.9999~ = 1. Prove me wrong

Status
Not open for further replies.
[quote name='tsheo']X9=9 this will only be true if your some how asuming that X=1 not .99~. .99~ isn't areal number so it can go either way for practicle purposes use .99999.... for theoretic proofs use .99999~[/quote]

How is 0.99~ not a 'real' number? It is computable and non-imaginary. BTW its good that the thread is back on track. :)
 
Any infinite value is nonreal.
It's like the square root of -1.

0.999999~ = 0.999999~
It's just one of those "duh" things.
1 is a different number.
 
[quote name='Quackzilla']Any infinite value is nonreal.
It's like the square root of -1.

0.999999~ = 0.999999~
It's just one of those "duh" things.
1 is a different number.[/quote]

I think we have a different definition of real numbers. My definition of a real number is a positive or negative non-complex i.e. doesn't involve the square root of -1. 0.99~ is nowhere near the jw-axis. I believe you are talking about rational numbers. According to these sites, infinite numbers are not irrational.

[quote name='WikiPedia']It is often erroneously assumed that mathematicians define "irrational number" in terms of decimal expansions, calling a number irrational if its decimal expansion neither repeats nor terminates. No mathematician takes that to be the definition, since the choice of base 10 would be arbitrary and since the standard definition is simpler and more well-motivated[/quote]

[quote name='http://www.cut-the-knot.org/do_you_know/numbers.shtml']However, there are numbers which are neither rational or irrational (for example, infinitesimal numbers are neither rational nor irrational[/quote]

Infer what you want from that, that's enough math for me today.. whew.
 
[quote name='SS4Brolly']
10X = 9.999~
-1X = -1X (.999~)
[/quote]

What the hell did you do? Either that doesn't make mathematical sense, or I'm going crazy.

No matter; there are mathematical theorems that both prove and disprove that .99.... = 1 and .99.... != 1

It's a question you'll have to be happy not knowing the answer to, because we'll never know if it does, in fact, equal 1.
 
[quote name='SS4Brolly']Let X = .99999~

X = .9999~
10X = 9.999~
-1X = -1X (.999~)
9X = 9
X = 1
[/quote]

i think cause you didn't read it correctly.
you have x, which = .9999~
so
x = .999~
then you times both sides by 10
10x = 9.999~
then subtract 1x (which is .999~) from both sides
10x - 1x = 9.999~ - .999~
which equals
9x = 9
and then
x = 1
 
No, if that was the case, he's subtracting 1x from the left, and 1x(.999~) which is less than 1 until proven to be one; so therefor making the entire proof moot. By proving that .999~ is 1, you can't assume that .999~ is 1 until it's proven.
 
[quote name='RichD1'][quote name='SS4Brolly']
10X = 9.999~
-1X = -1X (.999~)
[/quote]

What the hell did you do? Either that doesn't make mathematical sense, or I'm going crazy.

No matter; there are mathematical theorems that both prove and disprove that .99.... = 1 and .99.... != 1

It's a question you'll have to be happy not knowing the answer to, because we'll never know if it does, in fact, equal 1.[/quote]

Where is the proof that refutes their equality? I was under the impression that we (those of us that know) know that .999~=1 for a FACT.
 
Limit as X->Infinity is 1. The whole point of a limit is that it never approaches it. Basic calculus teaches us this.
 
[quote name='Quackzilla'].9~=.9~

1=1

1=/=.9~[/quote]

What kind of "proof" is that?

It's equivalent to saying:

2+3 = 2+3
1+4 = 1+4
2+3 =/= 1+4

I will now repeat the correct answer
( http://www.mathforum.org/dr.math/faq/faq.0.9999.html )

Why does 0.9999... = 1 ?
This answer is adapted from an entry in the sci.math Frequently Asked Questions file, which is Copyright (c) 1994 Hans de Vreught ([email protected]).

The first thing to realize about the system of notation that we use (decimal notation) is that things like the number 357.9 really mean "3*100 + 5*10 + 7*1 + 9/10". So whenever you write a number in decimal notation and it has more than one digit, you're really implying a sum.

