Is 0.999... Really Equal to 1? Exploring the Mathematical Proofs

[honestrosewater]

Leto,
I had conceptual problems with this too. Maybe I can help.
Conceptually, can you begin or end at infinity? Does that make sense?

If infinity at some point arbitrarily fills the gap then why can't .999 repeating = 2, or any other number? Certainly the repetition must continue to go on even after 1 is approached.

What do you mean when you say "infinity at some point"?

Happy thoughts
Rachel

 

[leto]

Conceptually, can you begin or end at infinity? Does that make sense?

No, you cannot. However, let's suppose something existed forever and never changed. I am able to observe that entity at a finite time in my existence. If I can do this, wouldn't that tie it to my concept of time regardless of how absurd it might be from the opposite point of view? Wouldn't I be able to observe it and be accurate in describing what it has and always will look?

What do you mean when you say "infinity at some point"?

I believe we have to break it to observe it.

 

[Integral]

I believe we have to break it to observe it.

Please make an effort to free your mind of physical objects when thinking of Mathematical infinity. Mathematical infinities do not exist in the the physical world, any Physical analogy will fail. You do not need to break anything to consider an infinite number of 9s after the decimal. Please go back to near the beginning of this thread where I posted a link to a Mathematical proof of this. Be sure to look at and read ALL 4 pages, not just the first.

 

[NateTG]

I don't doubt I may be wrong. What I will not do is mindlessly accept what someone tells me unless I understand the logic behind it.

Well, for rational numbers, bar notation can be defined to represent a particular fraction. It takes a bit of work, but it's quite easy to see that this approach leads to $$0.\bar{9}=1$$. Note that the decimal representations of all rational numbers eventually repeat, so they are all candidates for bar notation. (Although $$\bar{0}$$ is usually omitted when writing numbers.)

Now let's take a brief look at decimal notation for real numbers:

Real numbers are often written as unfinished decimals. A real number $$0\leq r \leq 1$$ might be written as:
$$0.d_1d_2d_3d_4d_5...$$
where each of the $$d_i$$ is a digit -- for now let's just use (0-9).

This decimal notation can be considered a shorthand for:
$$\sum_{i=0}^{\infty} \frac{d_i}{10^i}$$

Which leads to
$$0.9999... = \sum_{i=0}^{\infty} \frac{9}{10^i} = 1.0000...$$

Once again $$0.\bar{9}=1$$

Another option is to look at:
$$0.\bar{9} \times 0.\bar{9}=0.\bar{9}$$
but
$$0.\bar{9} \times x = 0.\bar{9}$$
and if you divide out by $$0.\bar{9}$$ you get
$$x=1$$
so now we have $$x=1$$ and $$x=0.\bar{9}$$. So either it's unsafe to divide by an unknown, or $$0.\bar{9}=1$$

Alternatively, let's take a look at the most likely reason that you wonder about $$1=0.9999...$$: the notion that each decimal sequence uniquely represents a real number in the usual representation. So, let's circumvent that notion with a slightly different approach to the real numbers that explicitly allows many different representatives for each real number:

(This may be a bit heavy)

Let's denote $$[x_n]=x_1,x_2,x_3,x_4...$$ to be a sequence of rational numbers. Then we say that $$[x_n]$$ is cauchy if given $$\epsilon > 0$$ there exists $$N$$ so that $$n_1,n_2 > N \rightarrow |x_{n_1}-x_{n_2}|$$.

Now, define $$\doteq$$ that $$[x_n] \doteq [y_n]$$ if the sequence $$x_1,y_1,x_2,y_2...$$ is cauchy. It's easy to show that $$\doteq$$ is an equivalence relation.

Then I can define the real numbers to be equivalance classes of these sequences under $$\doteq$$. Addition and multiplication is simply componentwise addition and multiplication. (I'm too lazy to prove that they work properly on equivalence classes right here.)

Now it's easy to see that a decimal representation of a real number $$0.d_1d_2d_3...$$ readily translates to a representative sequence $$[r_n]=\frac{d_1}{10}, \frac{10 d_1 + d_2}{100}, \frac{100 d_1 + 10 d_2 + d_3}{1000}...$$ that is cauchy.

Then the representations $$1.0...$$ and $$0.\bar{9}$$ lead to the sequenes $$1,1,1,1...$$ and $$0.9,0.99,0.999,0.9999,0.99999...$$ repsectivevely. But the sequence $$1,.9,1,.99...$$ is cauchy, so those two sequences represent the same real number. So $$0.\bar{9} = 1$$

 

[honestrosewater]

I agree with Integral, though Integral certainly doesn't need my agreement in order to be correct ;)

However, I know that reading a proof umpteen times will not necessarily make it any clearer. Understanding the proof should be your goal, but if you don’t understand it yet...

