Someone help me with the 0.9 repeating = 1 proof

[videogames101]

My favorite by far...



You accept 1/3=.333...? Yes

And you accept that 1/3*3=3/3=1? Yes

So, .333...*3=.999...=3/3=1

So simple...

 

[Eeezee]

Originally posted by: videogames101
My favorite by far...



You accept 1/3=.333...? Yes

And you accept that 1/3*3=3/3=1? Yes

So, .333...*3=.999...=3/3=1

So simple...

Without a doubt, that SHOULD work, and why it wouldn't is beyond my comprehension.

 

[Corolinth]

Originally posted by: DrPizzaYou're getting closer... yes, "1 over infinity" is frequently seen as a limit in calculus, and it evaluates to 0. Exactly 0. So, if we're looking at the value of 1/x as x approaches infinity, the limit = 0. (1/x "approaches" 0, but the limit *IS* zero.) I always have to give my calculus students counter-intuitive limits to show them why we can't simply "plug in" infinity (or plug in zero). That's because when doing limit problems, it's inappropriate to say "plug in" infinity, but unfortunately that's exactly what's often done out of laziness. Off the top of my head, here's an example showing this difference (although my example is one for plugging in 0)

In order for a limit to exist at any given point, the right-hand limit and left-hand limit must be the same. The left-hand limit of 1/infinity is 0, but since we can't evaluate it from the right, the right-hand limit does not exist. Since the left-hand and right-hand limits of 1/infinity are not the same....

 

[Born2bwire]

Originally posted by: Corolinth

Originally posted by: DrPizzaYou're getting closer... yes, "1 over infinity" is frequently seen as a limit in calculus, and it evaluates to 0. Exactly 0. So, if we're looking at the value of 1/x as x approaches infinity, the limit = 0. (1/x "approaches" 0, but the limit *IS* zero.) I always have to give my calculus students counter-intuitive limits to show them why we can't simply "plug in" infinity (or plug in zero). That's because when doing limit problems, it's inappropriate to say "plug in" infinity, but unfortunately that's exactly what's often done out of laziness. Off the top of my head, here's an example showing this difference (although my example is one for plugging in 0)

In order for a limit to exist at any given point, the right-hand limit and left-hand limit must be the same. The left-hand limit of 1/infinity is 0, but since we can't evaluate it from the right, the right-hand limit does not exist. Since the left-hand and right-hand limits of 1/infinity are not the same....

No, the limit of 1/x as x->+/-infinity is zero. Take a look in any single-variable calculus text. One simple proof is that for epsilon > 0, let N=1/epsilon. Then for x > N, we have,

x>N=1/epsilon so that 1/x<epsilon and thus |1/x-0|<epsilon. Thus the limit of x->+/-infinity of 1/x = 0.

Limits involving infinity are defined that for any number epsilon>0, there exists a number N such that,
|f(x)-L|<epsilon when x>N where lim x->infinity of f(x)=L.

 

[Rangoric]

Originally posted by: DrPizza
So, recapping: the limit of 1/x as x approaches infinity IS EXACTLY ZERO... it's not the smallest number greater than zero.

Thats right, the limit is 0. That is correct.

1/Infinity however, is not 0. It is the smallest value greater then 0. (Considering +Infinity, it would be the largest value below zero if it was -infinity).

BTW, I also don't like that 1/3 = 0.3333... Having to use "..." to me shows that the system you are using is unable to accuratly represent the translation. Just like somethings in Russian just don't make it into English without sounding rather silly. As such I see 0.333... as a very good and usable approximation of 1/3, its just missing something.

But alas when you use the flawed decimal system(See Below), this comes up now and then. However I don't mind because...

For normal everyday and even scientific purposes, it is so close, so very super duper crazy close, as to be perfectly usable as accepted. But that doesn't mean I'll close my mind to what is wrong with the inherent system.

(About Decimals, if the system was not flawed, there is no number that would be unable to be represented by a finite string. Also there would be no need to prove that something equals something that is not represented by the same string because it would be impossible for that to happen.)

 

[Howard]

Originally posted by: Rangoric

Originally posted by: DrPizza
So, recapping: the limit of 1/x as x approaches infinity IS EXACTLY ZERO... it's not the smallest number greater than zero.

Thats right, the limit is 0. That is correct.

1/Infinity however, is not 0. It is the smallest value greater then 0.

What is the smallest value greater than 0?

 

[blackllotus]

Originally posted by: Rangoric
1/Infinity however, is not 0. It is the smallest value greater then 0. (Considering +Infinity, it would be the largest value below zero if it was -infinity).

Infinity is a limit therefore 1/infinity must be evaluated as a limit. 1/infinity is 0.

