# .999999999999... = 1 ?

### #201

Posted 30 September 2005 - 01:10 PM

(e.g. object limited to a point) ...

and only under these specific conditions you could really call it true.

### #202

Posted 01 October 2005 - 10:53 PM

The role of the physician

I'm sure you meant "physicist" a physician is the one who heals you

### #203

Posted 02 October 2005 - 08:21 AM

Well you're rightI'm sure you meant "physicist" a physician is the one who heals you

### #204

Posted 03 October 2005 - 11:15 AM

I'm convinced there IS a mathematical model that "real universe" is based upon, even if our today's physicists don't know this model yet. However, they're looking for itI can't agree with that.

Like liquid said, Math and Physics are intensively used todescribethe world and the reality; they can only provide a description, a mathematical representation of a real phenomemon. This description is accurate more especially as the results we can get from it are close to the reality, but it will neverbethe reality.

The role of the physician is to find some new (simple) mathematical models to represent as better as possible a (complex) reality.

...

There is almost nothing "true" in physics. You can only say if a model is more or less accurate (and again, this depends strongly on the considered situation).

...

I agree that what e.g. Newton did and when he came to the conclusion that the formula "s = 0.5 a*t?" describes the physical process of accelerated move, he only

**described**more or less accurate what he was able to see (later Einstein proved that this description was NOT accurate, and maybe there will be another physicist one day how tells us that even Einstein description is not accurate).

Nevertheless (and regardless on if physicists will ever find it), you must agree there must be a mathematical model behind all this, how else could it work?

### #205

Posted 03 October 2005 - 06:41 PM

### #206

Posted 03 October 2005 - 07:56 PM

Even if it would be complex we could feed it to a computer and it could, maybe, draw conclusions or find connections we've never thought about.

### #207

Posted 04 October 2005 - 09:23 AM

Maybe such a model even would be much less complex than we would expect that. Imagine for example how simple Einstein's formulas are compared to what they are able to describe. You don't need inevitable a complex model to describe complex things...., but well, it would be so complex than we could never use it for concrete applications, thus it hasn't a great interest

And remember: there are some complex things that were not obeserved first and then described, but predicted by these formulas for the first time and observed only decades later then (black holes for example). So you should be carefull with statements like this.

### #208

Posted 24 October 2005 - 02:19 PM

Try this on a ClassPad 300

ExpToStr 0.99999999999999...,str1

ExpToStr 1,str2

StrCmp str1,str2,cmp

-> cmp<>0 !!!

so 0.99999999999999...<>0

### #209

Posted 29 October 2005 - 11:09 PM

### #210

Posted 01 November 2005 - 11:50 PM

Damn that was a good one

now if anyone tries to add an infinite number of nines it will never equal one, wich proves the point that 0.999(9) does not equal 1

### #211 Guest_Steve T_*

Posted 12 November 2005 - 12:13 AM

The number 0.9(repeating) can be expressed as the sum of an infinite series: (9/10)(1/10^n) adding it up for all values of n. This can be generalized to the form of a geometric series, the sum of ar^n, which converges to a certain value if the term a is less than 1. The value it would then converge to is : a/(1-r). In our particular series, (9/10)(1/10^n), the term a+(9/10), which is less than 1, so it converges.

using the formula for the sum which if converges to, we find that a/(1-r)=(9/10)/(1-(1/10))=(9/10)/(9/10)=1.

Another proof, based simply off algebra, is as follows.

x=0.99999....

10x=9.99999.... (multiply by 10)

9x=9 (subtract 0.99999... from 9.99999.... since the portion after the decimal in both numbers consists only of nines, they cancel eachother for ever)

x=1 (divide both sides by 9)

Steve

@Kilburn

Damn that was a good one

now if anyone tries to add an infinite number of nines it will never equal one, wich proves the point that 0.999(9) does not equal 1

### #212

Posted 12 November 2005 - 10:59 AM

### #213

Posted 12 November 2005 - 11:24 AM

*equals*1.

