0.99999999.......

[Citizen James]

Libertas pretty much nailed it, but I want to give a slightly more formal argument.

1/3 = .3333333333....

multiplying both sides by 3 we get

3(1/3) = 3(.3333333333....)

3/3 = .9999999999999999

Noting that 3/3 is equal to 1 we have:

1 = .9999999999999999...  QED.

Floating point computation is a whole other ball of wax because computers can only hold a finite number of places.    The number given is not the exact answer, but a (hopefully) good approximation of it.

[Vepres]

An easier way of thinking about it is that it's a flaw in our number system.

[Senator Libertas]

(3/3)=0.999999999....

Epic math fail. Three divided by three is one. Ridiculous to suggest that three thirds is anything but one...srsly...

Yes, your post is an "Epic math fail".

[Χahar]

(3/3)=0.999999999....

Epic math fail. Three divided by three is one. Ridiculous to suggest that three thirds is anything but one...srsly...

Precisely. Three divided by three is one. Therefore, three divided by three is 0.9999999...

[muon2]

(3/3)=0.999999999....

Epic math fail. Three divided by three is one. Ridiculous to suggest that three thirds is anything but one...srsly...

Precisely. Three divided by three is one. Therefore, three divided by three is 0.9999999...

The difference of opinions could be attributed to a difference in which set of numbers to consider. Describing a relationship like 3/3 = 1 could be a statement about the rational numbers of which 1/3, 3/3, and 345/86 are all examples. These are the numbers we are most comfortable with and results in statements like pi is approximately 22/7. The floating point numbers on a computer are a finite subset of the rational numbers.

If a person who uses 3/3 = 1 is grounded in the rational numbers, then equating either 3/3 or 1 to 0.999... would find the statement confusing or meaningless since 0.999... is not a rational number. It's only when 3 and 1 are used as real numbers that the relationship 3/3 = 0.999... is sensible. Since 3, 3/3 and 1 have more commonplace uses as rational numbers their ability to serve a real numbers doesn't often come into play.

[ChrisJG777]

It's certainly very close to 1, and if I was using it in some mathematical equation and didn't need to be too precise, I'd put it down as 1, but in the end though 0.9999999... ≈ 1, at the same time 0.9999999... ≠ 1 either.

It's infinitely close to 1, however, making it mathematically equal.

Quite frankly I fail to see how it's equal to 1, indeed it's very close, but just by looking at it you can see that it's not exactly equal.

Fun isn't it?

f(x) = 1 / x², a lot of people would say, never touches the x-axis, but only gets closer and closer to it.

It does, though, in theory. Under the same principle as the original question here, the function is considered to be infinitely long, and thus it also becomes infinitely close to the x-axis

Still, 0.9999999... is not equal to 1. 

It is though

What's the difference between 0.9 and 1?
What's the difference between 0.99 and 1?
What's the difference between 0.999999999999 and 1?

If you truly assume an infinite number of 9s behind the decimal point....there can't be any difference between the two numbers.

To claim that the two numbers are not equal, you would have to assume a finite number of 9s behind the decimal point. But even 0.99999999999999999999999999999999999999999999 (and a million more 9s) is not equal to "0.99....".

I do not dispute that 0.9999999... is indeed extremely close to one, but so say that it equals one make absolutely no sense whatsoever (but when has mathematics ever done that).  The difference will indeed be very (understatement) small, incalculable in fact, but no matter how much I look at it and think about it, it still strikes me as less than 1.  That said, if I don't need to be too precise when displaying the results of a calculation, I'll just treat it as one all the same (considering that when rounded you get 1), but I'll never be able to view it as equal to one though, that is, exactly equal.  I will look into this further.

It's certainly very close to 1, and if I was using it in some mathematical equation and didn't need to be too precise, I'd put it down as 1, but in the end though 0.9999999... ≈ 1, at the same time 0.9999999... ≠ 1 either.

It's infinitely close to 1, however, making it mathematically equal.

Quite frankly I fail to see how it's equal to 1, indeed it's very close, but just by looking at it you can see that it's not exactly equal.

