More on 0.999~ vs 1: Comparing Numbers

[HallsofIvy]

Am I incorrect in the premise that 0.9... is an irrational number?

scott_alexsk: any repeating decimal is rational. Even if you didn't know that 0.9...= 1, the fact that it is repeating tells us that it is rational.

 

[Curious3141]

Your ability to write nonsense is amazing.

Sorry for the OT remark, but I just wanted to say that *that* made me LOL. Literally. I'm still LOL-ing, I don't know why I find it so funny, it just seems so deadpan.

 

[scott_alexsk]

Well my thought was 0.3... and 0.6... are rational since they can be written as fractions, 0.9... cannot since 3/3=1. But I guess I was wrong.

-scott

 

[arildno]

Eeh? Isn't 3/3 a fraction??

 

[Robokapp]

Isn't an irrational number a number where the next decimal does not repeat in a pattern and there are infinitely many? or something like that?

that's not a deffinition, that's my interpretation but...a number that repeats 9s is rational. it's not natural, nor whole, but it belongs to rational numbers...right?

 

[StatusX]

Rational numbers are real numbers which are equal to p/q for some (and in fact infinitely many) integers p and q. That they have repeating or terminating decimal expansions is a secondary fact about them, but is not a good property to define them by, simply because base 10, or any other base, is completely arbitrary. 0.999... is a rational number because 1 and 1 are integers, and 0.999...=1/1. By the way, if you agree 0.999... is rational, what are some integers p,q with p/q=0.999...? Remember, all integers are finite numbers (neither infinity nor infinity-1 is an integer).

 

[d_leet]

Well my thought was 0.3... and 0.6... are rational since they can be written as fractions, 0.9... cannot since 3/3=1. But I guess I was wrong.

-scott

Ok but you do know that the sum of two rational numbers will still be rational right, so then notice that .9... = .6... + .3...

 

[chroot]

Well my thought was 0.3... and 0.6... are rational since they can be written as fractions, 0.9... cannot since 3/3=1. But I guess I was wrong.

-scott

That's sort of the point kiddo. The fraction 3/3 can also be written as 1, and can also be written as 0.999...

- Warren

 

[HallsofIvy]

Isn't an irrational number a number where the next decimal does not repeat in a pattern and there are infinitely many? or something like that?

that's not a deffinition, that's my interpretation but...a number that repeats 9s is rational. it's not natural, nor whole, but it belongs to rational numbers...right?

No, what just about every one here is saying is that 0.999..., infinitely repeating 9s is a natural number and a whole number (as well as an integer). It is exactly equal to 1.

 

[shmoe]

That they have repeating or terminating decimal expansions is a secondary fact about them, but is not a good property to define them by, simply because base 10, or any other base, is completely arbitrary.

To expand a little- whether a rational number is terminating or repeating depends on base. eg. 1/5=.2 in decimal but .001100110011... in binary.

However, non-terminating and non-repeating in one base, non-terminating and non-repeating in all bases, so you can use this to define irrational/rational, though the p/q version is nicer.

 

[HallsofIvy]

I remember, years ago, a person asking on a math forum, "how to prove that a rational number can be written as a fraction". I stared at that, dumbfounded, for a while before realizing that he must have learned "can be written as a repeating or terminating decimal" as the definition of rational number.

 

[Robokapp]

No, what just about every one here is saying is that 0.999..., infinitely repeating 9s is a natural number and a whole number (as well as an integer). It is exactly equal to 1.

I may lack the capability to explain it...but if it is true, assuming it is...you can't write your number. 0.99999 will eventually be rounded...up or down. if you round it down you end up with a lot of 9s...but not all of them. if you round it up you end up with a lot of zeros but not all of them...and a 1 in front. Out of comodity you don't write 1.00000...and that is correct. you don't have to. but you're not getting away when the 0.999... comes.

The way I see it is like that proof that 1 = 2. all you got to do is divide by (a-b) after you were told that a=b. yeah, 1 does = 2 once you assume 0 divided by 0 = 1.

So until you actaully write the 0.999... number it does not matter what it's equal to. It's not only unreal but inexistent, that's the point. You can write parts of it and talk about the hole, but what you are NOT writing is the tiny difference between many 9s and many 0s.

Well I probably pissed a lot of people off with what I just wrote but that's how i see it. You're comparing apples with photos of oranges and they're almoust the same...only problem is...one of them doesn't really exist.