Btw. it is the same as with always dividing a number by 2. It NEVER reaches 0 since it only converges.

a0 = 1

an=a(n-1)/2

or an=a0/(2^n)

-->an=1/(2^n)

you would never say that the term 1/(2^n) can

*equal*0 since no divison with a dividend <> 0 can ever be zero (or you have to show me one )

### #214

Posted 12 November 2005 - 12:04 PM

0.999...but you stated right, that it CONVERGES and that it NEVER

equals1.

**is**the limit of the serie (9/10)*(1/10^n), not a term of it, and this limit is equal to 1

### #215

Posted 12 November 2005 - 01:55 PM

### #216

Posted 12 November 2005 - 02:56 PM

How could you assume that?the limit is 1 not 0.999..

You are using an hypothesis saying that 1!=0.999.. , and you may not use it here, since this is what you want to demonstrate.

And about the calculator: if there are 2 equivalent solutions to a problem, the calculator will always give you the 'easiest' form for it; it's exactly the same situation as for +0 and -0 for example. Btw you may not use the results given by a computer here, since we are talking about pure algebra, not about systematic resolutions.

Btw, another demonstration I just thought about (I don't remember if we already said it in the past):

For any real numbers A & B such that A<B you can allways find a real C such that A<C<B. Here this no number between 0.999.. and 1 that would be different from 0.999.., and thus 0.999.. and 1 are the same number.

But now the question is: can we really consider the number 0.999.. as a real number?

The situation here hasn't change from the beginning: we are bringing plenty of proofs showing that

**we may admit**that 0.999..=1, and the only answers you give are "nah, you may not say it, because 0.999..!=1, that's all". I'm still not planning to change my point of view

*without introducing any error or incoherence*But since I belong to the people who disagree with you, I'm sure you won't change your opinion neither. But it's okay, I think that I can live without receiving our super admin's approbation once again

**Edited by Orwell, 12 November 2005 - 03:05 PM.**

### #217

Posted 12 November 2005 - 07:04 PM

### #218 Guest_Guest_*

Posted 12 November 2005 - 11:25 PM

Steve

### #219

Posted 13 November 2005 - 01:38 AM

Right, that's the definition of real numbers. A < B means that a C exists with A < C < B beacuse reals are not enumerable like natural numbers.For any real numbers A & B such that A<B you can allways find a real C such that A<C<B.

Meaning: Because there is no C with 0.9... < C < 1, it's 0.9... = 1. Both just describe exactly the same number two different ways (and for those who still don't believe: there are books )

But now the question is: can we really consider the number 0.999.. as a real number?

Well, tricky question As it equals to 1 and 1 is a real number, 0.9... should be so, too. 1 is 1 no matter how you represent it ("1", "0.9..." or "0.3+0.7"). The point is, though 0.9... with really

**infinite**9s is exactly 1, you can't use it this representation unless you find a way to read/write/proccess an infinite amount of digits "9"

To make clear what I mean (I mentioned it an earlier post already): of course you know "0.9... means 1" and whenever you see a term "0.9... + 2" you would calculate "0.9... means 1, 1 + 2 = 3, so 0.9... + 2 = 3". In fact, you didn't calculate with "0.9..." representing 1, but "1" representing 1 then. Calculating with "0.9..." representing 1 would mean (note that we always can calculate symbolic only, and we have an infinite amount symbols here) you do:

0.9...+2 = 2+9*10^-1+9*10^-2+...+9*10^-oo,

and as you see you can't do that.

**==>**

0.9... = 1, both just are representation forms of the same real number "1". The form 0.9... is useless for calculation though (-> it has no meaning). If you used it you needed to stop calculating after a concrete amount of digits, and then you would not have calculated with "1" but an approximation of it only.