So in modern mathematics, the string of symbols 0.9999... = 1 is understood to mean "the infinite sum 9/10 + 9/100 + 9/1000 + ...". This in turn is shorthand for "the limit of the sequence of numbers

9/10,
9/10 + 9/100,
9/10 + 9/100 + 9/1000,
...."

One can show that this limit is 9/10 + 9/100 + 9/1000 ... using Analysis, and a proof really isn't all that hard (we all believe it intuitively anyway); a reference can be found in any of the Analysis texts referenced at the end of this message. Then all we have left to do is show that this sum really does equal 1:

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


Not formal enough? In that case you need to go back to the construction of the number system. After you have constructed the reals (Cauchy sequences are well suited for this case, see [Shapiro75]), you can indeed verify that the preceding proof correctly shows

lim_(m --> oo) sum_(n = 1)^m (9)/(10^n) = 1

0.9999... = 1

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


Another informal argument is to notice that all periodic numbers such as 0.9999... = 9/9 = 1 are equal to the digits in the period divided by as many nines as there are in the period. Applying the same argument to 0.46464646... gives us = 46/99.

References

R.V. Churchill and J.W. Brown. Complex Variables and Applications. 0.9999... = 1 ed., McGraw-Hill, 1990.

E. Hewitt and K. Stromberg. Real and Abstract Analysis. Springer-Verlag, Berlin, 1965.

W. Rudin. Principles of Mathematical Analysis. McGraw-Hill, 1976.

L. Shapiro. Introduction to Abstract Algebra. McGraw-Hill, 1975.

Bottom line:

Lock this thread already! next thing people will start asking about the third common word ending with "gry". Enough!
 
[quote name='chunk'][quote name='RichD1'][quote name='SS4Brolly']
10X = 9.999~
-1X = -1X (.999~)
[/quote]

What the hell did you do? Either that doesn't make mathematical sense, or I'm going crazy.

No matter; there are mathematical theorems that both prove and disprove that .99.... = 1 and .99.... != 1

It's a question you'll have to be happy not knowing the answer to, because we'll never know if it does, in fact, equal 1.[/quote]

Where is the proof that refutes their equality? I was under the impression that we (those of us that know) know that .999~=1 for a FACT.[/quote]

It's not a fact. For every number you can conceive, you can always add a 9 in front of it. Saying .999~ is making the assumption that infinity actually exists; but as I said, you can ALWAYS add a 9 in front making the number. You can add a .9 in front of .999~, making the number .9infinity. As such, the number will always be less than 1 because you can always add .9 in front.
You will ALWAYS have a 1 there.

Since I suck at explaining this in words:

x = .99...
10x = 9.99....
9.00....1 != 9

Can always add an extra 9 to have the .1 show up. Hence; depends on whether or not you believe infinity exists or not.
 
[quote name='RichD1']Hence; depends on whether or not you believe infinity exists or not.[/quote]

By .999999... we mean an INFINITELY long series of nines. If your somehow don't believe that the concept of infinity exists (?????) (and again: ?????) you have no business even reading the rest of the problem.

Next we'll have people refusing to acknowledge the digit 6 exists on religious grounds.

For another example of mathematically-challenged freaks like the type you suggest exists, see:

http://www.snopes.com/religion/pi.htm
 
[quote name='eldad9'][quote name='RichD1']Hence; depends on whether or not you believe infinity exists or not.[/quote]

By .999999... we mean an INFINITELY long series of nines. If your somehow don't believe that the concept of infinity exists (?????) (and again: ?????) you have no business even reading the rest of the problem.

Next we'll have people refusing to acknowledge the digit 6 exist on religious grounds.

For another example of mathematically-challenged freaks like the type you suggest exists, see:

http://www.snopes.com/religion/pi.htm[/quote]

Mathematically-challenged? Because I can fuckin envision a number infinitely long + one more therefore not making it infinite? You're an idiot. No matter how infinitely long, there is always the possibility of one more. Get over yourself; you're not as smart as you think you are.
 