Well it does, you just can't express it. The difference between 0.999... and 1.000... is so far back in the decimals that you don't get to write it ever. Do you know what it looks like? No. You got no idea. It's like graphing an asymptote. It goes up and ends below the starting point. you can come infinitely close to it, but you're never gonig to reach it...

That's my best approach.

 

[FrogPad]

I may lack the capability to explain it...but if it is true, assuming it is...you can't write your number. 0.99999 will eventually be rounded...up or down. if you round it down you end up with a lot of 9s...but not all of them. if you round it up you end up with a lot of zeros but not all of them...and a 1 in front. Out of comodity you don't write 1.00000...and that is correct. you don't have to. but you're not getting away when the 0.999... comes.

The way I see it is like that proof that 1 = 2. all you got to do is divide by (a-b) after you were told that a=b. yeah, 1 does = 2 once you assume 0 divided by 0 = 1.

So until you actaully write the 0.999... number it does not matter what it's equal to. It's not only unreal but inexistent, that's the point. You can write parts of it and talk about the hole, but what you are NOT writing is the tiny difference between many 9s and many 0s.

Well I probably pissed a lot of people off with what I just wrote but that's how i see it. You're comparing apples with photos of oranges and they're almoust the same...only problem is...one of them doesn't really exist.

Well it does, you just can't express it. The difference between 0.999... and 1.000... is so far back in the decimals that you don't get to write it ever. Do you know what it looks like? No. You got no idea. It's like graphing an asymptote. It goes up and ends below the starting point. you can come infinitely close to it, but you're never gonig to reach it...

That's my best approach.

You have some weird logic my friend.
I am reminded of an event that recently happened. A friend of mine said to me that "a car is a car".
I told him,
"no, a car is not just a car. For examplle, there is a HUGE difference between a Ferarri and a Hyundai".
He said,
"well a car is something that gets you from point A to point B".

We were in the weight room at the time, so I said,
"Well that bench has wheels on it." As I pointed to a weightlifting bench. "I could push you from this spot here, to that spot over there." As I pointed with my finger from one carpet square to another. "I guess you would classify the bench as a car then?"

He said,
"a car is a car"

I just left it at that.

 

[d_leet]

I may lack the capability to explain it...but if it is true, assuming it is...you can't write your number. 0.99999 will eventually be rounded...up or down. if you round it down you end up with a lot of 9s...but not all of them. if you round it up you end up with a lot of zeros but not all of them...and a 1 in front. Out of comodity you don't write 1.00000...and that is correct. you don't have to. but you're not getting away when the 0.999... comes.

The fact is that we are not talking about.99999 or even .99999...99 where there are only a finite number of nines, we are talking about the number .999... where there ARE an infinite number of nines, we are not talking about rounding up or down and that really has nothing to do with what this number is EQUAL to. The thing is that we really don't need to write all of the nines because we've come up with notation that represents that this or any pattern will repeat forever and that is the use of ellipses(...).

The way I see it is like that proof that 1 = 2. all you got to do is divide by (a-b) after you were told that a=b. yeah, 1 does = 2 once you assume 0 divided by 0 = 1.

Can you explain in detail how we are doing something similar to this ( dividing by zero or some other mathematical fallacy) when it comes to showing that .999... is indeed equal to 1?

So until you actaully write the 0.999... number it does not matter what it's equal to. It's not only unreal but inexistent, that's the point. You can write parts of it and talk about the hole, but what you are NOT writing is the tiny difference between many 9s and many 0s.

Why do we need to write the whole number when we have something that represents this number perfectly which is .9... or
$$9\displaystyle\sum_{i=1}^\infty 10^{-n}$$
These two expressions represent the same thing, the same number, now do you care to explain how this is only part of the number, and not the whole? And how is .999... "unreal" and "inexistent"?

Well I probably pissed a lot of people off with what I just wrote but that's how i see it. You're comparing apples with photos of oranges and they're almoust the same...only problem is...one of them doesn't really exist.

Your analogy makes no sense for several reasons, first of which I'm sure all of the members on this forum would agree that apples are real and just as real as pictures of oranges are. Second can you explain how exactly the difference between 1 and .999... does not exist? Since it has been said and shown it this thread that it does exist and is equal to zero.

