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## #1051 2012-07-01 22:51:32

bob bundy
Moderator
Registered: 2010-06-20
Posts: 7,742

### Re: 0.9999....(recurring) = 1?

hi Stangerzv

when 0.999999..recurring=3x0.3333333..recurring or 3 x 1/3=1

That works for me.  When I calculate (on paper) 1 divided by 3,  I stop and write 'recurring' because I don't have any paper big enough.

So arithmetic only works for me if these are the same.

I edited my last post.  Did you get the bit about 13 not existing in some universes ?

Bob

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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## #1052 2012-07-01 23:00:25

bobbym
From: Bumpkinland
Registered: 2009-04-12
Posts: 107,135

### Re: 0.9999....(recurring) = 1?

Hi;

I like being able to use the formula for geometric series.

This proof is originally from Euler although anyone else could do it.

We have a common ratio r = 1 / 10.

Without the use of the this theorem practical mathematics would fall apart. .9999999... = 1 works for me.

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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## #1053 2012-07-18 11:44:06

MrButterman
Member
Registered: 2012-07-18
Posts: 3

### Re: 0.9999....(recurring) = 1?

Interesting how this thread got so long.

is a mathematical fact. The reason it is so difficult for people to understand may be due to confusion over the concept of infinity. Here are some different ways to think about it:
___________

1) pointed out above

___________

2)

is equivalent to
. But since the number of 0's are infinite, you never "reach" the 1; it is equivalent to 0!
___________

3) a popular proof

___________

Last edited by MrButterman (2012-07-18 11:44:44)

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## #1054 2012-07-18 11:56:52

bobbym
From: Bumpkinland
Registered: 2009-04-12
Posts: 107,135

### Re: 0.9999....(recurring) = 1?

Hi MrButterman;

Interesting how this thread got so long.

These type threads are on every forum. Mostly they are so long because the opponents of .9999... = 1 can not be convinced by any of those proofs or any others.

Welcome to the forum.

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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## #1055 2012-10-26 17:14:04

Calligar
Member
Registered: 2011-09-24
Posts: 272

### Re: 0.9999....(recurring) = 1?

To be fair, the proofs they offer can be argued from a logical standpoint (so long as you understand everything that is going on), but there is nothing in mathematics that can prove how they are different otherwise.  I myself do not...personally believe this as a mathematical "fact," but also realize how futile it is to argue against it.  So like those many, it is impossible to convince me as well, after all, there is a reason this idea is so highly controversial.

Last edited by Calligar (2012-10-26 17:16:01)

There are always other variables.  -[unknown]
But Nature flies from the infinite, for the infinite is unending or imperfect, and Nature ever seeks an end.  -Aristotle
Everything makes sense, one only needs to figure out how.  -[unknown]

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## #1056 2012-10-26 22:44:32

noelevans
Member
Registered: 2012-07-20
Posts: 236

### Re: 0.9999....(recurring) = 1?

Hi y'all

You might say that those on both sides of the argument tend to take it to the limit!

1/2 a > good day!

Writing "pretty" math (two dimensional) is easier to read and grasp than LaTex (one dimensional).
LaTex is like painting on many strips of paper and then stacking them to see what picture they make.

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## #1057 2012-11-20 08:40:53

jdgall
Member
Registered: 2012-11-20
Posts: 1

### Re: 0.9999....(recurring) = 1?

Calligar wrote:

I myself do not...personally believe this as a mathematical "fact," but also realize how futile it is to argue against it.  So like those many, it is impossible to convince me as well, after all, there is a reason this idea is so highly controversial.

The term mathematical fact might be a little vague.  First, not every number is rational -- not every number can be represented as the quotient of integers.  For example the width of a square whose area is 2 is not a rational number.  That is, we need the full blown set of real numbers.  Figuring out what the real numbers (really) look like is a hard challenge, and providing a description of them in set theory was a major challenge.  There are two main approaches: Dedekind's cuts and Cauchy sequences.  They produce the same set.  Essentially, take a bounded sequence of rational numbers, and identify a "number" L with this sequence.  The real numbers are the rational numbers with all these Ls.  Thus in the construction of real numbers, we see that every real number is the limit of a sequence of rationals.