 

[pcy]

Hi,


The problem here is all to do with matematics and in particular with what certain statements mean.

1/infinity = 0 ?????????????????

No it does not. It does not equal anything. It is not a numerical expression. Infinity is not a number in the normal sense.

1/infinity is shorthand for:
Lt {x -> infinity} 1/x
which clearly does equal 0, not least because mathematians, over the centuaries, have defined very clearly what things like:
Lt {x -> infinity} 1/x
mean, and how you evaluate them.


Thge original propblm i.e.
does 0.99999999... = 1
suffers from precisely the same problem. It's an infinite series i.e.
0.9 + 0.09 + 0.009...
and to determine what it's equal to, we have to accept what mathematitions say it means.

To do this we need to Define N(n) = -.999999999.... (i.e Nine repeated n times)

Clearly N(n) = 1 - 1/10*n


We next have to decide what we mean by 0.999999 repeated indefinately. It is a peice of mathematical notation. It means whatever mathematicians agree it means. They do agree, it is:
Lt {x -> infinity} N(n)
= Lt {x -> infinity} 1 - 1/10*n
= 1 - Lt {x -> infinity} 1/10*n
= 1
because of the laws that matematician define for calculationg such expressions.



All the way down the line the problems exist because many people find the concept of an infinite series rather counter-intuitive. This made worse by asserions like "1/infinfiy is 0", rather than "matematiciams have defined what we mean when we say 1/infinity, and according to that definition it does come to zero".


The definitions, are not peverse. In particular they serve to resolve Xeno paradox - Achilles and the Tortoise. In this verion Achiles runs 10 times as fast as the Tortoise for reasons that will become clear.

The Toriose starts at a distance .9 of the length of the race course from the start - i.e the point where Acchiles starts. Just for convenience lets say it will take Achilles 1 min to cover the course, arrioving at the finish at exactly the same time as the Tortoise.

So, after 0.9 mins Acchilles reaches the point where the Tortoise started. The Totoise is now 0.99 of the way down th course. In another 0.09 mins Achilles reaches that point....


What is happening here, is that the race is being cut up into smaller and smaller time intervals, so that an infinte number of these intervals are required before Achilles and the Tortoise both reach the finish simultaneously.

We all know that this tales one minute, and it's clear that this subdivision of time is entirely artificial, prioducing an infinte series of time intervals which nevertheless ad up to 1.

All that matematics has done is establish the notation and the rules so that the paradox collapses and we get the obvioulsy correct answers in these sorts of situations.



And in consequecne 0.99999999..... (repeated indefinately) does equal 1.





Peter

 

[Born2bwire]

Originally posted by: Rangoric

Originally posted by: DrPizza
So, recapping: the limit of 1/x as x approaches infinity IS EXACTLY ZERO... it's not the smallest number greater than zero.

Thats right, the limit is 0. That is correct.

1/Infinity however, is not 0. It is the smallest value greater then 0. (Considering +Infinity, it would be the largest value below zero if it was -infinity).

BTW, I also don't like that 1/3 = 0.3333... Having to use "..." to me shows that the system you are using is unable to accuratly represent the translation. Just like somethings in Russian just don't make it into English without sounding rather silly. As such I see 0.333... as a very good and usable approximation of 1/3, its just missing something.

There is no approximation here. We can fully equate the decimal of 1/3, or .9r, using summation series. 1/3 = sum(n=1, n->infty) (3*10^-n). The infinite sum implies that we are taking the limit of a sequence of numbers. Hence, the 1/infty here is taken to be a limit term and thus we do not have any approximations with the infinite summation.

 

[ReDx]

i don't think you can prove it with algebra, you need some understanding of limits, any actuaries out there? lol

 

[BrownTown]

its true by deffinition of real number.

 

[RaynorWolfcastle]

I didn't read through this whole thing because it's kind of ridiculous to go through it every couple of months, but I just did it by hand just for kicks (using geometric series, I'm sure someone mentionned it).

0.9999... = 9*limit [x -> infinity] {sum(10^-n,1..x)}
let's assume that S = limit[x ->infinity]{sum(10^-n,1..x}
so 0.999 = 9*S

Now we find S, which is a geometric series. I'll be a little careless with the way I move the limit around, but it works rigorously if there are any mathematicians in the house
S - 1/10*S = lim[x -> infinity] {sum(10^-x,1..x) - 1/10*sum(10^-n,1..x)}
9/10*S = lim[x -> infinity] {sum(10^-x,1..x) - sum(10^-n,2..x+1)}

All but the first and last terms of these series cancel each other out, so now we have:

9/10*S = lim[x -> infinity] {10^-1 - 10^(x+1)}
9/10*S = 10^-1 - 0
S = 1/9

so we go back above and say:
0.99999... = 9*S
0.99999... = 9*1/9
0.99999... = 1

QED

 

[alimoalem]

i think ed's post (his links) best describe it...the one above the wikipedia one is the one i read

 

[jagec]

Originally posted by: Howard

Originally posted by: Rangoric

Originally posted by: DrPizza
So, recapping: the limit of 1/x as x approaches infinity IS EXACTLY ZERO... it's not the smallest number greater than zero.