... Damn I did it again ... I posted in this thrad again. It will never end, it will survive every one of us

### #220 Guest_omg_*

Posted 03 July 2006 - 10:33 AM

### #221

Posted 09 July 2006 - 08:48 PM

### #222

Posted 28 September 2006 - 06:13 PM

have a nice maths...errr...sorry, days, casio people...........

protoman no one

### #223

Posted 30 September 2006 - 03:06 PM

### #224 Guest_Guest_protoman no one_*_*

Posted 30 September 2006 - 04:10 PM

a calculator can?t understand real numbers, or any number system. it is programed based on mathemathical definitions and properties....so......that picture did not tell me anything............

a calculator calculates based on its program, so, if you program a calc or even a computer based on 2+2=652516210, every time a 2 is + with a 2, the result will be 652516210..........

my trust in calcs is not complete (some people says that 0 is a natural number, others that not, who has the right answer? both and neither of them!!!!!!)

please see t.m.apostol

bye children

protoman no one

### #225

Posted 08 October 2006 - 12:06 PM

### #226

Posted 08 October 2006 - 03:52 PM

0.9999999999... is mean the are infinity "9" number, it will equa to 1, there is no 1 in 0.00000...001 because before we write 1 we must write infinity "0" number and never come to "1".-> That's logic?

The same with 0.999...<n<1, we cant find n because thre are inf. "9" and "inf+i=inf" is true.

0 is nature number? That false. ->That's logic?

I have a question "Universe is infinity?" that make me want to and crazy when think about it.

### #227

Posted 08 October 2006 - 04:28 PM

Let's say that people didn't giveNowadays, infinity didnt definition correctly, simplify that we cant count them. -> That is logic?

*you*a correct definition for infinity yet, you'll see later that all kind of infinities are well defined for a long time

### #228

Posted 09 October 2006 - 01:39 AM

A 1m ruler , we cut it into inf. part, but the sum always 1m.

### #229

Posted 09 October 2006 - 09:23 AM

Because there are some "concrete" numbers for you?

A number is just a number, nothing more

### #230

Posted 09 October 2006 - 11:53 AM

### #231

Posted 09 October 2006 - 02:06 PM

### #232

Posted 09 October 2006 - 02:20 PM

### #233

Posted 09 October 2006 - 03:18 PM

man....i?m a member!!!! what do you mean with "please login to post"?

my guest post was a little.....error on my memory circuits , because i forgotten to login beforeb write it...

i hope this never-ending topic will end now . machines........wata good creation, and error propagator!!!!!

excuse my english (foor ekzample, java nais dai )

protoman no one

### #234

Posted 11 October 2006 - 12:03 PM

### #235

Posted 11 October 2006 - 05:11 PM

i will ask my maths teacher, or mr hilbert

### #236

Posted 23 October 2006 - 02:32 PM

For the mathematical proof that 0.9999... = 1, I guess we all have our share.

Two questions remain though.

First, this implies that 0.0000...0001 = 0 and some here have troubles to admit it. Of course !

I read that the 1 at the end doesn't exist. Actually, it does, and doesn't depending on the point of view.

It does because it's written there, and we actually are speaking about representation of numbers.

It doesn't because what it represents is... nothing, I mean 0. Because our number writing systems says that the further on the right the digit is, the smallest the value it represents is. Here, it is infinitely far on the right, thus, infinitely small : this means it is equal to 0. And I am NOT speaking about a limit. Beacause writing "..." doesn't implie a limit toward infinity, it implies infinity.

And this is why we have this controversy. Our brain can admit infinity (and its consequences), but can not represent it. So we have two kinds of people, those who admit the equality because there is several mathematical proofs, and those who don't admit it because they don't SEE the logics of it.

Which brings me to my second point.

Logics.

Well, the world of science is full of examples where logics can't deal with infinity. "This sentence is not true", is a good one (though logics found an way to explain why it can't deal with it).

Let's go back to 0.000...001. Logic says : there is a "1", then, this thing is not 0.

My question is where is that 1 ? Infinitely far ? That is where we got a problem.

My answer ? Why do you think we write R=

**]**-00 , +00

**[**open ? Topology will sure give you the answer.

Or to use the argumentation of those who say "1 is

*the limit*of 9*10^-1+9*10^-2..., then never actually reached", that 1, far away after an infinity of zeroes can't be reached either. Since it can't be reached, its weight is never felt. It's value is 0.

That' is another way to explain it less mathematically and more intuitively.

An other argument is :

You can't multiply 0.3333... by 3 because you have to do an infinity of operations. Well, I can multiply the square root of 2 by itself or by another number - even pi -and it doesn't trouble me.

We don't multiply representations, we multiply the numbers they represent...