[quote name='RichD1']Mathematically-challenged? Because I can shaq-fu envision a number infinitely long + one more therefore not making it infinite? You're an idiot. No matter how infinitely long, there is always the possibility of one more. Get over yourself; you're not as smart as you think you are.[/quote]

You're not "envisioning" it correctly. Add one to a positive infinity, and it's still the same number. Maybe http://www.stormloader.com/ajy/hotel.html can clarify it.

The good news is that when you go to college and take more advanced math classes (calculus, etc) everything will start making sense to you.

In the meantime, I hope you'll forgive me for skipping the part of the post where I insult your intelligence, even though it is my turn.
 
Oh, you mean the intelligence insulting that you decided to start with your initial post?

I see infinity much different then you do; and as such, I see .999... as not equaling one.

Lets take 0 and .0000...1 for example. Do you believe that .0+infinity+1 = 0? I'd assume you do. But, alas, there IS always going to be a 1 at the end of that sequence thus not allowing it to =1. Just because there is no word to describe how small the number is, does not mean it equals one. Now take that infinitely smallest number imagineable that isn't =1 to win and subtract it from 1. You now have .99....; and you had to subtract something from 1 to get it. We can both agree on that, right? You just took 1, subtracted a number and came up with something different. If the logic was correct, wouldn't 1 - .0000...1 = 1 and not .99...?

Psst--this is more number theory than anything you can learn in a classroom. It can neither be proven for or against, since noone can ever accomplish infinity.
 
[quote name='eldad9']
You're not "envisioning" it correctly. Add one to a positive infinity, and it's still the same number.
[/quote]

That's the thing. I don't think it IS the same number. If you added one, then the original number HAD to exist and had to end at some point since you added 1 to it.

PS--agree to stop insults from both sides. This really is an interesting as hell topic, and I'm sure we can both agree.
 
[quote name='RichD1']It can neither be proven for or against, since noone can ever accomplish infinity.[/quote]

if you say that you can't prove that you are right or wrong, could you do me a favor and let this topic die?

there's not much that can be said that hasn't been already.
 
[quote name='RichD1']Oh, you mean the intelligence insulting that you decided to start with your initial post?

I see infinity much different then you do; and as such, I see .999... as not equaling one.

Lets take 0 and .0000...1 for example. Do you believe that .0+infinity+1 = 0? I'd assume you do. But, alas, there IS always going to be a 1 at the end of that sequence thus not allowing it to =1. Just because there is no word to describe how small the number is, does not mean it equals one. Now take that infinitely smallest number imagineable that isn't =1 to win and subtract it from 1. You now have .99....; and you had to subtract something from 1 to get it. We can both agree on that, right? You just took 1, subtracted a number and came up with something different. If the logic was correct, wouldn't 1 - .0000...1 = 1 and not .99...?

Psst--this is more number theory than anything you can learn in a classroom. It can neither be proven for or against, since noone can ever accomplish infinity.[/quote]

I'm not sure you fully understand the concept of infinity. Your value of .000...1 doesn't really exist because if you're looking for the smallest value greater than 0, you'll never get it. Even though you know that 1 has to go at the end, there is no end to infinity, and you can never pinpoint the exact location of where that 1 should go. Think in terms of scientific notation, where your value equals 1*10^x and x
 
I know that you can never pinpoint the place of the 1; but if infinity is in fact a number, you CAN place a number after it. If it's not a number, then, well, we're not making any sense. :p

No matter how many numbers you can amass, there is always room for another at the end, even if it's infinity. (Man, having an infinity symbol would make this so much easier)
.000infinity1 does exist, IMO. Like I said, it can't be proven to or to not exist, but IMO, it does. And as such, then there will always be an inequality. Think of it in reverse and you'll see where I'm coming from.

.1
.01
.001
.0001
Keep on going and you have the same number I'm trying to explain. You'll have an infinite amount of 0s ahead, but the 1 will always be at the end. Did that explain my thinking?


PS--your 1/3 problem is cool as hell and really gets me thinking, but I don't think anything is concrete evidence as to whether or not .99... = 1. I don't believe or disbelieve it. I'm just present a second side to the argument; as I can't prove either side correct. Who knows, maybe the answer to the age old question actually is: .999... does and does not equal 1.
 
