Is 1 = 0.9999......

[josphII]

Originally posted by: bleeb
Here is a prove that is somewhat feasible:

Given:

Two numbers are not equal if the difference doesn't equal zero...

1 - 0.9 = 0.1

1 - 0.99 = 0.01

1 - 0.999 = 0.001

1 - 0.9999 = 0.0001

1 - 0.99999 = 0.00001

so on and so forth. You can clearly see there will always be a difference. Now the main point that everyone is saying that as this difference a super infinitely small number, it will converge to zero. Thus proving that the numbers are equal. But that only occurs when we reach the end of infinity. Which you can obviously tell will never happen. >=)

none of those numbers will ever equal 0.999... so your argument is crap

 

[josphII]

Originally posted by: josphII
1 does equal 0.999..., it is mathematical FACT

heres the proof:

Proof: 0.9999... = Sum 9/10^n
(n=1 -> Infinity)

= lim sum 9/10^n
(m -> Infinity) (n=1 -> m)

= lim .9(1-10^-(m+1))/(1-1/10)
(m -> Infinity)

= lim .9(1-10^-(m+1))/(9/10)
(m -> Infinity)

= .9/(9/10)

= 1

after posting this i cant believe anybody would still argue that 0.999.... does not equal 1

 

[bigredguy]

If somehting is 99.99..% pure is it 100% pure? or is there an insignificant % of contamination? (stepping outside the math box)

 

[josphII]

Originally posted by: bigredguy
If somehting is 99.99..% pure is it 100% pure? or is there an insignificant % of contamination? (stepping outside the math box)

in the physical world nothing can ever be 99.99.... percent pure

 

[Woodchuck2000]

0.99999999999999...
Tends towards 1. It will never equal one for obvious reasons but for any/all practial purposes it is equal to 1.

 

[josphII]

Originally posted by: Woodchuck2000
0.99999999999999...
Tends towards 1. It will never equal one for obvious reasons but for any/all practial purposes it is equal to 1.

incorrect. as stated earlier a single number is in fact, a single number and nothing more. single numbers, such as 0.999..., tend toward nothing. only series of numbers, or sums, can trend toward something, or have a limit.

the proof of 0.999... being equal to 1 is stated 3-4 posts up

i find it interesting that those that think 0.999.... is not equal to 1 cant find fault with the proof but rather, state such obviously incorrect statements.

 

[Woodchuck2000]

incorrect. as stated earlier a single number is in fact, a single number and nothing more. single numbers, such as 0.999..., tend toward nothing. only series of numbers, or sums, can trend toward something, or have a limit.

Any how do you intend to define .99 reccuring without using some kind of series? As the floating point precision with which you define the number increases, it tends towards one.

As far as your 'proof' goes, I'd like to start by challenging

Proof: 0.9999... = Sum 9/10^n
(n=1 -> Infinity)

which makes no sense whatsoever.

followed by

= lim sum 9/10^n
(m -> Infinity) (n=1 -> m)

Where you've introduced an undefined variable m and done mysterious things with it.
I'll challenge the rest once you've explained the first premises.

BTW You're going to have to explain your notation a little as it's mathematical in no sense of the word.
Does m -> infinity mean 'as m tends towards infinity' or are you defining a range?

 

[zCypher]

Some things here are convincing, on BOTH sides. What to believe?

1 - 0.9 = 0.1

1 - 0.99 = 0.01

1 - 0.999 = 0.001

1 - 0.9999 = 0.0001

1 - 0.99999 = 0.00001

That one was convincing.. cause it's easy for my simple mind to grasp. but then there are some fairly simple arguments on the other side that make a lot of sense too. How to figure it out? Don't have the level of math/brain to figure it out and understand the complicated (or seemingly so, to me at least) ones posted here.

I guess it depends what you're applying it to? Maybe in some cases 1 can't equal 0.999... and in some cases it has to. Maybe for some purposes it's gotta and some it doesn't, I don't know. But there seems to be "proofs" on both sides of the argument so it makes it extremely difficult to choose a side to believe without fully comprehending all of it.

1=0.999... had me convinced for a while. Unsure, though. Entertaining to read, to say the least. hehe..

 

[Keego]

I round numbers, sue me.

 

[raptor13]

N/M

 

[spidey07]

OMFG!

I never realised just how ignorant folks are. That's not an insult, it just means you don't comprehend.\

Anyway you work it .999... is 1. They are the same number just different representations.

Thankfully nobody with a decent education has stepped in and offered reasons why it "couldn't"

Who's taken number systems as a graduate class?

RossGR - Where are you?

 

[spidey07]

, then does .9999.....98 = .99999

You can not arbitratrilly manipulate real numbers like you just stated. There is not such real number as .9999...98 in the real number system. If so please express it as a fraction.