Well it does, you just can't express it. The difference between 0.999... and 1.000... is so far back in the decimals that you don't get to write it ever. Do you know what it looks like? No. You got no idea. It's like graphing an asymptote. It goes up and ends below the starting point. you can come infinitely close to it, but you're never gonig to reach it...

Sure we can express it since it's equal to zero. It's not an assymptote because when we write .999... we are not saying that we are continually adding nines to .999 forever and ever but we are saying that THERE ARE AN INFINITE number of them, so we are never trying to reach anything in a sense we're already there.

I don't see you having any complaints in this thread about 1/3 being equal to .333... so why only a problem with .999...

 

[matt grime]

I may lack the capability to explain it...but if it is true, assuming it is...you can't write your number.

Why can't I? 0.999..., there I wrote it, or 1, I just wrote it again. You might mean that it doesn't have a terminating decimal expansion, but then almost no numbers do (only p/q in lowest terms where q is only divisible by 2 and 5 and no other primes, or p/q is an integer). In anycase, this has nothing to do with what 0.99... is.

0.99999

we aren't talking about 0.99999

will eventually be rounded...up or down.

That is not correct.

The way I see it is like that proof that 1 = 2.

No it is not. It is a prefectly valid statement, and easily proved, that as decimal representations of real numbers 0.999... and 1 are equivalent.

It's not only unreal but inexistent, that's the point.

THis is not a debate about platonism, is it?

You can write parts of it and talk about the hole, but what you are NOT writing is the tiny difference between many 9s and many 0s.

what does that mean? (Note: rhetorical question.)

Well I probably pissed a lot of people off with what I just wrote but that's how i see it.

You just demonstrated that you're not going to listen to reason, or read up on people's suggestions, and instead will just continue arguing for something from a poisition of ignorance. PLenty of people have expended time and considerable effort in this thread to explain what is going on and you've steam-rollered over it and ignored it all. If you think that pisses people off you're probably making a safe bet.

You're comparing apples with photos of oranges and they're almoust the same...only problem is...one of them doesn't really exist.

Well it does, you just can't express it.

You've just contradicted your own (bad) analogy.

The difference between 0.999... and 1.000... is so far back in the decimals that you don't get to write it ever.

this demonstrates you've not bothered to consider people's explanations of what decimal representations of real numbers are.

Do you know what it looks like? No. You got no idea. It's like graphing an asymptote. It goes up and ends below the starting point. you can come infinitely close to it, but you're never gonig to reach it...

That's my best approach.

your best approach would be to actually read what other people have read and consider the mathematics behind it: not what you think these things represent but what they actually do represent.

 

[uart]

ok Robokapp. If you want a really unsophisicated argument then think about it like this. Let's just say for one moment that 0.999... and 1 are different ok. Then by what amount do they differ?

The usual punter will think about this for a while and then reply that they differ by 0.000...1

Now let's look at the "number" represented by 0.000...1, that's a decimal point followed by an infinite number of zeros with a one placed right on the end. Really think about what that means, the zeros go on for ever, they never ever end, and then when they end you want to place a one. Can you see the logical falacy there, the one on the end is pointless, if the zeros never end then you simple *cant* place a one on the end. In other words 0.000...1 is not a valid representation of a number, there's really no point in even writting it, but if we do write it then we must at least agree the it equals zero exactly.

So 0.999... and 1 are different and the amount that they differ by is precisely zero. Ok well maybe we should just say they're equal then. You know it makes sense.

 

[Robokapp]

Well...I was thinking about just that. I started typing a message but I erased it...but here's basically what my answer is:

1/2=0.5
2/3=0.666...
3/4=0.75
4/5=0.8
(...)
99999/100000=0.99999

So...the limit of (x-1)/x as x approaches positive infinity is...1 becasue it's an equal heavy rational equation.
Point is...how do you obtain a 0.9999...?
You go as close as possible to infinity for x in this equation: (x-1)/x

But why I'm showing all this: (x-1)/x will always be subunitray...never equiunitary. (I hope those words actaully exist. I never used them in english. I am taking about top < bottom vs top = bottom).

So...you do pull out a 0.9999... if you go far enough, but you do not pull out a 1 no matter where you go. Isn't that what an assymptote is?
I hope that makes any sense. This topic is getting personal for me because it doesn't take much complex math, yet it's...a curiosity if you will.

