite potential barrier one moment then on the other side the next moment without ever having traveled through the barrier?
None of those questions make any logical sense, but they're all perfectly valid occurrences that are observable in nature. Quantum mechanics provides a whole host of crazy scenarios that defy common sense:
You may be familiar with Young's Double Slit experiment. This was what first conclusively proved that light was a wave (before Einstein proved that it was also a particle). Basically, if you shine light through two very closely spaced slits (or pinholes), it produces an interference pattern on an observation screen on the other side of the slits (basically you get bars of light spaced out with completely dark areas in between). This is much like if you drop two rocks in a pool of water at the same time and watch the resulting ripples. At some places you have two crests on top of each other making a bigger crest, while other places you have troughs overlapping making an even deeper one. But here's the crazy part: If you replace your light source in the experiment with a beam of electrons, you get the exact same interference pattern! It gets worse. If you slow the beam down to emit a single electron at a time, and observe where it hits the screen, it will always end up hitting in the light areas of the interference pattern and will never hit the screen in the previously observed dark areas. If you mark the screen where every electron hits and do this for several thousand electrons, you'll get the exact same interference pattern you had when you used light. Hold on though, it gets worse. If you then place a detector on one of the two slits so you can figure out which slit the electron actually went through, the interference pattern disappears! Conclusions: Particles are waves, but the very act of observing them collapses the wave function!

This lead to the proposal of a quantum Zeno paradox. Essentially, observing an unstable (or radioactive) particle mathematically reduces the probability that it will decay within a very short time span after the observation. Thus, a continuously observed unstable particle will never decay! This is essentially like claiming that if you can get a pencil to stand up on its end, it will never fall over as long as you keep watching it! Yet to a certain extent, this paradox has been experimentally verified.

Sorry Prowler, but there is a lot out there that doesn't make sense until you're intimately immersed in the field. Some of us have worked with the concept of infinity sufficiently that we are quite comfortable with it. Certainly in the field of computer programming, you won't have much use for the infinite. Even when we write scientific programs and mathematical simulations we're content to use something like MAX(double) to represent an effective infinity since it is the largest real number possible. But in more abstract mathematics, infinity is critically important, and it is essential that one be able to work with it in an exact manner (not just with approximations) in order to do higher mathematics.

[Timminz]

Seriously, why would the pattern all of a sudden end at infinity?

Because it is impossible to arrive "at infinity".

[pmchugh]

Doe sit really fukin matter? wierdos

[TheProwler]

Let's look at it as a variation on the "Achilles and the tortoise" conundrum.
Joe Swift is racing Speedy Fred.
Speedy Fred gets a 10-yard start.
Fred runs at 90% of Joe's speed.
By the time Joe reaches the point at which Fred started, Fred is 9 yards in front.
By the time Joe reaches there, Fred is 9/10's of that distance in front.
Logically, joe can never catch Fred, he can only get .999999.... of the way.
This is of course, nonsense, and it turns out he does pass him.
Because, as it happens, .999... is equal to 1. It's just a different way of expressing it.
If it weren't, overtaking anything would be physically impossible.

How Doc_Brown and others can read something this idiotic and not correct your mistakes is a testament to the lack of integrity people have when they've "picked a side".

This is a perfect example of someone who is patting himself on the back for "understanding" something when he doesn't have a clue.

[john9blue]

How Doc_Brown and others can read something this idiotic and not correct your mistakes is a testament to the lack of integrity people have when they've "picked a side".

This is a perfect example of someone who is patting himself on the back for "understanding" something when he doesn't have a clue.

You didn't disprove him...

[TheProwler]

What you have shown in your first paragraph is not a proof that 9/9 = 0.999... It only demonstrates that when you are using repeating decimals (of non-zeroes), you are only writing a good estimate. 21/99 is exact. 0.212121... is not.

So let's look at another pattern (try to keep an open mind now people):
0.9 + 0.9 = 1.8
0.99 + 0.99 = 1.98
0.999 + 0.999 = 1.998
...
When does it stop looking like this:
0.99999999999999999999999999999999999999999999
+0.99999999999999999999999999999999999999999999
=1.99999999999999999999999999999999999999999998

and start looking like this:
0.999...
+0.999...
=2

This is actually an excellent question. I think it's one of the first really valid paradoxes you've presented (it certainly is more valid than claiming that 0.333... is less than 0.333...!). But the answer is exactly what you hinted at. The pattern ends at infinity. Let's express the sum of the two numbers in mathematical terms:
S_N = 2 * sum(n=1 to N, 9*10^(-n))
Hopefully my notation makes sense. I'm expressing 0.999...9 as a finite sum containing N decimal digits. We can simplify the sum to:
S_N = 18 * sum(n = 1 to N, 10^(-n))
Using the formulas for finite sums, this simplifies to:
S_N = 18 * (1/9) * (1 - 10^(-N)) = 2 * (1 - 10^(-N)).
That is an exact value for any N. Notice that for any N contained in the natural numbers, S_N < 2. Yet S_N converges to 2. The sum of the infinite series version is also exact and is equal to the limit of S_N as N->infinity, that is, 2.