There is NEVER a point where you can stop the repetition and say that number = 1.

**The language really needs to be precise, and it isn't here, so we have to be especially careful and keep in mind that the conversation is informal and aimed at conceptual understanding.**

That said, the above quote is what I’ll try to get you to understand, because it doesn't seem like you actually understand it.

I understand that the repetitions never stop, however, the amount of places from the decimal point also never stop.

You don’t need the “however”, in fact, you need to get rid of the “however”. On the right side of the decimal point, every decimal place has a 9 in it. The “amount of places from the decimal point” is the same as “the repetitions”. See?

You are infinitely taking smaller and smaller steps closer to one. If infinity at some point arbitrarily fills the gap then why can't .999 repeating = 2, or any other number?

The gap is filled when infinity ends! :) And you already know that infinity doesn’t end. So the gap is never filled.

Another way of looking at it is that, if you are taking smaller and smaller steps, you never take a *last* step.

Note that, if the first step is .9, each successive step is 1/10 the size of the previous one, that is, multiply the size of the last step by 1/10 to get the size of the next step. 9/10, 9/100, 9/1000, 9/100000, ...
The size of each step is very important. Just getting smaller will not do, they have to get smaller by 1/10.

You cannot get to 2 because you never pass 1. You never pass 1 because the size of your steps is getting smaller by a very important amount, 1/10.

(Notice that you have already said "There is NEVER a point where you can stop the repetition and say that number = 1." You admit that .9 is less than 1. In order to get from .9 to 2, you must pass 1, since 1 is between .9 and 2. And so you must have assumed that the successive values 0.9, 0.99, 0.999, ..., "jump over" 1, or pass 1 without landing exactly on 1, or without being exactly equal to 1. Had you assumed that?)

You also never pass 1 because you never reach 1. This is the same as saying the gap is never filled or you never take a last step.

You can start at .9 then step to .99 then step to .999 and so on. Each step is closer to 1. And after each step, you only have to take a step “this size” to reach 1. But you can never take a step “this size” because you have to take a step 1/10 the size of your previous step, AND 1/10 the size of your previous step is always smaller than “this size”.
When you are at .9 you need to take a step of size .1 to reach 1.
But you can only take a step of size .09
See?

There is NEVER a point where you can stop the repetition and say that number = 1.

I hope that cleared up any misunderstanding you had so far.
The next thing to understand is why 0.9999... = 1.000...
Perhaps taking another look at Integral’s proof or reading through other's posts will make this clear.

Happy thoughts
Rachel

 

[leto]

honestrosewater, your reasoning for why .999.. =1 is my exact reasoning for why it isn't. I did fully understand all of the concepts you mentioned. I may not have been clear, but that was exactly what I was trying to express for why .999 would not equal 1. As you put it, "since the gap is never filled." How can the two be equal when the gap is never filled? I am actually kind of annoyed you walked me through my own reasoning, but I guess I didn't express it well enough.

I know that .999.. = 1 can be proven mathematically in many ways. I think my inability to grasp it is, as integral mentioned, I am thinking abstractly in physical terms. Although, the only value math has to me is in its ability to describe the physical world. I still don't understand why the two shouldn't correspond. Reading the link now.

Edit: I finally got done downloading acrobat so I could see the link. I've seen those proofs before and never qestioned whether it could be proven mathematically. I think I've mentioned I thought it was either the number system itself or my irrationality.

 

[honestrosewater]

Leto,
Are you faulting me for misunderstanding what you meant to express? (Please don't read any sarcasm into that question.) If so, I think that is unfair. I over-explained myself in order to avoid misunderstanding; it wasn't my intention to annoy you. I am sorry it made you feel that way.

To clarify- my comments were not aimed at convincing you that 0.999...=1.000...
"I hope that cleared up any misunderstanding you had so far.
The next thing to understand is why 0.9999... = 1.000..."

Do you still want to understand it? I will try to think of a way to explain why they are equal. Let me know if you're still interested :)

Happy thoughts
Rachel

 

[Hurkyl]

How do the decimals not correspond to the physical world?

Suppose I walk .9 meters, then .09 meters, then .009 meters, et cetera.
During this sequence of tasks, I've crossed every point that lies between my starting point and 1 meter from my starting point. (and nothing beyond)

From one point of view, I've traveled a distance of .9 + .09 + .009 + ... = .999... meters, as computed by adding up all of the distances I traveled during my tasks. From another point of view, I've traveled a distance of 1 meter, the size of the interval I've crossed. So how could suggesting .999... is unequal to 1 correspond to reality?

