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## #176 2006-12-13 07:13:19

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

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

Ok lets do this 1/3 = Infinite 0.33333333333333333333...................

Anthony, we are so close.  I just ask you to do one further thing.  We both agree to the above.  Just multiply both sides by 3 and tell me what you get.

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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## #177 2006-12-13 07:24:09

Devantè
Real Member
Registered: 2006-07-14
Posts: 6,400

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

That and the old classic proof is enough for me.

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## #178 2006-12-14 00:43:19

Anthony.R.Brown
Banned
Registered: 2006-11-16
Posts: 516

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

Quote Ricky!
Ok lets do this 1/3 = Infinite 0.33333333333333333333...................

Anthony, we are so close.  I just ask you to do one further thing.  We both agree to the above.  Just multiply both sides by 3 and tell me what you get.

The above Number has nothing to do with Infinite 0.9!
Any Calculations you put forward must be based around the number we are working on! ie Infinite 0.9

Its like me saying show me if you can divde 4 by 3 ? and make a whole number!

oh and by the way can you show me that is 7 - 4 = 3 its nothing to do with the above! I just want to see your answer!

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## #179 2006-12-14 02:29:51

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

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

1/3 = Infinite 0.33333333333333333333...................

1/3 * 3 = Infinite 0.33333333333333333333................... * 3

1 = Infinite 0.999999999999999.............

How is the above wrong?  Doesn't that show that 1 = Infinite 0.9?

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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## #180 2006-12-14 16:50:33

George,Y
Member
Registered: 2006-03-12
Posts: 1,306

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

I'm back

Ricky wrote:

George, we need to define what we mean by "decimal expansion" for the question, "Does 0.999... = 1?" to ever make sense.  You find my definition arbitrary, so can you please provide one that isn't?  In fact, all definitions are arbitrary.  That's why they are definitions and not theorems.

My definition is for defining the question, not the solution.  By intuition alone, I believe that:

1/3 = 0.333...
1/9 = 0.111...
square root of 2 = 1.41421...

I made a definition supporting this view.  My definition treats the digits of a decimal expansion as a sequence of numbers, and it is the limit of this sequence which we treat as the number itself.  With this definition, I claim that every real number has at least one decimal expansion.

You do not believe that such infinite decimal expansions exist.  As a result, not all real numbers, or even all rational numbers, have a decimal expansion.  You may not see this as a problem.  I do.

Now let me state the following very carefully because you have not been understanding my words.

Mathematical spaces are formed by definitions.  Every set of axioms can form a mathematical space, and each are just as valid as others.  Your rejection of my definition is just as valid of my acceptance of it.  Neither of us can say that one is right and one is wrong.  However, there is a trend in mathematics for definitions that get us somewhere.  When we find a set of definitions which can solve interesting and useful problems, people tend to flock to those definitions.  It does not mean they are any more valid, only popular.

The definition I use are the popular ones.  But I have also provided reasons for my definition.  I want a decimal expansion for every real number.  Frankly, I can't understand why you don't.  But that doesn't really matter in the end.

Now, as to your answer to the question, "Does 0.999... = 1", it should not be, "No."  Rather, is should just be, "The question does not make sense with my definitions of the real numbers and decimal expansion."  Those are two entirely different answers, and if you don't accept infinite decimal expansions, then how could you ever answer the question with anything other than "It doesn't make sense"?  And that is a perfectly valid answer.  For example, I have proved elsewhere that if you accept the real numbers as a field, then division by 0 just doesn't make sense.  And I find that as a valid conclusion.

But I'm curious.  Do you see any problems with my definition?  You have stated that you see problems with my use of infinity.  I use the same concept of infinity as used in the limit of an infinite sequence, and in fact, I set up the digits of a decimal expansion as an infinite sequence.  Does this make sense?

To be honest, I am rather insulted that you claimed that I was trying to, "suppress the minority's opinion by the majority's."  I come to this forum to help others, have fun, and because there are some rather smart people here worth talking to about math.  And I don't take claims that I'm trying to suppress someones opinion lightly.  I am not having this conversation/debate for my health, but because it interests me.  Do you really think I'm trying to suppress your opinion by having an open debate on a public forum?

"Most people accept this definition by intuition alone.  You're the first person I've ever seen which did not."-Ricky
Since you have explained this sentence enough,  I will no longer indicate that you were trying to judge me as "absurd".

I may also note you a phsycologic fact that people tend to stick to what they were told during their childhood with prejudices without enough questions, making it difficult for a heratic opionion to spread. For example, after my post exploring the inconsistency of infinity, still a lot of people tried to amend the "infinity" concept rather than to throw it away. Also some one sticked to 1-0.999...=0.00..1 where 0.00...1 is nothing or invalid, while in fact 0.00..9 is invalid according to the same logic.