 

[uart]

Ok, so you don't understand the concept of infity or limits. You don't understand the difference between infinity and a very large but finte number. That's ok some people just don't have the mental capacity to ever grasp some concepts, no body here will really care if you accept your limitations and move on. Just don't try and use you personal mental limitations to "prove" that accepted mathematical fact is wrong.

 

[StatusX]

You agree that (x-1)/x has a limit of 1 as x goes to infinity. Let's say we use the notation [f(x)] to designate the limit of f(x) as x goes to infinity, which is just a number (if the limit exists). Then [(x-1)/x]=1. It doesn't "approach" 1, it just is 1. (x-1)/x approaches 1 as x gets large, but we have defined [(x-1)/x] to be the limit, which we all agree is 1. So, with this notation I've just invented, you would agree that [(x-1)/x]=1, right?

Well 0.999... is the same thing. We use the ... notation for the limit as more and more 9's are added. At any stage, a number with many 9's will be less than one, but 0.999... does not refer to any of these stages. It refers to the limit, and this is exactly equal to 1.

 

[Hurkyl]

Point is...how do you obtain a 0.9999...?

Obtain... from what? Do you mean the function f(x) = (x-1)/x? You cannot "obtain" 0.999... from f. In fact... if you do a little algebra, you will find that 1 is the only number you cannot "obtain" from f, and thus 0.999... = 1.

You go as close as possible to infinity

There is no "as close as possible to infinity". There are only two possibilities in the extended reals: you are either "at infinity", or there is something between you and "infinity". Or, symbolically...

If $x \neq +\infty$, then $x < 2|x| < +\infty$, no matter what x is.



Can we agree on the following two things?

(1) $0.\bar{9} \leq 1$
(2) $y < 0.\bar{9}$, whenever y is equal to zero followed by a finite number of 9's after the decimal point

If we can agree on these, then it's easy to prove that $0.\bar{9} = 1$ as follows:

Let x be any number smaller than 1.
Then, $1 - 10^{-k} > x$ for some positive integer k. For example, we could choose $k = \lceil -\log(1 - x) \rceil$.

However, $1 - 10^{-k} = 0.99\cdots9$, where the r.h.s. has exactly k 9's. Therefore, $x < 0.\bar{9}$.

This proves that $0.\bar{9}$ is bigger than any number smaller than 1... and therefore $0.\bar{9}$ cannot be less than 1. Therefore, it must equal 1.

 

[gnomedt]

Criticize this:

To convert a decimal with a repeating pattern of period R into a fraction, the standard procedure is to form a natural number with the digits of the pattern, divide said number by (10^(R)-1) = 999...^{{R times}}999, and reduce the fraction. For instance:

0.2857142857142857...
=285714/999999 (confirm it)
=(285714/142857)/(999999/142857) (dividing top and bottom by gcd)
=2/7
which is the correction fractional equivalent of the decimal above.

Hence,
0.99999999...
=9/9
=1

 

[HallsofIvy]

Criticize this:

To convert a decimal with a repeating pattern of period R into a fraction, the standard procedure is to form a natural number with the digits of the pattern, divide said number by (10^(R)-1) = 999...^{{R times}}999, and reduce the fraction. For instance:

0.2857142857142857...
=285714/999999 (confirm it)
=(285714/142857)/(999999/142857) (dividing top and bottom by gcd)
=2/7
which is the correction fractional equivalent of the decimal above.

Hence,
0.99999999...
=9/9
=1

Perfectly valid demonstration, but to be a "proof" that 0.999...= 1, you would have to show that the calculation done in, for example, writing 10(0.999...)= 9.999..., is valid. To do that (it is true, of course) you would have to appeal to the definition of base 10 representation of a number less than 1 and the definitions of multiplication and subtraction of such things. It is simpler to use that definition to state that 0.999... means the infinite series $\sum_{n=1}^\infty (0.9)^n$ and then argue that that is a geometric series with sum $\frac{0.9}{1- 0.1}= 1$.

 

[Robokapp]

The demonstration I was presented with in my 7th grade I think is a little different.

let's say you want 20.05676767676767... the 67s repeating. I chose that example so it shows more than one single step.

you'd pick 200567-2005 (whole - the non-repeating part) and divide by as many 9s as the repeating decimals...in this case 99...and add as many zeros as the number of decimal spots non-repeating...in this case two zero.

so 20.05(67) can be written as...