 

[quartodeciman]

If someone actually tries to step off .9 meter, then step off .09 meter, then step off .009 meter and so on, then people tend to think the someone will be unable to proceed this way after a while. On the other hand, stepping off 1 meter is no problem. Therefore, they tend to think of them as two different operations. Fractionations of 1 stride are conceived as distinct from the summation of smaller strides. That is where both the Achilles and the Dichotomy get off to a bad start.

 

[leto]

Leto,
Are you faulting me for misunderstanding what you meant to express? (Please don't read any sarcasm into that question.) If so, I think that is unfair. I over-explained myself in order to avoid misunderstanding; it wasn't my intention to annoy you. I am sorry it made you feel that way.

You did misunderstand what I meant to express.

You don’t need the “however”, in fact, you need to get rid of the “however”. On the right side of the decimal point, every decimal place has a 9 in it. The “amount of places from the decimal point” is the same as “the repetitions”. See?

The however was meant to separate the two patterns. It was my attempt at expressing there was a gap because, not only was it getting closer, it was getting closer in smaller and smaller steps.

The gap is filled when infinity ends! :) And you already know that infinity doesn’t end. So the gap is never filled.

I never had a problem with this concept. I was trying to express that there was a gap, and if infinity somehow filled the gap then why couldn't it continue from there? (Since there is no end.) I was trying to express why I don't think the two are equal, and didn't understand you could somehow admit there was a gap and still believe the two to be equal. This seems irrational to me. Feel free to continue your explanation.

Edit: I still see no flaw in my reasoning, but I just thought about it from a different perspective and I can understand how the two could be equal. I may still just be crazy, but it's beginning to seem like a paradox to me. The gap between the two is never filled, but there is no measurable distance between the two. I am satisfied now, but I am still happy to hear why I am wrong with my initial reasoning.

 

[honestrosewater]

You did misunderstand what I meant to express.

I agree, my point was the misunderstanding was not intentional :)

The gap between the two is never filled, but there is no measurable distance between the two. I am satisfied now, but I am still happy to hear why I am wrong with my initial reasoning.

I don’t know that your initial reasoning was wrong. I noticed some things that seemed wrong to me, but, as we’ve agreed, I misunderstood and you have explained yourself.

Once you accept 1) there is no last step, 2) each step is smaller than the last, and 3) each step always gets closer to 1, then the conclusion is that they are equal (because there is no smallest step).
The only thing left for me to explain was why that conclusion follows, but you seem to have gotten it already.

Have you taken another look at the proofs Integral provided?

Happy thoughts
Rachel

 

[leto]

I don’t know that your initial reasoning was wrong. I noticed some things that seemed wrong to me, but, as we’ve agreed, I misunderstood and you have explained yourself.

What about my gap with no measurable distance?

 

[honestrosewater]

What about my gap with no measurable distance?

It seems to say the same thing as my "steps" explanation, and I granted you this already.
There are of course problems with both because they are not precise. They are just conceptual devices which need to be refined and put back into the mathematical context of the original problem. The language we're using is not precise enough for the problem.

The biggest problem with saying "there is no measurable distance between the two" is that it is also true if the distance is infinitely large, as well as infinitely small. That's why I added the "smallest step" bit- just to be clear. I also prefer the "step" explanation because it involves the idea of order (<, >, =). But it still has problems.

BTW A good example of a language goof is when I said: "2) each step is smaller than the last." Because "last" is ambiguous here, I should have said "each succesive step is smaller than the previous step" or something.

Happy thoughts
Rachel

 

[NateTG]

What about my gap with no measurable distance?

It's unclear whether you mean measurable in the technical sense, or you mean that the 'length of the gap' is zero.

You are indeed correct in your assessment that $$0.\bar{9} = 1$$ is in some sense an artifact of the real numbers. There are more exotic number systems where it's possible to have distinct numbers that are adjacent zero, but these number systems are more difficult to deal with than the real numbers.