### #237

Posted 24 October 2006 - 05:15 AM

### #238

Posted 24 October 2006 - 05:20 AM

- Took that quote from a friend.

"Infinity isn't really that far if you look at it from the other end."

There are many ways to show that 0.99999999999999999... = 1.

1.) Simple algebraic:

Variable: x is the variable

Function: f(x) = 10x-x

When x = 1: 10(1)-(1)=9

When x = 0.99999...: 10(0.9999999...)-(0.9999999...)=9

2.) Simple algebraic #2:

1/3 = 0.333333333...

(1/3)*3 = 0.99999... or 1

3.) Simple algebraic #3:

1/11 = 0.09090909...

+ 10/11 = 0.90909090...

= 11/11 = 0.99999999...

4.) Simple arithmetic #4:

2/7 = 0.2857142857...

+ 5/7 = 0.7142857142...

= 7/7 = 0.9999999999...

5.) Method of implying the definition of what makes two numbers equal:

Two numbers are equal if there is no number between the value of two: 0.9999999.... = 1

* EDIT * - It was suppose to say isn't:

6.)

*- 0.999999.... = 1 (Rounding isn't necessary. 0.9999... = 1)*

**"What if you're rounding?"**### #239

Posted 24 October 2006 - 11:57 AM

I dont think 0.99... is equa to 1 because of rounding. We dont need to round 0.99999999999... to make it become 1, just need saying- Took that quote from a friend.

6.)- 0.999999.... = 1 (Rounding is necessary. 0.9999... = 1)"What if you're rounding?"

**0.99999... is 1**.

### #240

Posted 24 October 2006 - 02:16 PM

Thanks for your argumentation !Sorry, you wrong.

Actually, what we are debating here is very close to Zeno's paradoxes. Since even Zeno himself knew where the reasonning failed, the problem has been solved 2500 years ago.

I recommend you this

http://en.wikipedia....eno's_paradoxes , and especially this part :

Status of the paradoxes today

Mathematicians thought they had done away with Zeno's paradoxes with the invention of the calculus and methods of handling infinite sequences by Isaac Newton and Gottfried Wilhelm Leibniz in the 17th century, and then again when certain problems with their methods were resolved by the reformulation of the calculus and infinite series methods in the 19th century. Many philosophers, and certainly engineers, generally agree with the mathematical results.

Nevertheless, Zeno's paradoxes are still hotly debated by philosophers in academic circles. Infinite processes have remained theoretically troublesome. L. E. J. Brouwer, a Dutch mathematician of the 19th and 20th century, and founder of the Intuitionist school, was the most prominent of those who rejected arguments, including proofs, involving infinities. In this he followed Leopold Kronecker, an earlier 19th century mathematician. Some claim that a rigorous formulation of the calculus (as the epsilon-delta version of Weierstrass and Cauchy in the 19th century or the equivalent and equally rigorous differential/infinitesimal version by Abraham Robinson in the 20th) has not resolved all problems involving infinities, including Zeno's.

As a practical matter, however, no engineer has been concerned about them since knowledge of the calculus became common at engineering schools. In ordinary life, very few people have ever been much concerned

But actually, if you have more time, you'd better check that : http://en.wikipedia.org/wiki/0.999... !

The equality 0.999? = 1 has long been taught in textbooks, and in the last few decades, researchers of mathematics education have studied the reception of this equation among students, who often vocally reject the equality. Their reasoning is often based on an expectation that infinitesimal quantities should exist, that arithmetic may be broken, or simply that 0.999? should have a last 9.

These ideas are false in the real numbers, as can be proven by explicitly constructing the reals from the rational numbers, and such constructions can also prove that 0.999? = 1 directly. At the same time, some of the intuitive phenomena can occur in other number systems. There are even systems in which an object which can reasonably be called "0.999?" is strictly less than 1.