On the popularity issue, I shall argue that infinite expansions never rivals sufficient but finite expansions when the real thing is clear. Pi is great, but it is more accurate to say that all "circles" we can encounter coincide with Only a finite digits of Pi. So my alternative to replace an infinite expansion is just sufficient expansions, or their group. Actually is my alternative so far that all the people of the world along with all the computers of the world can practice?

At last, a defination cannot be ARBITORY, it should be logicly consistent with itself and should mirror the reality. And these are far more important than intuition alone-people used to have many intuitions, including one that the sun rotates around the earth.

To Anthony:
You cannot agree that 1/3=0.333... for it is easy to use the "proof" with multiplying them with 3.
Instead, you should point out no one has calculated 1/3 out yet.(As I insisted in the previous thread, which has already been closed by Ricky)
Or simply further, 0.333... is inconsistent with itself. (As I insisted in this thread)

X'(y-Xβ)=0

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## #181 2006-12-14 17:25:27

Ricky
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Registered: 2005-12-04
Posts: 3,791

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

I may also note you a phsycologic fact that people tend to stick to what they were told during their childhood with prejudices without enough questions, making it difficult for a heratic opionion to spread

I have explained clear reasons as to why I accept this definition.  You seem to be implying that I am sitting here saying, "That's not what my teacher told me."  That is certainly not the case.

Pi is great, but it is more accurate to say that all "circles" we can encounter coincide with Only a finite digits of Pi.

Why is this?  How do you know a measurement, any measurement can be described in a finite amount of digits?  How do you know that it isn't just an artifact of our measurement technique which produces finite digits?

Even if the thing we are measuring is continuous (height, for example), because of our limited technology, we may only measure discrete distances.  You seem to be claiming you can accurately measure reality with no error.  Certainly, that is not the case.

At last, a defination cannot be ARBITORY, it should be logicly consistent with itself and should mirror the reality.

First off, show me how my definition is not logically consistent or does not mirror reality.  Actually, first, you can show me where infinity occurs in reality.  Or do you think infinity should not be part of mathematics too?  But I believe definitions are still arbitrary.  It is arbitrary is the same way that the word "chair" is arbitrary.

'Tis but thy name that is my enemy;
Thou art thyself, though not a Montague.
What's Montague? it is nor hand, nor foot,
Nor arm, nor face, nor any other part
Belonging to a man. O, be some other name!
What's in a name? that which we call a rose
By any other name would smell as sweet;

- Shakespeare

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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## #182 2006-12-14 17:51:57

George,Y
Member
Registered: 2006-03-12
Posts: 1,306

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

Why is this?  How do you know a measurement, any measurement can be described in a finite amount of digits?  How do you know that it isn't just an artifact of our measurement technique which produces finite digits?

Even if the thing we are measuring is continuous (height, for example), because of our limited technology, we may only measure discrete distances.  You seem to be claiming you can accurately measure reality with no error.  Certainly, that is not the case.
---I mean matters-wood, plastics, paper, to name a few. They are made of an integer amount of particles, disabling them to form any real circles.

First off, show me how my definition is not logically consistent or does not mirror reality.  Actually, first, you can show me where infinity occurs in reality.  Or do you think infinity should not be part of mathematics too?  But I believe definitions are still arbitrary.  It is arbitrary is the same way that the word "chair" is arbitrary.

--- I mean 0.999... with infinite digits is not, at least the form. So with a line formed by points.Secondly, I need not show you where is infinity. Instead, you need to show me how can a line be formed by "infinite" points in reality.
The English word "chair" may be arbitary, but the concept of "chair" is based on a repetitive object designed to  hold an math with a surface and usually 4 legs beneath the surface.

X'(y-Xβ)=0

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## #183 2006-12-14 18:12:41

George,Y
Member
Registered: 2006-03-12
Posts: 1,306

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

Even if the thing we are measuring is continuous (height, for example), because of our limited technology, we may only measure discrete distances.  You seem to be claiming you can accurately measure reality with no error.  Certainly, that is not the case.

---Great point. Time and space are the last havens for the concept of continuum. I am optimistic that some day physcians find them Discrete.

Still, I can show you if they are made of repetible basic elements- that is to say, the basic element is the smallest fraction among those of any given amount of time or length and can form the amount of time by Simply Reproducing itself, then they are not continous.

This assumption is what I believe as essential. Because if you interpret the length as arbitarily divisable ( on a continuum basis), it will be very hard to tell how "no length" continuously "grows" into arbitary small length. Or in other words, Zeno's paradox.

X'(y-Xβ)=0

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## #184 2006-12-15 01:46:13

Anthony.R.Brown
Banned
Registered: 2006-11-16
Posts: 516

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

The Biggest mistake Mathematicians make! Is that they believe an Infinite Endless number some how is! Or ends up a whole number!
The example below is the purest Math, with no assumptions, no beliefs, no forecasts, and no calculations other than what is the truth in front of ones eyes!!

