# How to represent a fraction infinitely close to 1?

## Main Question or Discussion Point

As a mathematical novice I have been following with great interest the recent debates (in more than one forum!) as to whether 0.999... = 1.

The question is whether two ostensibly different natural numbers are equivalent, equal and synonymous. (I quote the three terms because mathematicians and philosophers tend to attach different meanings to each, so I am just trying to cover all the bases). Now, there are few things in mathematics more fundamental than the series of the natural numbers, so, if we want to prove that two natural numbers are equal, the proof is unlikely to reside in a complex equation (unless we were resorting to a reductio ad absurdam, which is not the case which has been argued here). Our proof has to be expressed in terms which are logically antecedent to the terms in question. Since the series of natural numbers is established by definition, the only available logical antecedents are the primitive propositions and ideas (as presented, eg, by Peano).

This means we are effectively reduced to arguing in terms of the axioms of arithmetic. Within that context, the proposition that 0.999... = 1 obviously has difficult implications (especially to a layman like myself!). I'd like to ask some perhaps naive and obvious questions:

(1) Since we cannot use 0.999..., what is the correct method of representing a fraction which approaches infinitely close to 1?

(2) What is the product of 0.999... x 0.999...? Putting it another way, if 0.999...9 = 1, does that imply that 0.999...8 = 0.999...9?

(3) The number 0.999... consists of a decimal point followed (in this case) by an infinite series composed of the digit 9. Since this can be more economically represented - in fact, can ONLY be accurately represented - by the digit "1", would I be right to assume that the number 0.999... is logically redundant and meaningless, and should be replaced by "1" in every case? (vide Occam's Razor). Is there any conceivable situation, outside of this discussion, where 0.999... would be logically required?

(4) Can the principle be extended to every fraction, the final term of which is a single recurring digit (eg 0.1999... = 0.12)?

(5) How is the principle to be generalised to the case where the recurrent is not a single digit, but a group of digits (eg 0.13131313...)?

(6) '1' is a member of the set of whole numbers. How should we define 'whole number' in such a way that it includes some numbers which begin with '0.9'?

(7) The proposition entails, prima facie, the axiom that two different natural numbers may have the same value; is this axiom generally accepted?

These questions are posted in good faith, by one who has no mathematical aptitude, but an intense interest in mathematical philosophy... so if you could try to express your replies in lay terms, as far as is reasonable? I mean the kind of terms that Bertrand Russell would have used for a lay reader.

As a mathematical novice I have been following with great interest the recent debates (in more than one forum!) as to whether 0.999... = 1.

The question is whether two ostensibly different natural numbers are equivalent, equal and synonymous. (I quote the three terms because mathematicians and philosophers tend to attach different meanings to each, so I am just trying to cover all the bases). Now, there are few things in mathematics more fundamental than the series of the natural numbers, so, if we want to prove that two natural numbers are equal, the proof is unlikely to reside in a complex equation (unless we were resorting to a reductio ad absurdam, which is not the case which has been argued here). Our proof has to be expressed in terms which are logically antecedent to the terms in question. Since the series of natural numbers is established by definition, the only available logical antecedents are the primitive propositions and ideas (as presented, eg, by Peano).

This means we are effectively reduced to arguing in terms of the axioms of arithmetic. Within that context, the proposition that 0.999... = 1 obviously has difficult implications (especially to a layman like myself!). I'd like to ask some perhaps naive and obvious questions:

(1) Since we cannot use 0.999..., what is the correct method of representing a fraction which approaches infinitely close to 1?
I can never really make sense of sentences like this. What do you mean with "fraction approaching infinity?" A fraction is, and can never approach anything. It's the same as saying that 1 is approaching 2: it doesn't make any sense!

(2) What is the product of 0.999... x 0.999...? Putting it another way, if 0.999...9 = 1, does that imply that 0.999...8 = 0.999...9?
0.999...x0.999... =2.
And there is no such thing as 0.999...8, because that would imply an infinity of 9's and then an 8. But you cannot make sense of such a thing!

(3) The number 0.999... consists of a decimal point followed (in this case) by an infinite series composed of the digit 9. Since this can be more economically represented - in fact, can ONLY be accurately represented - by the digigt "1", would I be right to assume that the number 0.999... is logically redundant and meaningless, and should be replaced by "1" in every case? (vide Occam's Razor). Is there any conceivable situation, outside of this discussion, where 0.999... would be logically required?
No, 0.999.... is also an accurate representation of the digit. In fact, every decimal representation is accurate! And 0.999... is not at all meaningless. It is well defined as

$$0.999...=\sum_{k=1}^{+\infty}{\frac{9}{10^k}}$$

Just because some people don't like that it equals 1, doesn't mean that it's redundant or meaningless...

(4) Can the principle be extended to every fraction, the final term of which is a single recurring digit (eg 0.1999... = 0.12)?

(5) How is the principle to be generalised to the case where the recurrent is not a single digit, but a group of digits (eg 0.13131313...)?
Interesting question. Yes, in fact we can prove that the decimal representation of a number is unique EXCEPT for the numbers that end with a series of 9's or 0's.
For example 0.333... is the unique representation for 1/3. While 1.349999...=1.35.