That is a nice way of showing the equation for any N decimal digits. But I fear it is only to hide the fact that a pattern was noticed and it is going to be ignored.

What you have essentially shown is that as N approaches infinity, the limit of the equation is 2. I would never have argued against that. Just as I would never have argued against the fact that the limit of (1 - 10^(-N)) as N approaches infinity is 1.

I am arguing against the idea that N will ever reach infinity.

This is kinda funny:

Seriously, why would the pattern all of a sudden end at infinity?

Because it is impossible to arrive "at infinity".

He is trying to make a point against me....but he is actually defining the core problem when dealing with the concept of infinity.

If 0.999... never reaches an infinite number of decimal places, it will never equal 1.

Now I know, we are stating that there *are* an infinite number of decimal places. But the whole thing is ridiculous. Because the only time you or anyone will ever come across 0.999... is when it is either A) defined (by equation or simply declared), or B) calculated in err due to dealing with repeating decimals instead of fractions (as was done in the "proofs" presented earlier). It isn't real (in the non-number set sense of the word).


Doc_Brown, I find it interesting how you are willing to write several hundred words to me to get me to agree with the idea that 0.999... equals 1. Yet when there are obvious errors made by people who "have been assimilated", you are willing to overlook them.

When you taught, and you knew that some of the students were right out to lunch, but they sat their nodding their heads in agreement; even when you knew they really didn't understand, were you satisfied as long as they said they understood?

[TheProwler]

How Doc_Brown and others can read something this idiotic and not correct your mistakes is a testament to the lack of integrity people have when they've "picked a side".

This is a perfect example of someone who is patting himself on the back for "understanding" something when he doesn't have a clue.

You didn't disprove him...

Who? jonesthecurl?

How about he comes up with his proof to show how he came up with 0.9999...? You know, with a somewhat proper solution; not just using vague wording.

Let's call Fred's speed F.

Joe's speed is J.

F = .9J

To simplify things, pick actual speeds that these two travel that will satisfy the 90% condition.


I can go on naming variables and solving the equation for the time that Joe catches up to Fred.

But I'd like to see jones show how he is getting 0.999...

"He goes 90% of the way, then he goes 90% of the next distance, and so on."

And he comes up with 0.999...!!! Hahahaha!!! Plug in some real numbers and you can see there is no 0.999... forming. First snapshot, Fred went 9 yards. The second snapshot, Fred went 8.1 yards. The next, he went 7.29 yards. So where's the 0.999... coming from?


We should all know that if we plot this on a graph, time on the x-axis and distance on the y-axis, we are going to just have two lines with different slopes. And they are obviously going to intersect if you start the slower slope at a greater distance than the faster slope.

His explanation is bloody ridiculous!

"This is of course, nonsense, and it turns out he does pass him.
Because, as it happens, .999... is equal to 1. It's just a different way of expressing it.
If it weren't, overtaking anything would be physically impossible."

This has nothing to do with 0.999...!!!!

Does he not realize that his explanation of the "problem", he is only allowing smaller and smaller periods of time to elapse between taking snapshots? The first segment, Joe ran 10 yards. The next, 9 yards. The next, 8.1 yards. And so on. But he runs a consistent speed. So obviously we are dealing with smaller and smaller periods of time. And the periods of time will continue to get smaller and smaller.

We have two numbers that are of interest. The TIME that has to expire before Joe catches Fred (call that T). And the DISTANCE that Joe has to run before he catches Fred (call that D). As long as we continue to allow smaller and smaller segments of time to elapse...as long as we keep stopping to take measurements every time Joe goes 90% as far as the last time we stopped for a measurement...the limit of the time that will expire is T and the limit of the distance Joe has run is D. (As long as we are allowed to pause time to take our snapshots...ha!!)

So if we were mapping out these two lines on a graph, we'd just be slowing down...slower....slower....slower....slower....never allowing the two lines to intersect. Because we have to stop to take that snapshot.

But unfortunately for the artsy types that read this little "problem" and found it interesting, you can't slow down time. Oh no!!! Did I just say that?? I take it back!!! All of you can slow down time!!! You are so awesome!!!


So, john9blue, did you also think "Logically, joe can never catch Fred, he can only get .999999.... of the way."?