# Axioms, definitions, and 0.999...=1

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42 minutes ago, OldChemE said:

Again-- it is a Limit, not an equality.

Even in high school mathematics it is not a limit it is a sum.

There is a considerable difference.

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5 minutes ago, amplitude said:

The clear implication, though as far as I know Cantor doesn't appeal to it directly, is that the first number would be 0.000...1 and the last number would be 0.999... (0.000...1 being what you are left with, if you subtract 0.999... from 1, according to the axioms of arithmetic).

But that number doesn't exist. You can't have a never-ending sequence of numbers and then have a number at the end. It is paradoxical. That's why reals can't have successors.

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26 minutes ago, Strange said:

Sadly, there is no Nobel Prize for mathematics.

Then let me fix that: they are the same number, just written in different ways.

I suspect you are confusing "number" with "absolute value".  The number 1 is a familiar natural number;  0.999... is unintelligible without a coherent definition of what constitutes an infinite set.

Isn't there a Nobel Prize for mathematics?  That seems unfair.

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6 minutes ago, amplitude said:

I suspect you are confusing "number" with "absolute value".

Or are you confusing a number and its representation?

Quote

0.999... is unintelligible without a coherent definition of what constitutes an infinite set.

Well, we have a formal definition of infinity. Although, even an informal definition of infinity will show that the two represent the same number. As has already been pointed out, the difference between 1 and 0.999... is an infinite sequence of zeroes followed, according to some, by a 1. But if there are an infinite number of zeroes, then there cannot be a 1 at the end - wherever you attempt to append the 1, another zero can be inserted. Therefore there is no 1 and therefore the two numbers are equal.

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17 minutes ago, amplitude said:

t's worth pointing out, perhaps, that the decimal point is itself a red herring... it could occur at any position within the number, or there could be no decimal point at all;  the essential problem in mathematical philosophy remains unaltered.  So it doesn't matter if the equation is 0.999...=1, or 999...=1{000}... (pls allow for my probably-erroneous symbolism here).

Not really.

The sum could be divergent for numbers greater than 1.

The rules for manipulating infinite series only allow convergent series.

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12 minutes ago, Lord Antares said:

There are no successors to any real numbers. 0.99999... is no different than, say, 7 in that regard. Which real number comes after 7?

I would answer, if we stipulate  positive integers, then the real-number successor of 7 is 8.  Just as 8 is the natural-number successor of 7, according to the definitions of arithmetic, so +8 is the successor of +7 in the line of real integers.  That is not a matter of proof;  it is a matter of definition.

On the other hand, if we admit all of the possible numbers in the number plane, then the successor of 7 might be 7+h (h being a more respectable way to write 0.000...1).

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13 minutes ago, amplitude said:

On the other hand, if we admit all of the possible numbers in the number plane, then the successor of 7 might be 7+h (h being a more respectable way to write 0.000...1).

And therefore, because your h doesn't exist (or, at least, it is zero), there is no successor (or 7 is its own successor).

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34 minutes ago, amplitude said:

I would answer, if we stipulate  positive integers, then the real-number successor of 7 is 8.  Just as 8 is the natural-number successor of 7, according to the definitions of arithmetic, so +8 is the successor of +7 in the line of real integers.

No, it isn't really. Why would you reduce the set of real numbers to the naturals? It is uncalled for. There are infinitely many numbers in between 7 and 8 (and between any real numbers as well).

36 minutes ago, amplitude said:

On the other hand, if we admit all of the possible numbers in the number plane, then the successor of 7 might be 7+h (h being a more respectable way to write 0.000...1).

What you are trying to do is define an infinitesimal. But an infinitesimal cannot exist because, as has been noted, there cannot be an infinite amount of zeroes and then 1 at the end. If it ends in 1, then there MUST be a finite number of zeroes. Do you understand that a decimal number ending in any given number must, by definition, be finite?

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50 minutes ago, amplitude said:

I would answer, if we stipulate  positive integers, then the real-number successor of 7 is 8.  Just as 8 is the natural-number successor of 7, according to the definitions of arithmetic, so +8 is the successor of +7 in the line of real integers.  That is not a matter of proof;  it is a matter of definition.

On the other hand, if we admit all of the possible numbers in the number plane, then the successor of 7 might be 7+h (h being a more respectable way to write 0.000...1).

Yes, 8 would be the correct successor of 7.

But if (7+h) is a successor of 7 then so is (7+0.5h) and (infinitely) many others.

The point of successors are that they are unique so that

Every number has a successor, no two numbers have the same successor.

Two other axioms are necessary.

1 is a number

(I learned 1, modern treatments seem to use 0 which was originally avoided because it brings other difficulties with it)

1 is not the successor of any number.