Moreover, this situation occurs in EVERY number system. So even in binary we would have 1=0.1111... (thus instead of 9's, we work with 1's in binary).

(6) '1' is a member of the set of whole numbers. How should we define 'whole number' in such a way that it includes some numbers which begin with '0.9'?
The "whole numbers" (I think you mean the integers) will never include 0.9 or 0.99... It does, however, include 0.999....

(7) The proposition entails, prima facie, the axiom that two different natural numbers may have the same value; is this axiom generally accepted?
No, two different numbers cannot have the same value. The value of 1 and 0.999... are equal. However, two equal numbers can have other representations. This occurs in other situations too: for example 1/2=2/4, these are two numbers which are equal but have different representation. But somehow, that last example doesn't bother people as much...

I hope you are something with this. Ask if you want to know more!

Please also read https://www.physicsforums.com/showpost.php?p=3209034&postcount=35 by Frederik, which I think is one of the best posts on the topic.

The point is that 1=0.999... because we want it that way. This way, every thing which should be true, is true. If we would make a number system in which 1 and 0.999... are not equal, then this is possible, but the resulting number system will not be useful and will not have nice properties.

"I can never really make sense of sentences like this. What do you mean with "fraction approaching infinity?" A fraction is, and can never approach anything. It's the same as saying that 1 is approaching 2: it doesn't make any sense!"

Micromass, I think we should both pretend you didn't really write that! You know I said no such thing. The general concept of a series of fractions which approaches infinitely close to 1 is lifted from Russell, though offhand I can't quote a reference for you. The notion is that a fraction like 63/64 is closer to 1 than it is to 0, and 127/128 approaches closer still, and 255/256 still more.

Seriously though, thanks for your feedback. I'm a latecomer to mathematics and I have next to no formal training, which I guess lays me open to the accusation of wasting the time of guys like yourself, but I thank you for setting out some directions for me.

"The point is that 1=0.999... because we want it that way."

This does confirm a conclusion which seems to me inescapable: the truth-value of every mathematical proposition depends critically upon the choice of axioms. The history of the development of geometry in the 19th Century provides the definitive case study.

Meanwhile my original question remains unanswered! How would I represent a fraction which approaches infinitely close to 1 (assuming 0.999... is not allowable)? I trust you wouldn't deny that there is an infinite number of fractions between 0 and 1? Can you provide more specific answers for some of the other questions I raised?

gb7nash
Homework Helper
Meanwhile my original question remains unanswered! How would I represent a fraction which approaches infinitely close to 1 (assuming 0.999... is not allowable)?
What does this mean? A fraction is the ratio of two numbers. There's no such thing as a fraction that's infinitely close to a number. However, you can rewrite a fraction to represent the same number, i.e. 1 = .999... = (.999....)/1 = (1.999....)/2

I trust you wouldn't deny that there is an infinite number of fractions between 0 and 1?
There are an infinite number of numbers between 0 and 1, but this doesn't mean there's a fraction infinitely close to 1! Since if I assume there is, say a/b, by the midpoint formula, there exists the fraction:

$$(a/b + 1)/2 = (a + b)/2b$$

between the fraction and 1.

"I can never really make sense of sentences like this. What do you mean with "fraction approaching infinity?" A fraction is, and can never approach anything. It's the same as saying that 1 is approaching 2: it doesn't make any sense!"

Micromass, I think we should both pretend you didn't really write that! You know I said no such thing. The general concept of a series of fractions which approaches infinitely close to 1 is lifted from Russell, though offhand I can't quote a reference for you. The notion is that a fraction like 63/64 is closer to 1 than it is to 0, and 127/128 approaches closer still, and 255/256 still more.
Ah, I understand now! You didn't specified that you talked about a sequence of fractions. So I tought you talked about a fraction which approaches a number, which would make no sense. But indeed, a sequence of fractions approaching a number makes perfect sense!

Seriously though, thanks for your feedback. I'm a latecomer to mathematics and I have next to no formal training, which I guess lays me open to the accusation of wasting the time of guys like yourself, but I thank you for setting out some directions for me.
Don't worry, I think everybody who answers in this thread will be quite open to questions from laymen. The only problem with people discussing 1=0.999... is that they often are not willing to see the mathematician's point-of-view. This is what makes the entire discussion frustrating.

"The point is that 1=0.999... because we want it that way."

This does confirm a conclusion which seems to me inescapable: the truth-value of every mathematical proposition depends critically upon the choice of axioms. The history of the development of geometry in the 19th Century provides the definitive case study.
I think you made a very good observation here. Whether a statement is true depends on the axioms. However, it doesn't tell the entire story. You must also take into account why we have the axioms that we have. I don't know if you're aware about ZFC, but ZFC is the most commonly-used set of axioms in mathematics. But we didn't choose the ZFC-axioms for fun! We chose them because we think that they represent the real life in a good way, or that they at least approximate real life.
It's thesame way with Euclid's axioms. He chose those axioms because he tought they were true in real life. With Einstein, it was discovered that this is not the case. Thus other axioms were proposed which did confirm to reality. However, Euclid's axioms are still used because they form an approximation to reality.