Now you even know why you can't accept it...Skepticism in education

Students of mathematics often reject the equality of 0.999? and 1 for reasons ranging from their disparate appearance to deep misgivings over the limit concept and disagreements over the nature of infinitesimals. There are many common contributing factors to the confusion:

Students are often "mentally committed to the notion that a number can be represented in one and only one way by a decimal." Seeing two manifestly different decimals representing the same number appears to be a paradox, which is amplified by the appearance of the seemingly well-understood number 1.[12]

Some students interpret "0.999?" (or similar notation) as a large but finite string of 9s, possibly with a variable, unspecified length. If they accept an infinite string of nines, they may still expect a last 9 at infinity.[13]Intuition and ambiguous teaching lead students to think of the limit of a sequence as a kind of infinite process rather than a fixed value, since the sequence never reaches its limit. Those who accept the difference between a sequence of numbers and its limit might read "0.999?" as meaning the former rather than the latter.[14]

These ideas are mistaken in the context of the standard real numbers, although many of them are partially borne out in more sophisticated structures, either invented for their general mathematical utility or as instructive counterexamples to better understand 0.999?.

Many of these explanations were found by professor David Tall, who has studied characteristics of teaching and cognition that lead to some of the misunderstandings he has encountered in his college students. Interviewing his students to determine why the vast majority initially rejected the equality, he found that "students continued to conceive of 0.999? as a sequence of numbers getting closer and closer to 1 and not a fixed value, because 'you haven?t specified how many places there are' or 'it is the nearest possible decimal below 1'".[15]

Of the elementary proofs, multiplying 0.333? = 1/3 by 3 is apparently a successful strategy for convincing reluctant students that 0.999? = 1. Still, when confronted with the conflict between their belief of the first equation and their disbelief of the second, some students either begin to disbelieve the first equation or simply become frustrated.[16] Nor are more sophisticated methods foolproof: students who are fully capable of applying rigorous definitions may still fall back on intuitive images when they are surprised by a result in advanced mathematics, including 0.999?. For example, one real analysis student was able to prove that 0.333? = 1/3 using a supremum definition, but then insisted that 0.999? < 1 based on her earlier understanding of long division.[17]

Joseph Mazur tells the tale of an otherwise brilliant calculus student of his who "challenged almost everything I said in class but never questioned his calculator," and who had come to believe that nine digits are all one needs to do mathematics, including calculate the square root of 23. The student remained uncomfortable with a limiting argument that 9.99? = 10, calling it a "wildly imagined infinite growing process."[18]

As part of Ed Dubinsky's "APOS theory" of mathematical learning, Dubinsky and his collaborators (2005) propose that students who conceive of 0.999? as a finite, indeterminate string with an infinitely small distance from 1 have "not yet constructed a complete process conception of the infinite decimal". Other students who have a complete process conception of 0.999? may not yet be able to "encapsulate" that process into an "object conception", like the object conception they have of 1, and so they view the process 0.999? and the object 1 as incompatible. Dubinsky et al. also link this mental ability of encapsulation to viewing 1/3 as a number in its own right and to dealing with the set of natural numbers as a whole.[19]

In popular culture

With the rise of the Internet, debates about 0.999? have escaped the classroom and are commonplace on newsgroups and message boards, including many that nominally have little to do with mathematics. In the newsgroup sci.math, arguing over 0.999? is a "popular sport", and it is one of the questions answered in its FAQ.[55] The FAQ briefly covers 1/3, multiplication by 10, and limits, and it alludes to Cauchy sequences as well.

A 2003 edition of the general-interest newspaper column The Straight Dope discusses 0.999? via 1/3 and limits, saying of misconceptions,"The lower primate in us still resists, saying: .999~ doesn't really represent a number, then, but a process. To find a number we have to halt the process, at which point the .999~ = 1 thing falls apart.[56]

Nonsense."

The Straight Dope cites a discussion on its own message board that grew out of an unidentified "other message board ... mostly about video games". In the same vein, the question of 0.999? proved such a popular topic in the first seven years of Blizzard Entertainment's Battle.net forums that the company's president, Mike Morhaime, announced at an April 1, 2004 press conference that it is 1:

"We are very excited to close the book on this subject once and for all. We've witnessed the heartache and concern over whether .999~ does or does not equal 1, and we're proud that the following proof finally and conclusively addresses the issue for our customers."[57]

Blizzard's subsequent press release offers two proofs, based on limits and multiplication by 10.

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