In other words, the real number "1" is by definition the limit of the sequence
<0,9,0.99,0.999,0.9999,etc

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## #1058 2012-11-21 00:54:10

Calligar
Member
Registered: 2011-09-24
Posts: 272

### Re: 0.9999....(recurring) = 1?

Well, I guess I can try to be a little bit more clear...

jdgall wrote:
Calligar wrote:

I myself do not...personally believe this as a mathematical "fact," but also realize how futile it is to argue against it.  So like those many, it is impossible to convince me as well, after all, there is a reason this idea is so highly controversial.

The term mathematical fact might be a little vague.  First, not every number is rational -- not every number can be represented as the quotient of integers.  For example the width of a square whose area is 2 is not a rational number.  That is, we need the full blown set of real numbers.  Figuring out what the real numbers (really) look like is a hard challenge, and providing a description of them in set theory was a major challenge.  There are two main approaches: Dedekind's cuts and Cauchy sequences.  They produce the same set.  Essentially, take a bounded sequence of rational numbers, and identify a "number" L with this sequence.  The real numbers are the rational numbers with all these Ls.  Thus in the construction of real numbers, we see that every real number is the limit of a sequence of rationals.

In other words, the real number "1" is by definition the limit of the sequence
<0,9,0.99,0.999,0.9999,etc

Firstly, when I say mathematical fact, I am really just referring to what is currently accepted and arguably "known" in mathematics.  Secondly, the limit of sequence is just another way to represent it being infinitely close, but still not exactly equal to the number (unless I'm mistaken).  Just like for it representing 1/3 with <0.3,0.33,0.333,etc. (I might not have put everything in proper terms, sorry if that causes any confusion, wasn't sure how to say it simply off the top of my head).  Also would like to make a note, you messed up slightly when you posted; it should be 0.9, not 0,9 for the first one unless I'm mistaken (but doesn't really have any relevance to anything).  In other words, it is just more rules that exist that otherwise, as I was saying, prove it's a mathematically fact.  Remember when I said this...

Calligar wrote:

To be fair, the proofs they offer can be argued from a logical standpoint (so long as you understand everything that is going on), but there is nothing in mathematics that can prove how they are different otherwise.

In mathematics, there is no way to represent the difference between 0.¯9 and 1.  All proofs (including false ones) either assume things (for specifically this), or simply define it as one only because of the infinitely close distance (there might be a few other reasons, but those are the 2 major I see).  Even though some people will argue things like it is 0.0...1 away, or 1/10¯0 away, which might arguably seem right, one can argue about the infinite distance, therefore making it an impossible argument to win.  So this argument doesn't carry on (with me) over confusion, I'll explain in more detail.

Rules in math already say this is true (even though it is very controversial) for many reasons.  However, it largely also comes down to one major thing: saying that one number equals a number that isn't exactly the same.  In other peoples eyes, that would be like saying 2 = 1; the only reason for it not being like this is because of the rules for it (plus all of math as we know it would collapse).  Also, the whole reason for this controversy in the first place, is because of using decimals in a way that don't accurately represent what it is.  Decimals can't accurately represent everything, unfortunately, no system can (at least that I know of).  However, unfortunately in this case, things like hyper real numbers define this, which is where the controversy comes in (I am not saying hyper real numbers are a bad thing though).  There are other similar cases of this.  1/3 = 0.¯3, right?  But wait, there is a difference, isn't there (asking rhetorically)?  Why?  Because 1/3 can not be accurately represented in decimal form.  Changing 1/3 into a decimal will give you 0.3...3...3...forever.  However, there isn't that number at the "end" to...well end it.  It is just like pi in that sense; pi = 3.14159265..., however there is no end to it.  There is no way to accurate represent the number in decimal form, or in any form for that matter (that I know of), besides for just calling it pi.  It is that very reason I've seen people argue pi ≈ 3.14159265...instead of =; because no matter how long the decimals go, it doesn't exactly equal pi because there is no end to it (rather pi is mathematically = or ≈ to 3.14159265... in this case is irrelevant to what I'm trying to say, plus I never said which one it is).

Please just keep in mind I'm not exactly arguing for or against.  I'm trying to say overall that it is futile to argue.  I might be personally against the current idea in math, but I have learned to accept that this is currently mathematical fact.  I just wanted to make all that clear.