These four axioms are enough to set up self consistent finite and infinite sequences (originally called successions).

Incidentally set theory is normally reserved until University in the UK and it is not needed to introduce sequences.

By the way to get the infinity (and other useful symbols) on Windows,

type 'charmap.exe' into the windows run/search box depending upon which version you have.

Select the infinity symbol
chose copy
Paste in the symbol

Selerct the symbol and change the font size to suit

∞     is 36

@Old Chem Engineer.

You find the reason why we need a sum ( including to infinity) to work with to replace an unending sequence is that the unending sequence has a value if the sequence is convergent.
If that is the case we can do arithmetic with the sequence, and justify term by term actions

Edited by studiot

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12 hours ago, OldChemE said:

Let us assume the hypothesis that 0.99.... = 1

Then, by the rules of mathematics, 1-0.99.... = 0, which is a definable number in mathematics.  Or, to say it differently, a defined result proper to mathematics.

if we subtract 0.9 from 1.0, the result is 0.1, which is 1/10, which is 1/(10)^1

if we subtract 0.99 from 1.00 the result is 0.01, which is 1/(10)^2

Generalizing, 1 - 0.99.....  = 1/(10)^Infinity

But the result of division by an infinite number is undefinable in mathematics.

Therefore, 1 - 0.9999......  produces an undefined result

This falsifies the assumption that there is a definable result of zero

This falsifies the  original hypothesis.

OK; why is division by zero forbidden?

The usual answer is that if you try to allow it, you get inconsistencies.

We all know this sort of thing...

x=2/0
thus
0 times x =2
thus
2 = zero.

OK, by the same argument

" 1/(10)^Infinity" is forbidden because, as you point out, trying to do ordinary arithmetic with infinities gives silly results

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3 hours ago, Lord Antares said:

Those are not ''facts''. Those are just hypotheses or wild guesses. The majority consensus is not a fact. You need to understand what a fact is. Nothing (afaik) in science outside of math is considered factual. The highest degree is a theory, like relativity or evolution, hence the remodelling and re-thinking.

Also, I don't know what a greek elipse is. Google shows zero results.

I think this is the best example for what the OP is looking for, described with words and not symbols. There must be an infinite amount of real numbers between any two numbers. Seeing how no defined number is in between 0.999... and 1, one must conclude that 0.999 = 1.

Incorrect...

It is a FACT that it was ONCE considered a FACT the world was flat

It is a FACT that it was ONCE considered a FACT that Greek Ellipses where a model for the solar system

Aristotle thought the world was flat...for a FACT.  Socrates thought the ellipses proves the movement of the solar system...for a FACT.

Facts are nothing more than a objective decision by the majority that a particular subjective truth holds true.

FACTS change...even in science....

That said I like your response to the op...as such I will give you a +1....despite your trollishness.....

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53 minutes ago, conway said:

It is a FACT that it was ONCE considered a FACT the world was flat

I'm not sure that is true: https://en.wikipedia.org/wiki/Myth_of_the_flat_Earth

55 minutes ago, conway said:

It is a FACT that it was ONCE considered a FACT that Greek Ellipses where a model for the solar system

Do you mean epicycles?

But these examples are not really relevant because we are talking about mathematics. If something is true in mathematics then it is always true.

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This will be my last post on this topic.  I'd like to thank everybody for their input, I really appreciate it.  But it's obvious that we are not making progress with regard to my original question about developing a proof that 0.999...=1 by working upwards from the axioms and definitions, rather than working downwards from mid-level theorems.  You have all greatly helped me to clarify my own ideas in a number of important ways, but I've always believed in the principle that a thread should be terminated before it wanders off into areas of total irrelevance!

I might say that my interest in this question arises from the fact that another equally-simple proposition, namely 1+1=2, can be easily proved by arguing directly from the axioms and definitions, like so:

1 + 1 = 1+ S(0) = S(1 + 0) = S(1) = 2

Where the function S() represents "the successor of".

Of course, a great deal of creative insight went into making that argument possible, but the point is, it requires no knowledge of mathematics beyond an understanding of axioms, definitions, and logical operators.  That we cannot do the same for 0.999...=1, I assume, is connected with the fact that we cannot define the meaning of 0.999... without a prior definition of what constitutes an infinite set, which of course takes us into a whole alternative

1 hour ago, Lord Antares said:

But that number doesn't exist. You can't have a never-ending sequence of numbers and then have a number at the end. It is paradoxical. That's why reals can't have successors.

1 hour ago, Lord Antares said:

But that number doesn't exist. You can't have a never-ending sequence of numbers and then have a number at the end. It is paradoxical. That's why reals can't have successors.

Sorry, but I've just revisited this post, and I can't resist sticking in my two cents' worth once more!