A = 1
B = Infinite 0.9
C = 3

A + B + C = 4.99999999999999(NOT (5) ?

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## #185 2006-12-15 02:52:11

Patrick
Real Member
Registered: 2006-02-24
Posts: 1,005

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

Anthony - you're assuming that 4.(9) is not 5.

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## #186 2006-12-16 02:08:22

mathsyperson
Moderator
Registered: 2005-06-22
Posts: 4,900

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

x = 4.999...
10x = 49.999...
10x-x = 49.999...-4.999...
9x = 45
x = 5

4.999... = 5

Why did the vector cross the road?
It wanted to be normal.

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## #187 2006-12-16 03:38:06

Anthony.R.Brown
Banned
Registered: 2006-11-16
Posts: 516

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

The true value of the infinite number as in your post above is!

10 x 4.9........... =  49.99999999999999999999999999999999999999999999..

Now just as I have asked before!! if some can show how many decimal places it takes for  Infinite  4.9 to equal or become 5 then we might be getting somewhere!

Or Infinite 0.9 ( How many .9s  there are! Or it takes to = ) 1 ?

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## #188 2006-12-16 03:55:57

Anthony.R.Brown
Banned
Registered: 2006-11-16
Posts: 516

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

p.s I forgot to say!

"I'ts impossible to actually multiply an Infinite number! because to be able to multiply a number! first a person has to know the size of the number!"

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## #189 2006-12-16 05:03:25

Toast
Real Member
Registered: 2006-10-08
Posts: 1,321

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

If a circle is made up of the most basic of particles, which are indivisible, and you have the radius which is 10 particles long (the circle is 1 particle thick), and you want to find the area of the circle, you will have to multiply by pi, an irrational number. This must mean that even the most basic of particles must be divisible for fundamental maths to make sense. This is impossible however, so where does that leave us?

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## #190 2006-12-16 05:11:01

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

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

"I'ts impossible to actually multiply an Infinite number! because to be able to multiply a number! first a person has to know the size of the number!"

Contradiction.  If you agree that 1/3 is a real number, and that 1/3 = 0.3333..., then it must be that you are able to multiply it.  This is because the real numbers are a field under addition and multiplication.

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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## #191 2006-12-17 01:58:57

Anthony.R.Brown
Banned
Registered: 2006-11-16
Posts: 516

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

To Ricky first!

Yes 1/3 is a Number! but once it becomes Infinite 0.3 then you can't multiply the Number!

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## #192 2006-12-17 02:24:42

Anthony.R.Brown
Banned
Registered: 2006-11-16
Posts: 516

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

Lets have a look at why its impossible to Mutilply, Subtract, Divide Etc any Infinite Numbers!

For this I will use the Infinite 4.9 put forward as an example.

A = Infinite 4.9 (stage one) 1 decimal place.

B = Infinite 4.99 (stage two) 2 decimal places.

C = Infinite 4.9 (Actual Number) Endless decimal places!

10 x A = 49.9 we know this is a true Number because we know the size of the Number i.e. 1 decimal place!

10 x B = 49.99 we know this is a true Number because we know the size of the Number i.e. 2 decimal place!

10 x C = 49.9 ? Its impossible to know this because we dont know the size of the number and more important how many decimal places!

The only possible answer would be based on the facts! C started 0.1 less than 1.
In the calculations no other Numbers have been Add to C (or can be!).
No other Numbers have been multiplied,(or can be) the Number remains the Infinite!

So an Answer would be Infinite 4.9 = (0.1 < 5) this way we dont need to know how many .9s there are!

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## #193 2006-12-17 02:41:41

Devantè
Real Member
Registered: 2006-07-14
Posts: 6,400

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

I still don't agree with that.

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## #194 2006-12-17 03:29:49

luca-deltodesco
Member
Registered: 2006-05-05
Posts: 1,470

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

ok anthony, lets put it this way.

what do you do when you multiply a number by 10? you move the decimal place one place to the right.

so if we have 4.999999...

times it by 10, move the decimal place one place to the right

we have 4.(9) * 10 = 49.(9)

The Beginning Of All Things To End.
The End Of All Things To Come.

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## #195 2006-12-17 07:22:57

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

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

Yes 1/3 is a Number! but once it becomes Infinite 0.3 then you can't multiply the Number!

Anthony, this is incorrect.  Because the real numbers are a field, you may multiply any two real numbers and end up with a unique real number.

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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## #196 2006-12-17 08:52:37

MathsIsFun
Registered: 2005-01-21
Posts: 7,664

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

If you try to multiply 4.999... one digit at a time you won't ever get an answer, but if you step back and see the *idea* then you will find that you can.