Last edited by Calligar (2012-11-21 00:57:16)

There are always other variables.  -[unknown]
But Nature flies from the infinite, for the infinite is unending or imperfect, and Nature ever seeks an end.  -Aristotle
Everything makes sense, one only needs to figure out how.  -[unknown]

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## #1059 2012-11-21 03:25:36

Calligar
Member
Registered: 2011-09-24
Posts: 272

### Re: 0.9999....(recurring) = 1?

Unless I'm mistaken, it's more then simply for convenience.  Also, 0.¯9 doesn't end, because the ¯ over any number (which I put before the repeating number because otherwise I'd have to show in a picture), means it goes on forever.  If it had an end, that means we'd be able to put something after it, therefore, there'd be no reason for this controversy in the first place.

There are always other variables.  -[unknown]
But Nature flies from the infinite, for the infinite is unending or imperfect, and Nature ever seeks an end.  -Aristotle
Everything makes sense, one only needs to figure out how.  -[unknown]

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## #1060 2012-11-21 05:10:47

anonimnystefy
Real Member
From: Harlan's World
Registered: 2011-05-23
Posts: 16,018

### Re: 0.9999....(recurring) = 1?

0.999... doesn't exist. Recurring 9's aren't allowed.

Here lies the reader who will never open this book. He is forever dead.
Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

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## #1061 2012-11-21 05:19:17

Calligar
Member
Registered: 2011-09-24
Posts: 272

### Re: 0.9999....(recurring) = 1?

???

Last time I checked, hyper real numbers still existed.  Just because you can't get to an actual answer of it, doesn't mean it doesn't exist.  What do you mean by that?

There are always other variables.  -[unknown]
But Nature flies from the infinite, for the infinite is unending or imperfect, and Nature ever seeks an end.  -Aristotle
Everything makes sense, one only needs to figure out how.  -[unknown]

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## #1062 2012-11-21 22:35:05

SMBoy
Banned
Registered: 2012-11-17
Posts: 13

### Re: 0.9999....(recurring) = 1?

But if 1/3 = 0.333... Then Why does 0.333... Not become = to 0.4 Because that would then be the Same Infinite Calculation as...

0.999... becoming = to 1

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## #1063 2012-11-21 22:49:45

bobbym
From: Bumpkinland
Registered: 2009-04-12
Posts: 107,135

### Re: 0.9999....(recurring) = 1?

Hi;

Intuitively when you take

1 - .9 = .1
1 - .99 = .01
1 - .999 = .001
1 - .9999 = .0001
1 - .99999 = .00001

you can see it is getting smaller and smaller, we say it is approaching 0.

When you do

.4 - .3 = .1
.4 - .33 = .07
.4 - .333 = 0.067
.4 - .3333 = 0.0667
.4 - .33333 = 0.06667

notice that it is not getting smaller it seems to be approaching .06666666666... which is equal to 1 / 15. So .33333333... can never be equal to .4.

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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## #1064 2012-11-21 22:54:50

SMBoy
Banned
Registered: 2012-11-17
Posts: 13

### Re: 0.9999....(recurring) = 1?

But approaching 0 and seems to be approaching Are both not Actually ever going to get there ?

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## #1065 2012-11-21 22:57:56

bobbym
From: Bumpkinland
Registered: 2009-04-12
Posts: 107,135

### Re: 0.9999....(recurring) = 1?

Hi;

We are not talking about that. We are talking about your question.The 1 - .999999... keeps getting smaller the more nines we add. the .4 - .3333333... does not get smaller the more threes we add. If .333333333... was equal to .4 then when we subtract them we would get 0, but we do not. The isn't even a hint that it is getting to 0.

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Online

## #1066 2012-11-22 01:53:07

anonimnystefy
Real Member
From: Harlan's World
Registered: 2011-05-23
Posts: 16,018

### Re: 0.9999....(recurring) = 1?

But the number 0.9999... itself doesn't exist...

Here lies the reader who will never open this book. He is forever dead.
Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

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## #1067 2012-11-22 02:02:23

bobbym
From: Bumpkinland
Registered: 2009-04-12
Posts: 107,135

### Re: 0.9999....(recurring) = 1?

I don't look at it that way. To me it is shorthand for a series that thank the Lord, sums to 1.