Fun isn't it?

f(x) = 1 / x², a lot of people would say, never touches the x-axis, but only gets closer and closer to it.

It does, though, in theory. Under the same principle as the original question here, the function is considered to be infinitely long, and thus it also becomes infinitely close to the x-axis

Still, 0.9999999... is not equal to 1. 

(1/3)=0.333333333....

(2/3)=0.666666666....

(3/3)=0.999999999....


Gotta wonder whether that excellent public school education you kept telling me about is to blame for an 18-year-old being ignorant of a basic mathematical fact...

Being a dick again?  You don't have a proper argument against public schooling so you resort to insulting those who support it.  Above, I'm having a quiet debate with Franzl, and though he may not agree with me on the point at hand, at least he's not acting like a git over it.  Besides, your description of 1 being equal 0.99999... as mathematical fact isn't quite true, considering that mathematics is not a "closed book" or so to speak, if anything it's closer to a mathematical theory- commonly accepted and is backed up by proof and evidence, but still open to debate and amendment pending the surfacing of any proof/evidence in favour or to the contrary.

Anyway, tell me, why do you act like an obnoxious arsehole towards people who have differing opinions to you?  Come back when you have a credible response outside of calling everyone who disagrees with you "brainwashed statist zombie sheeple".  I'm open to discussing things but no personal insults, and you've just gone made them part of your initial response.  I've personally had enough of your egotism, and that "holier than thou" attitude of yours.

[Sibboleth]

GO AWAY MATHS

[I left.]

I do not dispute that 0.9999999... is indeed extremely close to one, but so say that it equals one make absolutely no sense whatsoever (but when has mathematics ever done that).  The difference will indeed be very (understatement) small, incalculable in fact, but no matter how much I look at it and think about it, it still strikes me as less than 1.  That said, if I don't need to be too precise when displaying the results of a calculation, I'll just treat it as one all the same (considering that when rounded you get 1), but I'll never be able to view it as equal to one though, that is, exactly equal.  I will look into this further.

OK, despite Libertas extremely nasty behavior....his math isn't wrong.

We agree that any number divided by itself equals 1, right? That can't be disputed! So 3/3=1.

Alright, now if you had to write 1/3 as a decimal number, what would it be? Is it 0.33? or 0.(lots of 3s?). It's clear that it's an infinite number of 3s behind the decimal point, correct?

But now if you multiply that number by 3, what do you get? wouldn't that be 0.333.... x 3 = 0.999...?


To argue that 0.999... is not equal to 1, you would have to dispute that 3 x 1/3 is equal to 1

[muon2]

I do not dispute that 0.9999999... is indeed extremely close to one, but so say that it equals one make absolutely no sense whatsoever (but when has mathematics ever done that).  The difference will indeed be very (understatement) small, incalculable in fact, but no matter how much I look at it and think about it, it still strikes me as less than 1.  That said, if I don't need to be too precise when displaying the results of a calculation, I'll just treat it as one all the same (considering that when rounded you get 1), but I'll never be able to view it as equal to one though, that is, exactly equal.  I will look into this further.

As I posted earlier, the numbers 0.999... and 1.000... are just decimal representations of the real number 1. Real numbers as a set are defined by convergence, and since both 1.000... and 0.999... converge to the same quantity they represent the same number. The fact that 1.000... converges faster than 0.999... doesn't change the application of the definition, both still converge.

The notion of convergence is critical to the calculus. It shows up in the use of limits. However, it doesn't get covered much except by mathematics majors in college, so the concept is not strongly part of most school curricula.

[ChrisJG777]

I do not dispute that 0.9999999... is indeed extremely close to one, but so say that it equals one make absolutely no sense whatsoever (but when has mathematics ever done that).  The difference will indeed be very (understatement) small, incalculable in fact, but no matter how much I look at it and think about it, it still strikes me as less than 1.  That said, if I don't need to be too precise when displaying the results of a calculation, I'll just treat it as one all the same (considering that when rounded you get 1), but I'll never be able to view it as equal to one though, that is, exactly equal.  I will look into this further.

