Curse

Rayndeon · Feb 05, 2006, 12:14 AM // 00:14

Tur713, 0.9 recurring DOES EQUAL 1. Allow me to reiterate the proof:

Let x = 0.999999999....
x = 0.999999999999.....
10x = 9.9999999999.....
10x-x = 9.99999999..... - 0.9999999999....
9x = 9
x = 1

That is the proof via deductive reasoning. You can also go about by inductive reasoning. As Dr. Math said, try viewing the pattern:

1/9 = 0.111111111.....
2/9 = 0.222222222.....
3/9 = 0.333333333.....
4/9 = 0.444444444.....
5/9 = 0.555555555.....
6/9 = 0.666666666.....
7/9 = 0.777777777.....
8/9 = 0.888888888.....

But then, what does 9/9 equal? Surprisingly, the inductive reasoning leads us to believe that 9/9 = 0.999999999..... ! As Dr. Math said, you have to understand what it means to go on forever instead of a long time.

In the end, by deductive and inductive reasoning, 0.99999999..... = 1

PieXags · Feb 05, 2006, 12:31 AM // 00:31

Quote:

Originally Posted by Rayndeon

Tur713, 0.9 recurring DOES EQUAL 1. Allow me to reiterate the proof:

Let x = 0.999999999....
x = 0.999999999999.....
10x = 9.9999999999.....
10x-x = 9.99999999..... - 0.9999999999....
9x = 9
x = 1

That is the proof via deductive reasoning. You can also go about by inductive reasoning. As Dr. Math said, try viewing the pattern:

1/9 = 0.111111111.....
2/9 = 0.222222222.....
3/9 = 0.333333333.....
4/9 = 0.444444444.....
5/9 = 0.555555555.....
6/9 = 0.666666666.....
7/9 = 0.777777777.....
8/9 = 0.888888888.....

But then, what does 9/9 equal? Surprisingly, the inductive reasoning leads us to believe that 9/9 = 0.999999999..... ! As Dr. Math said, you have to understand what it means to go on forever instead of a long time.

In the end, by deductive and inductive reasoning, 0.99999999..... = 1

That pattern doesn't make 0.999... equal one buddy. It makes it so that it should equal one.

0.999(...) is not equal to one, but is 0.000(...)1, away from one. Making it infinately as close to one as you can get without being one as an exact number. I understand that 0.000(...)1 doesn't make a whole lot of sense, but it makes just enough sense to equal half the other senseless crap that is mathematics.

Rayndeon · Feb 05, 2006, 12:39 AM // 00:39

However, you have completely ignored the proof. The wonderful thing about mathematics is that provided one goes through mathematical rigor, any proofs and theorems are ETERNAL. Statements are immortal and are always true, within the accepted set of axioms.

PieXags, you fail to acknowledge the proof. It is mere elementary algebra, it should not be hard to absorb. Statements and reasons below:

*recurring

Let x = 0.9999999999* (assignment of a variable)
x = 0.9999999999* (given)
10x = 9.99999999999* (Multiplication Property of Equality)
10x-x = 9.9999999999* - x (Subtraction Property of Equality)
10x-x = 9.9999999999* - 0.9999999999* (Substitution Property of Equality)
9x = 9 (Subtraction Property of Equality)
x = 1 (Division Property of Equality)
0.99999999999* = 1 (Substitution Property of Equality)

I have observed mathematical rigor, working within the accepted axioms. And I have reached the conclusion that 0.999999999* = 1. Since I proved this, it becomes ETERNALLY a true statement within this axiomatic system. It is entirely irrefutable.

Also, the pattern does work. Inductive reasoning shows this.

Racthoh · Feb 05, 2006, 12:45 AM // 00:45

Quote:

Originally Posted by Rayndeon

PieXags, you fail to acknowledge the proof. It is mere elementary algebra, it should not be hard to absorb. Statements and reasons below:

*recurring

Let x = 0.9999999999* (assignment of a variable)
x = 0.9999999999* (given)
10x = 9.99999999999* (Multiplication Property of Equality)
10x-x = 9.9999999999* - x (Subtraction Property of Equality)
10x-x = 9.9999999999* - 0.9999999999* (Substitution Property of Equality)
9x = 9 (Subtraction Property of Equality)
x = 1 (Division Property of Equality)
0.99999999999* = 1 (Substitution Property of Equality)

Ugh... I remember that. Makes me so angry.

Rayndeon · Feb 05, 2006, 12:49 AM // 00:49

Also, by the way, to state that 1 - 0.99999999... = 0.000000(...)1 is to state that infinity has an end. Which is absurd. Ergo, PieXags, your argument is false by the classical reductio ad absurdum.

Mentalmdc · Feb 05, 2006, 01:00 AM // 01:00

Quote:

Originally Posted by Rayndeon

However, you have completely ignored the proof. The wonderful thing about mathematics is that provided one goes through mathematical rigor, any proofs and theorems are ETERNAL. Statements are immortal and are always true, within the accepted set of axioms.