198562/9900. I have never met this process anywhere outside my country in any class...and i have never had to learn or use it again. However...it's been fustrating me for a lnog time that...exactly like this topic's subject here, it shows that 0.9999=1

it's just...(9-0)/9
I guess I'm terribly wrong. The problem I'm facing is...the word "equal" to me signifies perfect identic things on both sides of the sign, even in different forms. Without going into the math, one would immediatelly assume that 0.999... comes on the number axis a little ahead of the number 1. However I've been proven wrong so...I'm wrong.

I'm sorry if i put too much effort in this topic...it has been on my mind for a while. I had some "personal" struggles with exact form answers of problems like 1/2+2.(9)*5 or something like that.

One final question. Can you really say that you can be 'at infinity'? Reason I'm asking is in my 8th grade, same math teacher told me you never write infinity with a ] and never consider it a solution because it is never going to be reached and if it is it is not applicable. A comment on this?

Thank you for your time.
~Robokapp

 

[d_leet]

198562/9900. I have never met this process anywhere outside my country in any class...and i have never had to learn or use it again. However...it's been fustrating me for a lnog time that...exactly like this topic's subject here, it shows that 0.9999=1

It definitely does not show that 0.9999=1, .9999 = 9999/10000 which is not equal to one, but if we take the notation that .999... represents the infinite geometric series with .9 as the first term and 1/10 as the common ratio then this is equal to one .999... = 1.

it's just...(9-0)/9
I guess I'm terribly wrong. The problem I'm facing is...the word "equal" to me signifies perfect identic things on both sides of the sign, even in different forms. Without going into the math, one would immediatelly assume that 0.999... comes on the number axis a little ahead of the number 1. However I've been proven wrong so...I'm wrong.

i don't understand this. I'm sure that at some point in your mathematical experience that you've had to solve equations such as
3x - 7 = 7x -5
As you can see these two expressions on each side of the equals sign are exactly equal if x = -3 but these two expressions look nothing like each other, so it seems to me that the sense of equality you are thinking of is more "cosmetic" than mathemtical. You also say even in different forms and that is pretty much what .999... is, it is another form of 1 since the two are exactly equal.

One final question. Can you really say that you can be 'at infinity'? Reason I'm asking is in my 8th grade, same math teacher told me you never write infinity with a ] and never consider it a solution because it is never going to be reached and if it is it is not applicable. A comment on this?

Definitely don't take me as the expert on this because I'm not entirely sure, but in my complex analysis class we did make use of a point at infinity and when doing mobius transforms actually mapped this point at infinity to some complex value in the plane, but that could just be an severe informatlity that we are allowed to use because of more advanced mathematics, or something like that.

 

[matt grime]

The reason you don't write [0,oo] is because there is no point on the real line that is (labelled) infinity: it is redundant to allow it as a possibility in the interval. So, in the real numbers, you are, in some sense 'never at infinity'. However, I would absolutely refuse to use such terminology. Just stick to mathematics, and the definitions of the objects at hand.

It is perfectly possible, even useful to include extra symbols such as oo, or i, and extend the reals to other sets which have other/extra properties. Indeed, one of the most useful objects in mathematics is the complex sphere, or the complex plane with the point at infinity added (this is P^1(C) as well), and we can do projective stuff with point(s) 'at infinity'. You just have to stop thinking of these things in the physical manner in which you evidently want to.

You must stop thinking about the real numbers, whatever they may be, as decimal expressions. Those are just representations of real numbers. 1/2=2/4 causes you no problems, does it? Why does 0.999...=1 cause issues?

 

[Robokapp]

It definitely does not show that 0.9999=1, .9999 = 9999/10000 which is not equal to one, but if we take the notation that .999... represents the infinite geometric series with .9 as the first term and 1/10 as the common ratio then this is equal to one .999... = 1.

I wanted to say 0.9999...
I always talked about 0.9999... but i forgot the ... because that's not the notation I learned.
I learned that if you want to write...let's say 0.6767 repeating you write
0.(67) in parenthesys like that. But you all use the ... instead so I tried to use yours but at times I forgot it.

0.9999 is a banality to convert...also an irrelevance.
However 0.(9) using the method I did with the long number as example would look like this:

(the whole number - the non-repeating number)/as many 9s as different repeating decimals followed by as many zeroes as the decimals not repeating.

So...

(09-0)/9 is just 9/9 or 1.