Thats right, the limit is 0. That is correct.

1/Infinity however, is not 0. It is the smallest value greater then 0.

What is the smallest value greater than 0?

0.00000...(infinity-1 times)...1:laugh:

You're not going to convince a Jehovah's Witness that Xenu doesn't exist, and you're not going to convince the .99999... people that it is, in fact, equal to one. The proof requires an understanding of mathematics that, once possessed, makes you no longer a believer.

 

[darkhorror]

you can't do operations on infinity 1/infinity can not be definied as infinity is not a number. This is a reason we use limits that way we can deal with infinity, we never actually get 1/infinity but we can get the limit as 1/x goes twards 1/infinity.

In order to understand .999... you must understand infinity. And saying that you can put another number after the 9's means that they don't, thus they will not understand why .999... = 1 untill they do. One of the things I hear from people is that they try and take .9, .99,.999, .9999,... and keep on doing then say that no matter how many 9's you add you will never "reach" 1. Which is completely true, but then take there same .9, .99, .999, .9999, ... no matter how many 9's you add you will never "reach" .999... ether, because at no point will you "reach" infinity. Which would mean that .999... is not part of the .9, .99, .999,... that they said wasn't equal to 1. That means that the .9, .99, .999, ... that they were thinking of as being not equal to 1 is also not equal to .999... So they would have to think about it another way to get what it really means.

Once they understand infinity and that there is no end to the 9's then they should be able to understand why it's equal to 1.

 

[RapidSnail]

Marked for later. I have questions from a sceptic in the family.

 

[kevinthenerd]

Originally posted by: blackllotus

Originally posted by: Rangoric
1/Infinity however, is not 0. It is the smallest value greater then 0. (Considering +Infinity, it would be the largest value below zero if it was -infinity).

Infinity is a limit therefore 1/infinity must be evaluated as a limit. 1/infinity is 0.

lim (x->inf), x = inf
therefore inf = lim (x->inf), x

therefore

1 / inf = 1 / lim (x->inf), x

1 / inf = lim (x->inf), 1/x = 0

 

[darkhorror]

Originally posted by: kevinthenerd

Originally posted by: blackllotus

Originally posted by: Rangoric
1/Infinity however, is not 0. It is the smallest value greater then 0. (Considering +Infinity, it would be the largest value below zero if it was -infinity).

Infinity is a limit therefore 1/infinity must be evaluated as a limit. 1/infinity is 0.

lim (x->inf), x = inf
therefore inf = lim (x->inf), x

therefore

1 / inf = 1 / lim (x->inf), x

1 / inf = lim (x->inf), 1/x = 0

I am not exactly sure what you are trying to show here.

Lim(x->infinity) of x does not mean that you can take x = infinity. As x is never infinity.

lim(x->inf) of x is infinity
lim(x->inf) of x^2 is infinity along with an infinite more posibilities.

this means that you can't use infinity as a number since it isn't.

 

[DrPizza]

Originally posted by: Rangoric

Originally posted by: DrPizza
So, recapping: the limit of 1/x as x approaches infinity IS EXACTLY ZERO... it's not the smallest number greater than zero.

Thats right, the limit is 0. That is correct.

1/Infinity however, is not 0. It is the smallest value greater then 0. (Considering +Infinity, it would be the largest value below zero if it was -infinity).

BTW, I also don't like that 1/3 = 0.3333... Having to use "..." to me shows that the system you are using is unable to accuratly represent the translation. Just like somethings in Russian just don't make it into English without sounding rather silly. As such I see 0.333... as a very good and usable approximation of 1/3, its just missing something.

(About Decimals, if the system was not flawed, there is no number that would be unable to be represented by a finite string. Also there would be no need to prove that something equals something that is not represented by the same string because it would be impossible for that to happen.)

Again, 1/infinity is meaningless. Infinity is not a number. 1/infinity only arises in situations where you are evaluating a limit, and is merely shorthand for the denominator is getting larger (in either the positive or negative direction) without bound.

.333... works just fine as notation for 1/3. It's accurate: a decimal and an endless string of 3's following it.

Your claim about the decimal system being flawed makes no sense. There are an infinite number of "sizes" between 0 and 1. It's somewhat reminiscent of the ancient greek's belief that all numbers had special meaning and all values could be written as the ratio of two integers. Do you have some way of proposing that a system could exist to describe an infinite quantity of values with finite strings? This idea seems that it has been outdated ever since the first proof of the existence of irrational numbers.