[quote name='RichD1']I know that you can never pinpoint the place of the 1; but if infinity is in fact a number, you CAN place a number after it. If it's not a number, then, well, we're not making any sense. :p

No matter how many numbers you can amass, there is always room for another at the end, even if it's infinity. (Man, having an infinity symbol would make this so much easier)
.000infinity1 does exist, IMO. Like I said, it can't be proven to or to not exist, but IMO, it does. And as such, then there will always be an inequality. Think of it in reverse and you'll see where I'm coming from.

.1
.01
.001
.0001
Keep on going and you have the same number I'm trying to explain. You'll have an infinite amount of 0s ahead, but the 1 will always be at the end. Did that explain my thinking?


PS--your 1/3 problem is cool as hell and really gets me thinking, but I don't think anything is concrete evidence as to whether or not .99... = 1. I don't believe or disbelieve it. I'm just present a second side to the argument; as I can't prove either side correct. Who knows, maybe the answer to the age old question actually is: .999... does and does not equal 1.[/quote]

I think you're confusing yourself by classifying infinity as a number, not a concept. So really, you can't place a number after a concept, per se. Think of it this way: what's the greatest value less than 1? You know the answer should be .999..., but since that is only a concept and not a value, there is no correct answer.
 
[quote name='RichD1']No. it's interesting. Don't like it? Don't enter. Not that hard.[/quote]

the reason i said stop is because you are wrong, and there's not really much of an argument for you to make. you are saying that in your opinion infinity+1 is a valid statement, but it's not. there is nothing larger than infinity. you can say "x+1 > x" or maybe "n < n+6" but infinity+1 is still infinity.

you have the earlier argument that:
x = 0.9999...
10x = 9.9999...
10x - x = 9.9999... - 0.9999...
9x = 9
x = 1.
and you reply with "What the hell did you do? Either that doesn't make mathematical sense, or I'm going crazy. "

it's a valid statement, and it's got the same theory behind it as factorials. for instance, "(N+1)!/N! = (N+1)" because (N+1)! = (N+1)(N!), so you then cancel out the N! and you are left with (N+1). the reason for my bringing this up is because that it is the same thing as dividing by .9999~, since neither is a definitive number. so, if you look at it that way, .999~ has to equal 1, no matter what our better judgement may tell us.

a big problem here is that you keep saying "in my opinion", but in math, there isn't room for opinions. it's like going to a 15% off sale and picking up an item for $20, then you walk to the registers, the employee scans it, and tells you that it will cost you $17 (before they add tax to it), to which you reply "that doesn't sound right, i should get a larger discount." math is based on facts, not opinions.
 
But it's the concept OF a number.

con·cept (
n.
2. Something formed in the mind; a thought or notion. See Synonyms at idea.

Yup, something we can only form in our mind. But that doesn't mean that this isn't true:
.1
.01
.001
...
.0...1
equals
1 - .99...

Infinite numbers; only a concept with an answer that's only a concept. But that second concept is no less real than infinity or 1. Oh fuck, I just confused myself. Don't mind anything I just said.


Answer to the question? Nope, there's .00...1 difference between .999~ and 1.That's my take on things. I'm going to do some SERIOUS pondering about this tomarrow. My head hurts and I need sleep now.

1 - .99... = .00...1 makes just as little sense as the 1/3 problem, and quite frankly, it sucks knowing I'll never be able to explain it. :(
 
[quote name='thatstoobad'] math is based on facts, not opinions.[/quote]

And it's a FACT that infinity can always be expanded upon. It's the very definition of infinity. But no matter how much you expand upon infinite .9, it will still end in a 9 and thus, when subtracted from 1, will give you an answer of .infinity1
However, since no mathematician can properly solve equations with infinity, whether they'd like to or not, they can only give an opinion on the matter. There's no telling if infinity abides by the same number laws as every other number; which is where the opinions come in. It's YOUR opinion that infinity abides by all laws; it's mine that it doesn't.
 
[quote name='RichD1']But it's the concept OF a number. [/quote]

a concept is not the same as an opinion. inifinity is a concept. "green's a pretty color" is an opinion.