The idea that an infinite set cannot have finite limiting terms is a common misconception, which arises perhaps from the equally-common misconception that "infinity" can be defined as "a number greater than any natural number".

That is an inevitable property of any infinity, you might even say it's a criterion for identifying an infinity, but it's not a definition in the sense required by  mathematics.

There is an infinite number of different-sized infinities, which the foregoing "definition" would clearly fail to account for.  Furthermore, it purports to define "infinity" in terms of the natural numbers, which is a serious logical error.

An infinity may terminate in a finite number (eg the set of negative integers which terminates with the number -1), or may begin with a finite number (eg 0, the starting point for the set of the natural numbers) or may both begin and end with finite numbers (eg the set of numbers which are >=0 and <=1).  It is easy to prove that this last set is a much bigger infinity than the infinite set of the natural numbers, nevertheless, paradoxically, when you have listed all of the numbers >=0 and <=1, you still have not listed all of the numbers >=0 and <=1.  To do that, you need to calculate the power set of the numbers >=0 and <=1, which has the cardinality of 2 to the power of infinity.  But only God (or some such being) could do that...

Goodnight, everybody, and thank you again for a most rewarding debate.

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14 minutes ago, amplitude said:

This will be my last post on this topic.  I'd like to thank everybody for their input, I really appreciate it.  But it's obvious that we are not making progress with regard to my original question about developing a proof that 0.999...=1 by working upwards from the axioms and definitions, rather than working downwards from mid-level theorems.

It wasn't quite sure what you were looking for. I was going to comment on John's very first reply where he referred to "elementary school maths" to point out that that was what you mean by "mid-level mathematics" but I wasn't sure if that was what you meant.

Anyway, obviously it is possible to derive all the properties real numbers from first principles. It would however, be a huge amount of work. You might want to use an automated theorem proving system to help. But there really isn't any point.  0.999... is the same as 1.

21 minutes ago, amplitude said:

That is an inevitable property of any infinity, you might even say it's a criterion for identifying an infinity, but it's not a definition in the sense required by  mathematics.

We are not dealing with infinities, so this isn't really relevant.

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1 hour ago, studiot said:

Yes, 8 would be the correct successor of 7.

But if (7+h) is a successor of 7 then so is (7+0.5h) and (infinitely) many others.

The point of successors are that they are unique so that

Every number has a successor, no two numbers have the same successor.

Two other axioms are necessary.

1 is a number

(I learned 1, modern treatments seem to use 0 which was originally avoided because it brings other difficulties with it)

1 is not the successor of any number.

These four axioms are enough to set up self consistent finite and infinite sequences (originally called successions).

Incidentally set theory is normally reserved until University in the UK and it is not needed to introduce sequences.

By the way to get the infinity (and other useful symbols) on Windows,

type 'charmap.exe' into the windows run/search box depending upon which version you have.

Select the infinity symbol
chose copy
Paste in the symbol

Selerct the symbol and change the font size to suit

∞     is 36

@Old Chem Engineer.

You find the reason why we need a sum ( including to infinity) to work with to replace an unending sequence is that the unending sequence has a value if the sequence is convergent.
If that is the case we can do arithmetic with the sequence, and justify term by term actions

Thanks for the tip about charmap.exe, Studiot!  I blush to think that after all these years of using Windows, I still wasn't aware of that...

As to the special difficulties which attach to the number 0 - material for a whole new thread there perhaps!

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2 hours ago, Strange said:

I'm not sure that is true: https://en.wikipedia.org/wiki/Myth_of_the_flat_Earth

Do you mean epicycles?

But these examples are not really relevant because we are talking about mathematics. If something is true in mathematics then it is always true.

Very interesting article.  Perhaps I chose my examples poorly.  And yes I did mean epicycles...my apologies....I concede to you here as you are clearly correct....+1 for being nice about it....however I stand by my point that facts change. Including mathematics.  The very fact that math has evolved over time shows a "changing" of the "facts".

I will not take up further time from the op in this matter...my apologies.

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Folks

This has been a fun discussion.  There are a number of threads in different forums (sp?) on the internet where this is debated.  But, I think we've beat the subject to my satisfaction, I'll bow out here (not agreeing, but agreeing to disagree)

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amplitude:

A proof from definitions that 0.99999.... = 1.

The definition of the decimal 0.a1 a2 a3 ... where each of the ai is a digit from 0 to 9 is the limit of the series $\sum_{i = 1}^\infty \frac{a_i}{10^i}$. There is a relatively simple proof that any such series will converge, using the basic comparison test.

So 0.999999... is the limit of the series $\sum_{i = 1}^\infty \frac{9}{10^i}$.