 

[pcy]

Hi,

Agreed...


Infinity is not a number.


So, 1/Infinity is indeed meaningless, unless we can all agree that it is actually shorthand for:
Lt {x -> infinity} 1/x

If we can accept that, the next issue is, what is the value of:
Lt {x -> infinity} 1/x

The answer is zero. We know this because of the mathematical definition of a limit. You can argue that you don't like the definition, but the fact remains that:
1. Lt {x -> infinity} 1/x is a piece of mathematical notation
2. mathematics defines what that notation means
3. according to that definition the result exists and it is zero.


The same arguement applies to the original question of 0.9999* (i.e. the 9s repeating indefinately). 0.9999* = 1 because it is a piece of mathematical notation and the rules of mathematics say it does.

The point here is that we all agree what the meaning of any finite decimal expansion means. But infinity is not just a very big number, its well... infininite - it goes on for ever. An infinite decimal expansion means whatever mathematics says it means.

In fact the definition is pretty obviious:

0.9999* means Lt {n -> infinity} 0.99999...9 (9 repeated n times)

Nobody, I hope is going to deny that the value, for all finite n is 1 - 1/10*n (10 to the power of n)

So 0.9999* is 1, by virtue, purely, of what mathematics says it means, if and only if:
Lt {n -> infinity} 1/10*n = 0

As n increases 1/10*n steadily approches zero, getting arbitrarily close to zero. This (in non technical terms) is the definition of a limit; so the limit exists and it is zero.

Any other definition would be perverse. I hope we can all agree that if 0.9999* means anything at all, and has a value, then that value must be closer to one than any possible value other than one. A moment's though will convince you that no value excpet one can possibly meet that criteria.


Finally, may I just observe that there is no such thing as the "smallest number" greater than zero. If there were, then 1/10th of it would be even smaller, but still greater than zero.


That tells us the the nummber 0.0000*1 does not exist. This expression supposedly means an infinite number of zeros folloed by a 1. But infinity does not end, so it cannot be followed by anything. the final 1 does not exist because there is no place for it to be. In fact, and for precisely this reason, mathematics does not admit to this expression - it is simply not a number.




Peter

 

[CSMR]

Originally posted by: DrPizzaInfinity is not a number... This idea seems that it has been outdated ever since the first proof of the existence of irrational numbers.

When you say number you seem to be thinking of "real number" but the greeks did not mean real number by number but rational number. Isn't that right?

 

[pcy]

Hi,

I don't know what DrPizza was thinking of, but the fact remains that Infinity is not a number



Peter

 

[videogames101]

Here I go again...

1/3=.333...
then

3/3=.999...


It's not rocket science.

 

[PowerEngineer]

It still remember my high school freshman algebra test in which we were asked to rationalize 2.9999...

I ran through the process: X=2.9999... 10X=29.9999... 9X=27.0 X=3.0

My teacher saw the stunned look on my face, and after seeing my answer asked me to think about why I was surprised. It dawned on me then that this proved that 0.9999... was actually equal to 1.0! (It seems to me that these kind of learning experiences while taking a test are the most powerful.) I can't understand why others aren't completely convinced by this demonstration.

 

[pcy]

Hi,


That proof is invalid, IMO.

3 x 0.3* = 1, not 0.99999*, if it is defined/permitted at all.

Infinity is getting in the way here - and the rules for performing arithmetic on infinte decimal expansions are certainly couter-intuitive, if they exist.

The problem is that the rules for multiplying decimal expansions require that you start at the least significant end, and work up the line, in order to correctly handle any carries. With infinite decimal expansions there is no end - the least significant digit does not exist, so you cannot even start, far less complete, the operation.



In addition, even if it were valid, this proof fails to help anybody who denies that 0.9* =1, because it uses pricisely the notations (i.e. an infintie repeating decimal expansion) that is the cause of all the trouble.

Anybody who denies that:
0.9* = 1 is having problems with the definitaion of an infinite repeating decimal expansion, and therfore might also deny that:
0.3* = 1/3

It's the same problem:
1/3 - 0.33333..3 (3 repeated n times)
is clearly equal to 1/3x10*n
jist as 1 - 0.99999..9 (9 repeated n times)
is equal to 1/10*n

Accepting thet 1/3 = 0.3* demands acceptance of the definition of an infinte decimal expansion, and also demands acceptance that:
Lt {n -> infiinty} 1/3x10*n = 0

If you accept those two things it's hard to see how you can fail to accept that:
Lt {n -> infiinty} 1/10*n = 0
which is the key step to accepting that 0.9* = 1




Peter