OK, despite Libertas extremely nasty behavior....his math isn't wrong.

We agree that any number divided by itself equals 1, right? That can't be disputed! So 3/3=1.

Alright, now if you had to write 1/3 as a decimal number, what would it be? Is it 0.33? or 0.(lots of 3s?). It's clear that it's an infinite number of 3s behind the decimal point, correct?

But now if you multiply that number by 3, what do you get? wouldn't that be 0.333.... x 3 = 0.999...?


To argue that 0.999... is not equal to 1, you would have to dispute that 3 x 1/3 is equal to 1

Even though it has moments like this that just don't make any sense on the outside, I actually quite enjoy reading about mathematics.


On the outside, 0.999... and 1 look completely different to each other, and would be seen as not exactly but certainly nigh on equal to each other.  When I first go into the subject, I knew it wouldn't be consistent, but there are something that really do take the cake.  Makes me wonder what it'd be like if we used base 6 numbering instead, for one thing.

I can indeed see where you're coming from here, I'd put most certainly 3 multiplied by 1/3 as equal to 1, such as when you take a circle, slice it into three equal slices and put them together again you get 1 circle.  So, 3 by 1/3 = 1, no argument from me there.  Tough one.  Anyway, you do indeed have a point there, unlike Libertas, who just trolls.  0.999..., weird number, cosmetically unequal to 1, yet for the purposes of most equations the former's treated just like the latter.

[So the Heroes Fall]

Actually for all intents and purposes this IS mathematical fact. It's universally agreed upon by mathematicians, probably more so than evolution is agreed upon by those in that field.

[ChrisJG777]

Actually for all intents and purposes this IS mathematical fact. It's universally agreed upon by mathematicians, probably more so than evolution is agreed upon by those in that field.

With mathematics and the sciences it's generally safer to assume that things like this are not solid facts, only well supported theories, as there's always the potential that something else could come along and convincingly contradict the matter in question.  However I don't doubt that this matter agreed upon more by mathematicians than evolution is by biologists.  But still, for example, no matter how obvious it is in our day to day lives, gravity is still a theory, if you catch my drift.

[muon2]

Actually for all intents and purposes this IS mathematical fact. It's universally agreed upon by mathematicians, probably more so than evolution is agreed upon by those in that field.

With mathematics and the sciences it's generally safer to assume that things like this are not solid facts, only well supported theories, as there's always the potential that something else could come along and convincingly contradict the matter in question.  However I don't doubt that this matter agreed upon more by mathematicians than evolution is by biologists.  But still, for example, no matter how obvious it is in our day to day lives, gravity is still a theory, if you catch my drift.

Mathematics and science are fundamentally different in this respect. Scientific views change in light of new facts based on measurements and observations. Scientists views can differ based on interpretation of the data.

Mathematics is based on a set of definitions and axioms that lead to logical proofs. When I specify a set of definitions then I can draw inescapable conclusions. There may be other definitions, but within the confines of one set of definitions there is complete agreement as to the conclusions drawn form those definitions. At times there are mathematical conjectures not yet proved or disproved, but that is different from a difference in interpretation that one sees in science.

In this case of real numbers we are talking about proved statements. If you wish to claim use only of the definitions for rational numbers then your skepticism about this thread's subject is based on that different use of definitions. It doesn't affect the conclusions drawn for real numbers.

[Senator Libertas]

It's certainly very close to 1, and if I was using it in some mathematical equation and didn't need to be too precise, I'd put it down as 1, but in the end though 0.9999999... ≈ 1, at the same time 0.9999999... ≠ 1 either.

It's infinitely close to 1, however, making it mathematically equal.

Quite frankly I fail to see how it's equal to 1, indeed it's very close, but just by looking at it you can see that it's not exactly equal.

Fun isn't it?

f(x) = 1 / x², a lot of people would say, never touches the x-axis, but only gets closer and closer to it.