And it's a FACT that infinity can always be expanded upon.
Click to expand...

you can expand infinity as far as you like, it's still infinity.
 
[quote name='RichD1'][quote name='eldad9']
You're not "envisioning" it correctly. Add one to a positive infinity, and it's still the same number.
[/quote]

That's the thing. I don't think it IS the same number. If you added one, then the original number HAD to exist and had to end at some point since you added 1 to it.
[/quote]

Adding a constant number to infinity doesn't change it. Them's the rules.

Do a search on Aleph-Null, if you will.

And/or
http://en.wikipedia.org/wiki/Infinity
more of Hilbert's hotel and an intuitive explanation
http://en.wikipedia.org/wiki/Hilbert's_paradox_of_the_Grand_Hotel
 
[quote name='thatstoobad'][quote name='RichD1']But it's the concept OF a number. [/quote]

a concept is not the same as an opinion. inifinity is a concept. "green's a pretty color" is an opinion.[/quote]

I edited my last post where I explained what is opinionated. Granted, chances are you won't agree, but quite frankly I'm done with this argument now.
 
[quote name='video_gamer324']I think you're confusing yourself by classifying infinity as a number, not a concept. So really, you can't place a number after a concept, per se. Think of it this way: what's the greatest value less than 1? You know the answer should be .999..., but since that is only a concept and not a value, there is no correct answer.[/quote]

No, the "greatest value less than 1" doesn't exist any more than "the largest prime number" does.

Look, suppose such a number existed. Let's mark it as X

since X
 
[quote name='thatstoobad'][quote name='RichD1']But it's the concept OF a number. [/quote]

a concept is not the same as an opinion. inifinity is a concept. "green's a pretty color" is an opinion.

And it's a FACT that infinity can always be expanded upon.
Click to expand...

you can expand infinity as far as you like, it's still infinity.[/quote]

And you can make something infinitely less smaller, but that doesn't mean it will cease to exist. Which is the point I've been trying to make. By subtracting .99... from 1, you are making the difference INFINITELY less smaller, but alas, still existant.

PS--appreciate all of the explanations and links, eldad. It's nice to see someone not be an ass and still debate something. Odd thing on the internet. :)
 
[quote name='RichD1'][quote name='thatstoobad'] math is based on facts, not opinions.[/quote]

However, since no mathematician can properly solve equations with infinity, whether they'd like to or not, they can only give an opinion on the matter. There's no telling if infinity abides by the same number laws as every other number; which is where the opinions come in. It's YOUR opinion that infinity abides by all laws; it's mine that it doesn't.[/quote]

Wha? Where did you get this? What do they teach in American schools?

First meeting somebody who's convinced viruses can be written to target any computer, and now an infinity-denial.

(sorry if this sounded personal - it's not).

Please, don't accept the word of people who don't understand math. You're young, you're curious - learn about these things yourself.
 
[quote name='eldad9'][quote name='video_gamer324']I think you're confusing yourself by classifying infinity as a number, not a concept. So really, you can't place a number after a concept, per se. Think of it this way: what's the greatest value less than 1? You know the answer should be .999..., but since that is only a concept and not a value, there is no correct answer.[/quote]

No, the "greatest value less than 1" doesn't exist any more than "the largest prime number" does.

Look, suppose such a number existed. Let's mark it as X

since X
 
[quote name='eldad9'][quote name='RichD1'][quote name='thatstoobad'] math is based on facts, not opinions.[/quote]

However, since no mathematician can properly solve equations with infinity, whether they'd like to or not, they can only give an opinion on the matter. There's no telling if infinity abides by the same number laws as every other number; which is where the opinions come in. It's YOUR opinion that infinity abides by all laws; it's mine that it doesn't.[/quote]

Wha? Where did you get this? What do they teach in American schools?

First meeting somebody who's convinced viruses can be written to target any computer, and now an infinity-denial.

(sorry if this sounded personal - it's not).

Please, don't accept the word of people who don't understand math. You're young, you're curious - learn about these things yourself.[/quote]

How can ANYONE correctly assess the laws of infinity if they can't envision infinity?