Mathematicians do not (or rarely) choose axiom systems if they are useless and do not represent something in real life. These things must be taken into account when discussing the axioms of mathematics. That things are true depend crucially on the axioms, however it is still the mathematician's hope that the axioms represent something in reality!

Meanwhile my original question remains unanswered! How would I represent a fraction which approaches infinitely close to 1 (assuming 0.999... is not allowable)?
This is exactly the thing I don't like to see. I know what you mean, but it makes no sense that a fraction approaches 1. You should talk about sequences of fractions, then it makes sense.

Giving a sequence of fractions which come close to 1 is easy:

$$0.9,~0.99,~0.999,~0.9999,~\hdots$$

But I'm not sure if this is what you want.

I trust you wouldn't deny that there is an infinite number of fractions between 0 and 1? Can you provide more specific answers for some of the other questions I raised?
Sure, in what questions was I not specific enough?

gb7nash
Homework Helper
He's talking about a sequence of fractions? I'm completely off.

He's talking about a sequence of fractions? I'm completely off.
No, I don't really get it as well. I'm just waiting for some more clarifications...

Mark44
Mentor
Something like this:
$$\{\frac{n-1}{n}\}_{n=1}^{\infty}$$

HallsofIvy
Homework Helper
Mark44 has given a sequence of fractions that approach 1. Unfortunately, in his first post Allen1000 asked about "representing a fraction which approaches infinitely close to 1".

That is, again, a fraction that is in some way approaching 1. As micromass told him, that is non-sense. A fraction is not "approaching" any thing- it just is.

God, I hate the 0.999...=1 thing, people make it more complicated than it is.

If we are considering these "numbers" as possibly infinite sequences, then yes, the two are different. If we are considering the two as REAL NUMBERS, we must ask:

"what does 0.999.... mean?"

Well, by the way we construct real numbers, the expression 0.999.... represents the unique real number which is the limit of the sequence 0,0.9,0.99,0.999,... This is the definition of what the number 0.999... is, there is nothing philosophical here. The above limit is 1, so 0.999.... is just another representation of the real number 1.

It's literally that simple, there is nothing complicated or philosophical going on at all.

Oh, and to answer your question in the title "How to represent a fraction infinitely close to 1?" how about 1/1?

gb7nash
Homework Helper
Oh, and to answer your question in the title "How to represent a fraction infinitely close to 1?" how about 1/1?
Unfortunately, the question the OP asks has no answer. There's no such thing.

Hurkyl
Staff Emeritus
Gold Member
the truth-value of every mathematical proposition depends critically upon the choice of axioms.
Yes!

You say this like it's a fact we grudgingly accept because there's no other choice, but this is basic logic. You can't assert the conclusion of a theorem, except when the hypotheses are satisfied. A theorem of group theory probably isn't true when applied to things that aren't groups. And so forth.

Some technicalities, though.

If we don't include logic itself in "the choice of axioms", then the truth value of tautologies and contradictions do not depend upon the choice of axioms. Theorems relative to a particular set of axioms can always be converted into a tautology. For example, the theory of Euclidean geometry proves
The angle sum of a triangle is 180 degrees​
and we also have a tautology:
If P is a model of Euclidean plane geometry, then the angle sum (in P) of any triangle (in P) is 180 degrees (in P).​
This last statement is simply true -- it's a tautology. However, it's not a very useful tautology unless we are studying something that is a model of Euclidean plane geometry, or we are studying a triangle whose angle sum is not 180 degrees.

Also, "the choice of axioms" isn't quite as relevant as it might sound. One thing that people often don't understand is that axioms are not part of a mathematical theory -- they are part of a way to present a mathematical theory.

It doesn't actually make any sense to say that "two distinct points determine line" is an axiom of Euclidean geometry. It does, however, make sense to say that it is an axiom of Hilbert's axiomatization of Euclidean geometry.

Hurkyl
Staff Emeritus
Gold Member
Unfortunately, the question the OP asks has no answer. There's no such thing.
Actually, 1/1 is infinitely close to 1. It's the only rational number with that property.

gb7nash
Homework Helper
Actually, 1/1 is infinitely close to 1. It's the only rational number with that property.
My whole foundation is broken. How is this true?

Hurkyl
Staff Emeritus
Gold Member
My whole foundation is broken. How is this true?
The first statement, or the second one?

The first one is because 0 is an infinitesimal real number. (of course, it is not a nonzero infinitesimal)
The second one is because 0 is the only infinitesimal real number. This is a restatement of the Archimedian property.

I don't understand the issue. If .99... does not equal 1, there must be some number in between. Nothing is in between so .99... = 1.

Why is this an issue?

People's confusions with this are always the same- they agree that 0.999... is infinitely close to 1 but say that it never quite gets there, as if the number 0.999... is in some sort of unstatic state, approaching something. This is not how you should view things, 0.999... is just a number like any other, whose value, by the definition of the notation of a never ending decimal, is the UNIQUE limit of the sequence 0,0.9,0.99,0.999,....

I'm curious about how axioms have been established and have changed with time. Are there any good books or articles on this subject that someone might recommend?