It does, though, in theory. Under the same principle as the original question here, the function is considered to be infinitely long, and thus it also becomes infinitely close to the x-axis

Still, 0.9999999... is not equal to 1. 

It is though

What's the difference between 0.9 and 1?
What's the difference between 0.99 and 1?
What's the difference between 0.999999999999 and 1?

If you truly assume an infinite number of 9s behind the decimal point....there can't be any difference between the two numbers.

To claim that the two numbers are not equal, you would have to assume a finite number of 9s behind the decimal point. But even 0.99999999999999999999999999999999999999999999 (and a million more 9s) is not equal to "0.99....".

I do not dispute that 0.9999999... is indeed extremely close to one, but so say that it equals one make absolutely no sense whatsoever (but when has mathematics ever done that).  The difference will indeed be very (understatement) small, incalculable in fact, but no matter how much I look at it and think about it, it still strikes me as less than 1.  That said, if I don't need to be too precise when displaying the results of a calculation, I'll just treat it as one all the same (considering that when rounded you get 1), but I'll never be able to view it as equal to one though, that is, exactly equal.  I will look into this further.

It's certainly very close to 1, and if I was using it in some mathematical equation and didn't need to be too precise, I'd put it down as 1, but in the end though 0.9999999... ≈ 1, at the same time 0.9999999... ≠ 1 either.

It's infinitely close to 1, however, making it mathematically equal.

Quite frankly I fail to see how it's equal to 1, indeed it's very close, but just by looking at it you can see that it's not exactly equal.

Fun isn't it?

f(x) = 1 / x², a lot of people would say, never touches the x-axis, but only gets closer and closer to it.

It does, though, in theory. Under the same principle as the original question here, the function is considered to be infinitely long, and thus it also becomes infinitely close to the x-axis

Still, 0.9999999... is not equal to 1. 

(1/3)=0.333333333....

(2/3)=0.666666666....

(3/3)=0.999999999....


Gotta wonder whether that excellent public school education you kept telling me about is to blame for an 18-year-old being ignorant of a basic mathematical fact...

Being a dick again?  You don't have a proper argument against public schooling so you resort to insulting those who support it.  Above, I'm having a quiet debate with Franzl, and though he may not agree with me on the point at hand, at least he's not acting like a git over it.  Besides, your description of 1 being equal 0.99999... as mathematical fact isn't quite true, considering that mathematics is not a "closed book" or so to speak, if anything it's closer to a mathematical theory- commonly accepted and is backed up by proof and evidence, but still open to debate and amendment pending the surfacing of any proof/evidence in favour or to the contrary.

Anyway, tell me, why do you act like an obnoxious arsehole towards people who have differing opinions to you?  Come back when you have a credible response outside of calling everyone who disagrees with you "brainwashed statist zombie sheeple".  I'm open to discussing things but no personal insults, and you've just gone made them part of your initial response.  I've personally had enough of your egotism, and that "holier than thou" attitude of yours.

You're the one who claimed to be the poster boy of public school education and said I was somehow "jealous" of your public schooling. And here you are arguing against basic mathematics you should have learned when you were ten years old.

[Fmr. Emperor PiT]

I do not dispute that 0.9999999... is indeed extremely close to one, but so say that it equals one make absolutely no sense whatsoever (but when has mathematics ever done that).  The difference will indeed be very (understatement) small, incalculable in fact, but no matter how much I look at it and think about it, it still strikes me as less than 1.  That said, if I don't need to be too precise when displaying the results of a calculation, I'll just treat it as one all the same (considering that when rounded you get 1), but I'll never be able to view it as equal to one though, that is, exactly equal.  I will look into this further.

OK, despite Libertas extremely nasty behavior....his math isn't wrong.

We agree that any number divided by itself equals 1, right? That can't be disputed! So 3/3=1.

Alright, now if you had to write 1/3 as a decimal number, what would it be? Is it 0.33? or 0.(lots of 3s?). It's clear that it's an infinite number of 3s behind the decimal point, correct?

But now if you multiply that number by 3, what do you get? wouldn't that be 0.333.... x 3 = 0.999...?