By the logic presented to have the 2 numbers equal then my example of
.1, .01, .001 will eventually equal 0, correct? If it does, then what happens to that 1 at the end of the infinity? Does it just stop existing? It will always be there, correct? It will always be the difference of the 2 original numbers; thus making it equal to 0 since 1-1=0. But the 1 is there. How can anyone dispute that?

(I'm not accepting anyone word on any of this. This is just me trying to rationalize. I'm not even saying that .9999 doesn't equal 1; I'm just trying to find something that would prove it, but I'm just finding more things that disprove it. I realise half of this is coming out of my ass or making no sense; but I'm here trying to find answers, not really trying to argue, and if making a fool of myself is a byproduct of the quest for knowledge, then so be it.)
 
[quote name='video_gamer324']Yes, I realize that there is no greatest value less than 1, hence my earlier statement that "there is no correct answer," but perhaps I did not explain correctly. My apologies, and thanks for providing the correct mathematical support.[/quote]

Dood you were like there is no correct answer and then I was like there is a correct answer: no such number exists. But yes, we understand each other and vice versa.

I think RichD1's problem may be that he thinks he can get rid of the "..." at some point; he can't.
 
[quote name='eldad9'][quote name='RichD1'][quote name='thatstoobad'] math is based on facts, not opinions.[/quote]

However, since no mathematician can properly solve equations with infinity, whether they'd like to or not, they can only give an opinion on the matter. There's no telling if infinity abides by the same number laws as every other number; which is where the opinions come in. It's YOUR opinion that infinity abides by all laws; it's mine that it doesn't.[/quote]

Wha? Where did you get this? What do they teach in American schools?

First meeting somebody who's convinced viruses can be written to target any computer, and now an infinity-denial.

(sorry if this sounded personal - it's not).

Please, don't accept the word of people who don't understand math. You're young, you're curious - learn about these things yourself.[/quote]

Don't get me wrong; I'm NOT denying infinity's existance so much as I'm denying that by expanding upon infinity, you have to have something tangible to expand upon. 1+1+1+1 infitely, but when you get to an infinity plus 1s and add another +1, then the number you just added 1 to wasn't infinity. Oh hell, I don't even know what I'm talking about anymore; I just defined my logic with something that noone is even disagreeing with. I'll shut up now.
 
[quote name='eldad9'][quote name='video_gamer324']Yes, I realize that there is no greatest value less than 1, hence my earlier statement that "there is no correct answer," but perhaps I did not explain correctly. My apologies, and thanks for providing the correct mathematical support.[/quote]

Dood you were like there is no correct answer and then I was like there is a correct answer: no such number exists. But yes, we understand each other and vice versa.

I think RichD1's problem may be that he thinks he can get rid of the "..." at some point; he can't.[/quote]

No, I know the ... is infinite and such. But .0...1 exists, it's just not freaking palpable. As such, .999... and 1 are seperated by an untangible number, but one that exists. When I said it MIGHT not adhere to mathematically laws: there may quite possibly be a 4th dimension (time) involved in the equasion that can never be accounted for. The difference between the 2 is getting smaller for infinite amounts of time, but that doesn't take away from its existance. Should I shut up now? I'm not making sense.
 
OK people, i just saw this thread and thought i would provide a DIFFERENT explanation as to why 1 is equal .999... that we talked about in my conceptual mathematics course at ohio state.

bascially one of the many ways to differentiate two numbers from one another is that you can find another number that lies between them. for instance .333... is equal to 1/3 because there are no possible numbers between them, but 1/3 and say 1/4 are not equal because you can find another number that lies between them. therefore, .999... is equal to 1 because of its infinite nature it is logically impossible to find a number that will differentiate 1 from .999..., therefore they are the same number. this also works with numbers like 4.999... = 5 and such.

in case your still confused with decimals, a number in between .50003 and .50004 would be .500032, as you can always extend the decimal place to find another number that would be in between two others, except obviously in an infinite case like .999... or .333... listed above
 
[quote name='RichD1']By the logic presented to have the 2 numbers equal then my example of .1, .01, .001 will eventually equal 0, correct? If it does, then what happens to that 1 at the end of the infinity? Does it just stop existing? It will always be there, correct? It will always be the difference of the 2 original numbers; thus making it equal to 0 since 1-1=0. But the 1 is there. How can anyone dispute that?
[/quote]

I agree that dealing with infinites can be counter-intuitive - if you don't have the background for it you may not be able to grasp it (kinda like quantum physics). My problem is now to find an intuitive explanation to try and get you to understand it; if I'll fail in my task I'll try to confer with people who actually deal with math. Let's see...