To give you an idea of the kinds of problems this leads to, consider, for a moment, the following:

Let's assume, for a moment that $$0.\bar{9}$$ is not equal to 1. Then $$\frac{0.\bar{9}}{2} = 0.4\bar{9} \neq 0.5$$.
Now, if you convert this into base 3, you get $$0.\bar{9}_{10}=0.\bar{2}_3$$ and $$0.5_{10}=0.\bar{1}_3$$ and $$0.4\bar{9}_{10}=\frac{0.\bar{9}_{10}}{2_{10}}=\frac{0.\bar{2}_3}{2_3}=0.\bar{1}_3$$ so we have $$0.4\bar{9}_{10} \neq 0.5_{10} \rightarrow 0.\bar{1}_3 \neq 0.\bar{1}_3$$

 

[Grizzlycomet]

I spoke to my math teacher about this issue today, and he was quite firm in his belief that $$0.\bar{9}$$ does not equal 1. He didn't provide any proof though.

 

[BoulderHead]

I spoke to my math teacher about this issue today, and he was quite firm in his belief that $$0.\bar{9}$$ does not equal 1. He didn't provide any proof though.

The public should understand education. And it would do no harm if teachers and professors understood it, too.
-Hutchins

 

[Grizzlycomet]

The public should understand education. And it would do no harm if teachers and professors understood it, too.
-Hutchins

Indeed. I am planning to make a small document with various proofs that 0.999... does equal 1. Any links anyone could provide would be most helpful.

 

[quartodeciman]

I spoke to my math teacher about this issue today, and he was quite firm in his belief that $$0.\bar{9}$$ does not equal 1. He didn't provide any proof though.

It would be a good idea to first ask teacher what endless repeating decimal expressions actually mean, inasmuch as addition is a finite (originally binary) operation between numbers.

 

[Grizzlycomet]

It would be a good idea to first ask teacher what endless repeating decimal expressions actually mean, inasmuch as addition is a finite (originally binary) operation between numbers.

Further discussion with my teacher would indeed be a good idea. However, my school year ends this week so there's not exactly a lot of time for this.

 

[TENYEARS]

I was posting on the math thread before it was closed. .99999 can only be 1 if indeed it was generated by the division of three equal parts of a whole. It would preclude you knowing that the .999999 was generated by this act and then could be equated to one. Other wise no one has a right to make .999999 = 1 because it is not so and non relative at this point. So it could have been generated by three parts of a whole or it was a selected number. If it is just a selected number it is not equal to 1.

 

[ahrkron]

Tenyears, as I told you in the math thread, the fact that 0.999...=1.00... is not a matter of convention, "acceptance", or authority. It is a logic inevitability from the definitions of real numbers. The issue is not controversial at all among professional mathematicians, and is based on solidly established branches of math (in particular, Real Analysis).

 

[europium]

quick random q
where do you get all your mathematical symbols from on the keyboard... thanks

K_

 

[Grizzlycomet]

quick random q
where do you get all your mathematical symbols from on the keyboard... thanks

K_

The matematical symbols is created using a code in the forum knows as Latex. It's usage is described in This Thread

 

[Njorl]

I am firmly in the .999...=1.0 camp, where 1.0 equals the real number one. Is it allowable to say that the real number one is equal to the integer number one? Is this like mixing apples and oranges (one apple does not equal one orange)?

I feel that the integer one does equal the real one, but I don't know of a rigorous logical foundation for it. Is there one? Is it just defined to be so? Am I wrong?

Njorl

 

[honestrosewater]

I am firmly in the .999...=1.0 camp, where 1.0 equals the real number one. Is it allowable to say that the real number one is equal to the integer number one? Is this like mixing apples and oranges (one apple does not equal one orange)?

I feel that the integer one does equal the real one, but I don't know of a rigorous logical foundation for it. Is there one? Is it just defined to be so? Am I wrong?

Njorl

I have only seen two contructions of R (Dedekind cuts & Cauchy sequences) and both construct R from Q. And Q is defined by Z and Z by N. So I would think that is rigorous logical foundation, but what do I know?

Happy thoughts
Rachel

 

[TENYEARS]

I don't care what proof you come up with.

1 - I take a geometric object a cicle or a square and divide it into three parts and then add the decimal values the number is .999999 but is 1 relative to the object as a whole. This is correct for it is the totality of the object.

2 - I take the number .9999... out of the blue with no reference to a geometric representation of an object this is not 1. If I take a geometric reference to the universe, divide it into three equadistant rays starting from a central point and extending into infinity running a wall of ray for the length of extension so you have three defined parts, make that a decimal value of .333... added together, then I will equate .99999... to one with repsect to the universe.

If there is no geometric reference, and it is just a number, it is not equal to 1.

 

[NateTG]

Really, this seems like a funky philosophy of math question but, there is an obvious subset of $$\Re$$ that is isomorphic to the integers with the appropriate operations. As long as you think of it more as an instatiation of the integer 1 rather than as the only integer 1, you should be fine.