To argue that 0.999... is not equal to 1, you would have to dispute that 3 x 1/3 is equal to 1

Even though it has moments like this that just don't make any sense on the outside, I actually quite enjoy reading about mathematics.


On the outside, 0.999... and 1 look completely different to each other, and would be seen as not exactly but certainly nigh on equal to each other.  When I first go into the subject, I knew it wouldn't be consistent, but there are something that really do take the cake.  Makes me wonder what it'd be like if we used base 6 numbering instead, for one thing.

I can indeed see where you're coming from here, I'd put most certainly 3 multiplied by 1/3 as equal to 1, such as when you take a circle, slice it into three equal slices and put them together again you get 1 circle.  So, 3 by 1/3 = 1, no argument from me there.  Tough one.  Anyway, you do indeed have a point there, unlike Libertas, who just trolls.  0.999..., weird number, cosmetically unequal to 1, yet for the purposes of most equations the former's treated just like the latter.

     Well what happens is that the difference is infinitely small. As we know, 1/infinity=0. Because of that, any two numbers where the distance in between them is infinitely small are equal, because an infinitely small number is just equal to zero.

[Peter]

I'm a "high school" maths teacher in the UK, with a degree in Mathematics. I have to agree with 0.999999999999999... = 1 being true.

Another argument I am yet to see in this thread is the following:

Let us say that:

x = 0. 999...
then
10x = 9.999... since the decimal is recurring.

Finding the difference between these:
10x-x = 9.999... - 0.999...
so
9x = 9
so
x = 1
QED.

[Sibboleth]

QED.

Ah, finally I can comment in a non "numbers scare me" way - the place where they do endoscopies in the old Q.E. building is called QED.

[Bacon King]

.9999999999999999999999999999... repeating equals 1. It took me a few years to understand it as a kid but then I realized the nature of infinity.

That is, 1 - .999999999 repeating equals something INFINITELY small, and something infinitely small is, quite literally, nothing.

[Peter]

Quod erat demonstrandum, or alternatively Quite Easily Done.

[Lief]

Al is the only one in this entire thread making any sense.

[phk]

Question for the math and number geeks out there.

Could 1 - .9999(repeating) = *, be regarded as true?
* defined as an infinitesimal

[So the Heroes Fall]

Question for the math and number geeks out there.

Could 1 - .9999(repeating) = *, be regarded as true?
* defined as an infinitesimal

As discussed in this thread no. Basically "infinitesimally small" does not exist, it equals zero.

[Citizen James]

Question for the math and number geeks out there.

Could 1 - .9999(repeating) = *, be regarded as true?
* defined as an infinitesimal

If you really want to get your geek on, Wikipedia has a quite well documented page on the subject.

I suspect part of the confusion is in comprehending the concept of infinity.  And that it is a  concept - not a number.  A phrase such as (infinity plus one) has no real meaning beyond humor value.

The Zeno's paradox mentioned earlier gets to the heart of the concept.  Consider the following situation - a runner is heading toward the finish line.   In a certain amount of time, he will be 9/10ths of the way there (.9), later he will close 9/10 of the remaining distance (reaching .99) this continues for another 9/10ths of the remaining distance (.999) and so on.  Will the runner ever cross the finish line? (try this yourself if you aren't sure).  Infinitely close (and infinitesimal distance) is nothing.  The great and mighty wiki also has an article on the concept of infinitesimal (as well as noting that the concept is different than the common usage - merely meaning 'very small').

[A18]

Question for the math and number geeks out there.

Could 1 - .9999(repeating) = *, be regarded as true?
* defined as an infinitesimal

As discussed in this thread no. Basically "infinitesimally small" does not exist, it equals zero.

There are no non-zero infinitesimal real numbers. There are non-zero infinitesimal hyperreal numbers, however.

[dead0man]

I have nothing to add to the thread, but I'd like to point out that if somone gets the answer "wrong" and it pisses you off and makes you think less of them, YOU ARE AN ASSHOLE!  Period, end of story.