In your example, you start with 1 and divide it by 10, an infinite amount of time. You can't say the last number in the series is or isn't zero, because there is no last number. What we mean by

lim (x-->infinity) 1/10^x

is the number 1/10^x approaches, as x itself approaches infinity (that is, grows and grows since you're essentialy counting the element you reached in the infinite series.

For every positive number you could think of, you'll eventually reach a number in the series which will be smaller (closer to 0), so the limit of this infimite series is zero. That's not to say there is an element in it that exactly equals zero, or that there is a last element.

Think of the "lim" function (or, as you called it, "...") as an additional function, like sine or log or whatever, that you just haven't learned about yet.

Makesense?
 
Calculus next year. Thank god. I honestly can't wait for college (such a short distance away. :( ) where I can actually take classes like scar said; conceptual math.

That's not to say there is an element in it that exactly equals zero, or that there is a last element.
Click to expand...

That's just it, though; there is no element that exactly equals zero. If there was, then that means 1/10^x would eventually diminish itself into nothingness. (infinity and natural laws statement a whiles back) But if 1/10^x, as it approaches zero, keeps getting closer without ever actually reaching it, then that means 1-.99.. would approach zero but never hit it, thus always having SOME substance within itself, because if it ever hit 0, then that means 1/10^x as x approached 0 would have just diminshed itself into nothing, and therefore making .99... not true because 1-1 = 0. If only someone could get into my head and see what I'm thinking so they could properly explain it.

pssst, eldad, you rock. This is one of the best conversations I've been apart in in quite a while.
 
Thanks, but I think most people who have taken calculus can explain it in some way - even though there are so many ways to explain it.

OK, think of it this way. On one hand, yes, the value of the number never reaches zero. But on the other hand, the index of the number you're looking it never reaches infinity. Those two nevers magically cancel each other out.

The "eventually" reaching zero doesn't make sense - because there is no eventuality, just an infinite series.
 
[quote name='RichD1']...Who knows, maybe the answer to the age old question actually is: .999... does and does not equal 1.[/quote]

Sounds like Schrödinger's cat...

Like you said, there's no way to know. So it does in a way parallel quantum mechanics.
 
[quote name='Wet Ninja'][quote name='RichD1']...Who knows, maybe the answer to the age old question actually is: .999... does and does not equal 1.[/quote]

Sounds like Schrödinger's cat...

Like you said, there's no way to know. So it does in a way parallel quantum mechanics.[/quote]

You're wrong. There is a way to know; I know; it is. The answer is that they're equal. If you don't understand that, you really don't understand what we mean by the "...".

The similarity to quantum physics is that it's very counter-intuitive; until your brain understands it, your mind will keep protesting, making understanding almost impossible. It is only when you allow yourself not to be bound by your intuition that you can actually begin to understand.
 
Are you all stupid? All you uneducated morons talking about how cool this problem is, and how cool the guy who posted it must be to have come up with it. This is a BASIC concept is series. You learn it in any Discrete Math or second-semester Calculus class. .9999etc. is equal to one because it is the series (9/(10^(x+1))) from zero to infinity. This is equal to 9 tenths plus 9 hundreths, plus 9 thousandths, etc. It is INFINITELY close to one and thus converges to one. You cannot truly understand this problem if you haven't studied series. You cannot argue that the series never reaches one. It's not a numeric evaluation of an infinite sum. Because it gets infinitely close to one, it is equivalent to one, because there is no finite difference between the two. Leave math to the mathematicians. Thanks.
 
Status
Not open for further replies.
bread's done
Back
Top