For example

4.9 × 3 = 14.7
4.99 × 3 = 14.97
4.999 × 3 = 14.997

So, the pattern is that 4.999... = 14.999...

And what about that last 7? You will *never* reach it. Infinity is endless.

"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

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## #197 2006-12-17 14:13:33

George,Y
Member
Registered: 2006-03-12
Posts: 1,306

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

MathsIsFun wrote:

If you try to multiply 4.999... one digit at a time you won't ever get an answer, but if you step back and see the *idea* then you will find that you can.

For example

4.9 × 3 = 14.7
4.99 × 3 = 14.97
4.999 × 3 = 14.997

So, the pattern is that 4.999... = 14.999...

And what about that last 7? You will *never* reach it. Infinity is endless.

So is it endlessly growing? A growing number should be termed more likely as a variable than a number.

If infinity is indeed endless, infinity is never reachable, for there is no end. As infinity is not reachable, 4.999... with infinite amount of digit has no sense.

X'(y-Xβ)=0

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## #198 2006-12-17 15:39:47

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

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

George, with my definition on what an decimal expansion is, it does in fact make sense.

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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## #199 2006-12-17 15:48:51

George,Y
Member
Registered: 2006-03-12
Posts: 1,306

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

Ricky wrote:

George, with my definition on what an decimal expansion is, it does in fact make sense.

You first assumed r to be a decimal expansion that can has infinite decimals.

As I had disproved, the concept of infinite digits itself is plausible, but you assumed it.

Your defination just hided what you cannot prove inside the premise and then you "proved" the result you want.

X'(y-Xβ)=0

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## #200 2006-12-17 16:50:12

George,Y
Member
Registered: 2006-03-12
Posts: 1,306

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

Ricky wrote:

George, as promised, I will be answering your posts either tomorrow or Saturday, so don't worry, they won't go ignored.  I ask you (as a friend in a debate, not as a moderator) not to post until I have, otherwise I fear there will just be utter confusion.

Till now, you have disagreed on decimal expansion on the real numbers, and I always figured the problem was higher up, for example definition of the convergence of an infinite series, which would intuitively come after a consensus of real numbers and decimal expansions have been reached.  Let me address this now.

I believe we all agree on the following things:

1. A decimal expansion of any real number r takes the form of:

2. For any real number r, given an epsilon (e), we can find a finite number of decimal digits d such that:

But we have problems with this.  Specifically we can't write all real numbers or even all rational numbers with a finite number of digits. So what do we do about this?

One thing is to do nothing.  Accept this failure of decimals.  Personally, I feel unsatisfied with this.  Is there no way we can save decimal expansion of infinite decimals?  Certainly, we can't "see" infinitely many decimals.

But wait!  We have this really cool method of dealing with infinite things invented/discovered by Cauchy et. al.  Limits.  They allow us to handle infinite sequences of numbers.  And in the end, isn't this all we're dealing with?

So let's just explore this possibility for the time being.  How can we use limits?  Certainly it may not be possible, but lets just give it a try.

For any real number r, let us define a sequence of real numbers (using the same notation as before):

It should be clear that:

For any integer n.  So this sequence is monotonely increasing, as well as bounded.  By the monotone-bounded convergence theorem, this sequence must converge.  As we (hopefully) agreed in #2 (see above), it does in fact converge to r.

So where are we at?  Given any real number, we can use the decimal expansion to approach it, and get arbitrarily close, a limit per say.

It is with this in mind that we make the following definition:

Note that the notation above simply means an infinite amount of decimal points.

The results of this definition are the following:

1. No contradiction with any math that I'm aware of
2. Every real number has at least one decimal expansion equal to it.
3. Some decimal expansions are not unique.

3 can be considered a problem.  I certainly do.  But I argue (with only opinion, not pure logic) that having multiple decimal expansions are a lesser problem than not being able to represent some real numbers with a decimal expansion.

An immediate consequence of this definition is that 0.999... = 1.

Most people accept this definition by intuition alone.  You're the first person I've ever seen which did not.

Edit: I don't think my signature has every applied to any one of my posts greater than this one.

Specifically we can't write all real numbers or even all rational numbers with a finite number of digits.

we can't "see" infinitely many decimals
-the problem I proposed in this thread is that infinite decimals, if existed, would have contravasory among themselves. Not as simple as we cannot see.

it does in fact converge to r.
- do you interpret "converge" or abitarily proximate as "reach"? The same old topic again.

an infinite amount of decimal points.
-again contravasory and no sense.

The most important thing I insist is that an amount that you can tell as a determined number, such 2, 3.5, has no way to transform into either infinitesimal or infinity. Infinite sequences are a dilema as well. Infinite sequences mean only you can add in new entries without a stop.

If you mean an infinite series itself can be defined as a decimal expansion, then infinite digit numbers are at least non-static.

X'(y-Xβ)=0

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