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Online

## #1068 2012-11-22 04:26:52

SMBoy
Banned
Registered: 2012-11-17
Posts: 13

### Re: 0.9999....(recurring) = 1?

.9 + .1 = 1
.99 + .01 = 1
.999 + .001 = 1
.9999 + .0001 = 1
.99999 + .00001 = 1

But the a Two differences remain constantly the same! just another 9 and 0 ...further down the line does not make them nearer to being equal to 1

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## #1069 2012-11-22 05:09:40

bob bundy
Moderator
Registered: 2010-06-20
Posts: 7,742

### Re: 0.9999....(recurring) = 1?

Stefy wrote:

But the number 0.9999... itself doesn't exist...

How many times must I say this?  No numbers exist.  They are all just elements in an imaginary set that mathematicians have invented.

You cannot have a 3             =>            it doesn't exist.

But it's a jolly useful concept (especially for describing how many apples I have today) so let's go on using it.

Clearly this argument is going to continue for ever.

So:

assign the number 0.9 to post #1
assign the number 0.99 to post #2
.........
assign the number 0.999......9999999 (n 9s) to post #n
.........

Bob

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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## #1070 2012-11-22 05:24:04

anonimnystefy
Real Member
From: Harlan's World
Registered: 2011-05-23
Posts: 16,018

### Re: 0.9999....(recurring) = 1?

Hi Bob

I didn't mean it in that sense. Such a number isn't allowed in mathematics.

Here lies the reader who will never open this book. He is forever dead.
Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

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## #1071 2012-11-22 05:29:46

noelevans
Member
Registered: 2012-07-20
Posts: 236

### Re: 0.9999....(recurring) = 1?

Hi!  Let's reduce the .9999... down to its "roots" .1111...  which is supposed to be equal to 1/9.
So the question becomes "Why is .1111... equal to 1/9?"

Simple solution:  Choose base 9 instead of base 10.  Then 1/9 = .1  Now we don't have to
deal with an infinite repeating decimal.  Of course, now representing 1/10 in base 9 is an
infinite repeating decimal.  So to eliminate all these problems just scrap the "decimal systems"
and go back to using only the fractions.

The basic problem (in base 10) is that any unit fraction 1/N where N has prime factors other
than 2 or 5 cannot be written as a finite sum of fractions whose denominators only have
powers of 10.

Suppose we have an N with prime factors not twos or fives.  Then the algorithm for dividing
1 by N would never have a remainder of zero since products of 3, 7, 11, 13, 17, 19, ... never
end in zero.  So at every stage of the division we have non-zero remainders.  Hence the
"decimal representation" of 1/N would have to be "infinite."

Any time the base b and N are relatively prime (this doesn't cover all cases for N's that
cause an "infinite repeating b-esimal") the division algorithm always has a remainder at
each stage.  If we stop at any stage, then we have to add in the remainder to get the
original number exactly.

In applications all we ever use in "real world" problems are approximations to "some number
of decimal places."  Take pi for instance.  We use so many decimal places to approximate
pi in applications.

BUT it sure is CONVENIENT to "use" infinite decimals in mathematical expositions.

Writing "pretty" math (two dimensional) is easier to read and grasp than LaTex (one dimensional).
LaTex is like painting on many strips of paper and then stacking them to see what picture they make.

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## #1072 2012-11-22 06:50:37

bob bundy
Moderator
Registered: 2010-06-20
Posts: 7,742

### Re: 0.9999....(recurring) = 1?

Stefy wrote:

I didn't mean it in that sense. Such a number isn't allowed in mathematics.

What number is that? 3 ?

I think you'll find it is used in quite a few areas of mathematics.

And so is 0.999999999 recurring.

To save me some time, I refer you back to my post # 999.

Bob

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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## #1073 2012-11-22 07:12:18

anonimnystefy
Real Member
From: Harlan's World
Registered: 2011-05-23
Posts: 16,018

### Re: 0.9999....(recurring) = 1?

No, 3 is allowed, 0.(9) isn't. There cannot be recurring 9's after the decimal point.

Here lies the reader who will never open this book. He is forever dead.
Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

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## #1074 2012-11-22 07:27:28

bob bundy
Moderator
Registered: 2010-06-20
Posts: 7,742

### Re: 0.9999....(recurring) = 1?

Why not?

Bob

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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## #1075 2012-11-22 08:51:49

bobbym
From: Bumpkinland
Registered: 2009-04-12
Posts: 107,135

### Re: 0.9999....(recurring) = 1?

Hi SMboy;

But the a Two differences remain constantly the same!

Is .1 the same as .00001 or .0000000001 or ...

Those differences are not the same. They seem to be vanishing.

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Online