Winding back one definition, that means that 0.999... is the limit of the sequence $\sum_{i = 1}^n \frac{9}{10^i}$

Using induction (I will provide the full proof on request), we can see that $\sum_{i = 1}^n \frac{9}{10^i} = 1 - 10^n$

Therefore, we are looking for $\lim_{n \rightarrow \infty} 1 - 10^n$

Using the epsilon-delta definition of limits, we can see that this limit is exactly 1 (again, proof provided on request).

Therefore, by using these definitions, 0.9999... = 1.

This is a proof from the definitions (although I have skipped two large steps). Is this what you wanted?

Edited by uncool

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7 hours ago, OldChemE said:

While it is a theoretical nit (perhaps), it should be noted that the infinite sum of 9/10 + 9/100 .... does not actually reach 1, it converges toward 1.[/quote]

And you are wrong about the "theory".  Saying that series "converges to 1" means that sum is equal to 1.  An infinite convergent sum has a single correct value, it is not "converging toward" that number.  You are confusing the infinite sum with its partial sums.  The partial sums of $$\sum_{n=1}^\infty \frac{9}{10^n}$$,  0.9, 0.99, 0.999, ---

Those partial sum converge to 1 so $$\sum_{n=1}^\infty \frac{9}{10^n}$$ because an infinite sum is defined as the limit of the sequence of partial sums, not the that sequence itself.

9/10 is 9 tenths of the distance on the number line from 0 to 1

9/100 is nine tents of the distance on the number line from 9/10 to 1

Every term in the infinite sum adds to the sum 9/10 of the remaining distance on the number line between its previous term and the total of 1.  Because bo term ever adds more than 9/10 of the remaining distance on the number line, we never actually reach 1.

This is the situation of the old puzzle about a person who in each unit of time walks exactly half the remaining distance to his/her destination.  With each succeeding term we add to the infinite sum we travel 9 tenths of the remaining distance to 1.  Again-- it is a Limit, not an equality.

and the infinite sum is defined to be that limit- so it is an equality.

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On 10/17/2017 at 8:02 AM, amplitude said:

So my question is:  is there any way by which we can mount an argument that 0.999...=1 by arguing "upwards" from the axioms and definitions of arithmetic, rather than "downwards" from mid-level mathematics?

If we know that some a does not equal b (let a > b), then there always exists some number c such that a > c > b.

If 0.99999... is not equal to 1, what is that number that comes between them?

I have never seen someone who 'doesn't believe' 0.99999... = 1 give any kind of meaningful answer to this.

Edited by Bignose

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On 10/17/2017 at 0:50 PM, OldChemE said:

The fallacy in 10x-x = 9x is that truly infinite numbers do not behave properly in mathematics.  In order to use mathematics for this you have to truncate the number someplace.  When you do that, and multiply a finite version of 0.999... by 10, then try to subtract the original 0.999..., you have a result a tiny bit smaller than 9 because the first digit is 8 and the last is 1.

for example:    lets use 0.999999999999999.  10 x = 9.99999999999999  (14 digits to the right of the decimal).  The original number, 0.999999999999999, has 15 digits to the right of the decimal.  When you do the subtraction, the result is 8.999999999999991, which is not 9.

Why would ever truncate a 9?

The proper practice would be to round it off to 10.

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13 hours ago, Bignose said:

If we know that some a does not equal b (let a > b), then there always exists some number c such that a > c > b.

If 0.99999... is not equal to 1, what is that number that comes between them?

I have never seen someone who 'doesn't believe' 0.99999... = 1 give any kind of meaningful answer to this.

I thought that was a great answer. Why on Earth would anyone give it a down vote!?

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3 hours ago, Strange said:

I thought that was a great answer. Why on Earth would anyone give it a down vote!?

A fit of pique no doubt (I've upvoted both posts for balance).

Edited by dimreepr

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19 hours ago, Bignose said:

If we know that some a does not equal b (let a > b), then there always exists some number c such that a > c > b.

If 0.99999... is not equal to 1, what is that number that comes between them?

I have never seen someone who 'doesn't believe' 0.99999... = 1 give any kind of meaningful answer to this.

I do think (and can prove) that 0.999... = 1, but in the mindset of someone who doesn't:

Most people who disagree think of 0.999... as a "process", based around the sequence 0.9, 0.99, 0.999, ... As such, I would hazard a guess that they would say the "number" given by the process 0.95, 0.995, 0.9995, ... would be between the two.

And yes, this does run into the problem of whether 0.9, 0.99, 0.999, ... is the same "process" as 0.99, 0.999, 0.9999, ..., but most people don't think that far.

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For the crowd that believes 0.999 = 1.

Does this mean 0.999 is useless?

And.

Does .0991 also equal 1?

This is not an argument I'm just curious. I have no idea who is right and I'm ok with that.