 

[Woodchuck2000]

Originally posted by: spidey07
OMFG!
I never realised just how ignorant folks are. That's not an insult, it just means you don't comprehend.\
Anyway you work it .999... is 1. They are the same number just different representations.
Thankfully nobody with a decent education has stepped in and offered reasons why it "couldn't"
Who's taken number systems as a graduate class?
RossGR - Where are you?

Cheers Spidey, nice to see you're providing a proof to back up the insults you're chucking at people

I've seen a really good proof using Cauchy numbers, I was just playing devils advocate because josphII is pissing me off... His little proof was a cut-and-paste job from this page. and I'm willing to bet that he doesn't really understand it so I thought I'd probe a little.

 

[ElFenix]

gah! people who know nothing of number theory trying to bend their minds around it is horrible! this is as bad as the "you're on a game show and you've shown 3 doors and behind 2 are nothing and behind the other is a million bucks" question

 

[spidey07]

Not trying to insult.

Just trying to point out that folks are trying to manipulate a repeating decimal real number using arithmatic.

 

[oLLie]

I thought I remember my professor saying infinity is not a number...
1 is surely a number, but it doesn't seem clear to me that infinity (or a number with infinite digits) is a number.

Could someone please explain to me how infinity is a number?

Use as many insults as you feel is necessary

Also, some of the people who have a secure grasp of this, could you please explain to me why Skyclad1uhm1's argument is invalid? It seems to be along the same vein of logic.

 

[oLLie]

Originally posted by: spidey07


Not trying to insult.

Just trying to point out that folks are trying to manipulate a repeating decimal real number using arithmatic.

Do you mean like in the proof?

x = 0.9999...
10x = 9.9999...
10x - x = 9.9999... - 0.9999...
9x = 9
x = 1.

Looks like he is manipulating a repeating decimal using arithmetic.

 

[MadRat]

We've covered every aspect of this argument in a thread. Search the archives.

 

[MadRat]

The truth is they are not the same. Your pathetic attempts to prove .999...=1 are wrong, especially the illusion of dividing .999... by 10 since it is not possible. You cannot equate objective values with subjective ones.

 

[DoNotDisturb]

in the regular real number system, no it is not.

1 would be its own number; considered a whole number.

*edit, saw the proof, i guess i've learned something*

 

[bleeb]

They are not the same because you can NEVER reach infinity. And the only way for 0.9999... to converge to 1 is if you reach infinity, but there is no way to reach infinity. THERFORE 0.9999.... != 1. (hhaha i love it)

 

[DoNotDisturb]

Originally posted by: bleeb
They are not the same because you can NEVER reach infinity. And the only way for 0.9999... to converge to 1 is if you reach infinity, but there is no way to reach infinity. THERFORE 0.9999.... != 1. (hhaha i love it)

that is true. anything that is considered "infinite" is theoretical, the LIMIT approaches 1 as n -> infinity, HOWEVER, there is theoretically no "end point", thus you cannot say it's exactly equal to 1. it gets closer and closer to 1, but will never equal it because infinity is not attainable. if you don't know why its theoretical, read on infinity. if you look at the proof in that dr.math site, n -> infinity.

 

[DoNotDisturb]

Originally posted by: oLLie
I thought I remember my professor saying infinity is not a number...
1 is surely a number, but it doesn't seem clear to me that infinity (or a number with infinite digits) is a number.

Could someone please explain to me how infinity is a number?

Use as many insults as you feel is necessary

Also, some of the people who have a secure grasp of this, could you please explain to me why Skyclad1uhm1's argument is invalid? It seems to be along the same vein of logic.

infinity is not a number.. for more math info check out the link on my profile....

 

[Yomicron]

Originally posted by: Woodchuck2000

As far as your 'proof' goes, I'd like to start by challenging

Proof: 0.9999... = Sum 9/10^n
(n=1 -> Infinity)

which makes no sense whatsoever.

in the decimal system, the number 325.7 is defined as:

3*10^2 + 2*10^1 + 5*10^0 + 7*10^-1

so 0.9999... would be:

9*10^-1 + 9*10^-2 + 9*10^-3 + ...

this can be represented as the sum of 9*10^-n with n going from 1 to infinity (You may be familiar with this notation)

followed by

= lim sum 9/10^n
(m -> Infinity) (n=1 -> m)

Where you've introduced an undefined variable m and done mysterious things with it.
I'll challenge the rest once you've explained the first premises.

inorder to take a sum that goes to infinity you need to intorduce another variable, in this case 'm', you then take the sum upto m and have m approach infinity (pretty picture)

 

[DoNotDisturb]

that is correct.... theoretically, 0.9999 would equal it, but thats assuming that infinity ends, remember -> means "approaches"... does it reach infinity? you tell me.