I have only seen two contructions of R (Dedekind cuts & Cauchy sequences) and both construct R from Q. And Q is defined by Z and Z by N. So I would think that is rigorous logical foundation, but what do I know?

You can also construct the real numbers from things like the set of all countable sequences of zeroes and ones that do not end in reapeating 1's, or something, but a construction like that has a PITA factor while Cauchy sequences and Dedekind cuts can readily be shown to have the desired properties. I bring this up because it allows for $$0.\bar{1}_2$$ <binary notation>, not to be a real number.

I expect that, the (mistaken) notion that $$0.\bar{9}$$ and $$1$$ are distinct is a result of the mistaken assumption that decimal representations are unique.

 

[honestrosewater]

I don't care what proof you come up with.

I think you are giving ahkron too much credit.
If ahkron has seen further it is by standing on the shoulders of Giants.

Now, if you cut a giant into 3 parts... only kidding, in good fun

Okay, now I can't help myself ;)

"If I have not seen as far as others, it is because giants were standing on my shoulders." -- HalAbelson

"In the sciences, we are now uniquely privileged to sit side by side with the giants on whose shoulders we stand." -- GeraldHolton?

"If I have not seen as far as others, it is because I was standing in the footprints of giants"

"If I have seen farther than others, it is because I was standing on a really big heap of midgets." -- EricDrexler (Nice for those of us who believe the inspiration of giants isn't the only engine of progress.)

"If I have seen further than others, it is because I was surrounded by dwarves." -- attributed to MurrayGellMann?, possibly maliciously.

"I cannot see very far, because my eyes are full of midgets."

 

[honestrosewater]

You can also construct the real numbers from things like the set of all countable sequences of zeroes and ones that do not end in reapeating 1's, or something, but a construction like that has a PITA factor while Cauchy sequences and Dedekind cuts can readily be shown to have the desired properties. I bring this up because it allows for $$0.\bar{1}_2$$ <binary notation>, not to be a real number.

Great, now I'm confused too Is there a quick way to explain how that construction proceeds? Oh, countable is a clue methinks. No, the set is countable? Or the sequences are countable? Yeah, What is this PITA factor you speak of? (I get the PITA, but what is it?)

 

[Hurkyl]

Basically, you define all the arithmetic operations by the method of elementary school arithmetic, but the trick is that you have to perform addition from left to right. The "PITA" factor is in cases where 'ambiguous'
whether you should have a carry or a borrow when doing an operation.

E.G. when adding 0.1100... and 0.011000..., you can "look ahead" to see that the second place to the right of the decimal point generates a carry, and it's propagated through the next place, so you can set the one's digit to be a 1, and then so on.

However, when adding 0.101010... and 0.010101..., there's nothing to say that there should or should not be a carry. Thus, you make a definition; you either say that in this situation you will always consider there to be a carry, never consider there to be a carry, or define both options as being equal.

 

[honestrosewater]

However, when adding 0.101010... and 0.010101..., there's nothing to say that there should or should not be a carry. Thus, you make a definition; you either say that in this situation you will always consider there to be a carry, never consider there to be a carry, or define both options as being equal.

1) carry-> 1.000...
2) no carry-> 0.111...
3) equal-> 1.000...=0.111...

Yes? No? So why can .111... not be real? I am still missing something; hopefully I will see clearer after I some shuteye.

Happy thoughts
Rachel

EDIT- Oh duh- if you decide to carry, you cannot ever get the "noncarry" number, and vice versa.

 

[Swigs]

My understanding of infinite numbers is that you can not treat them like any other number. Like what 'tenyear" was talking about.

Infinite numbers are just a mathimatical idea and don't really exist in the real universe as we know it. I thought that is why "limits" were created anyway.

 

[mikesvenson]

infinity, as I've been thinking about it, is whithin every real number as a result of mathamatical processes. e.g, divide to infinity, add to infinity, subtract to infinity, ...etc. Even whithin 0, there is infinity.

Infinity is the true nature of all things from greatly massive to micro. "Limits", are just easier ways of looking at things and are nothing more than a generalization of reality.

 

[lvlastermind]

infinity, as I've been thinking about it, is whithin every real number as a result of mathamatical processes. e.g, divide to infinity, add to infinity, subtract to infinity, ...etc. Even whithin 0, there is infinity.

every real number has an infinite number of digits but no real number has an infinite magnitude. e.g. ...000000001.0000000...

 

[mikesvenson]

it has an infinite divisible magnatude