PieXags, you fail to acknowledge the proof. It is mere elementary algebra, it should not be hard to absorb. Statements and reasons below:

*recurring

Let x = 0.9999999999* (assignment of a variable)
x = 0.9999999999* (given)
10x = 9.99999999999* (Multiplication Property of Equality)
10x-x = 9.9999999999* - x (Subtraction Property of Equality)
10x-x = 9.9999999999* - 0.9999999999* (Substitution Property of Equality)
9x = 9 (Subtraction Property of Equality)
x = 1 (Division Property of Equality)
0.99999999999* = 1 (Substitution Property of Equality)

I have observed mathematical rigor, working within the accepted axioms. And I have reached the conclusion that 0.999999999* = 1. Since I proved this, it becomes ETERNALLY a true statement within this axiomatic system. It is entirely irrefutable.

Also, the pattern does work. Inductive reasoning shows this.

QFT!!

-THE M-

karunpav · Feb 05, 2006, 01:44 AM // 01:44

Quote:

Originally Posted by Shattered Self

x = 0.9999999...(etc.)

10x = 9.99999999...(etc.)

10x-x = 9.999999 - 0.999999
9x = 9
x = 1
0.999999999...(etc.) = 1

Quoted for truth and...just...O.o

PieXags · Feb 05, 2006, 01:46 AM // 01:46

Quote:

Originally Posted by Rayndeon

Also, by the way, to state that 1 - 0.99999999... = 0.000000(...)1 is to state that infinity has an end. Which is absurd. Ergo, PieXags, your argument is false by the classical reductio ad absurdum.

A lot of math is absurd, personally I find the reasoning behind something like "imaginary numbers" to be just about the stupidest thing I've ever seen in my life. I find the IDEA of infinite to be entirely absurd. I never said you would enjoy what you get with 1 - 0.9(...) = 0.0(...)1, I'm just saying that's what it is. To comprehend infinite takes about as much reasoning as it does to believe that there is something after infinite.

If you can accept that 1 - 0.9(...) = 0.0(...)1, then no 0.9(...) does not equal one. If you can't accept that, then 0.9(...) = 1 so far as we care to understand. Unfortunately we can't show that infinite exists, therefore we can't show a number like 0.0(...)1 anymore than we can show 0.9(...), it's because we can't accurately show 0.9(...) that I also believe 0.0(...)1 to be just as reasonable as 0.9(...) in itself. We believe that pi extends forever, don't we? And doesn't pi change numbers along the way? Absolutely. Think of 0.0(...)1 as pi. The pattern will never repeat itself, and we'll never find the end of it. As was stated in my post, several posts up in about the middle of this page, you can either accept it or you can't. And it's because you have the choice to do so, that makes this such a topic worthy of discussion.

Rayndeon · Feb 05, 2006, 02:40 AM // 02:40

Quote:

A lot of math is absurd, personally I find the reasoning behind something like "imaginary numbers" to be just about the stupidest thing I've ever seen in my life. I find the IDEA of infinite to be entirely absurd... To comprehend infinite takes about as much reasoning as it does to believe that there is something after infinite.

Whether or not you find the idea to be absurd (in terms of opinion) does not have any bearing on its validity. Also, these "absurd" concepts have proven to be quite useful. Indeed, mathematicians remain astonished as to how even the purest branches have applications of their own.

I do not understand the dilemma: All infinity means is that there is an unbounded limit. Numbers do not end. What is the problem here? If you are attempting to view it and comprehend it as something finite, then that would be the problem. And you can't have something after infinity in these terms. In fact, that isn't even a correct mathematical statement. Infinity may not be used, strictly speaking, in arithmetic. It is a concept. In certain cases however, mathematicians have, albeit rarely, used infinity as a number. You can have orders of infinity however.

Quote:

I never said you would enjoy what you get with 1 - 0.9(...) = 0.0(...)1, I'm just saying that's what it is.

You have messed up the math. You have to understand what it means for a number to be interminably repeating. Also, this is an infinite geometric series converging to 1, ergo 0.99999999(...) = 1.

Quote:

If you can accept that 1 - 0.9(...) = 0.0(...)1, then no 0.9(...) does not equal one. If you can't accept that, then 0.9(...) = 1 so far as we care to understand.

It is not a matter of acceptance. The only things that are accepted in mathematics are axioms. Everything else is PROVED.

Quote:

Unfortunately we can't show that infinite exists, therefore we can't show a number like 0.0(...)1 anymore than we can show 0.9(...), it's because we can't accurately show 0.9(...) that I also believe 0.0(...)1 to be just as reasonable as 0.9(...) in itself.

We can and have accurately shown that 0.99999(...) = 1. It is an infinite geometric series which converges to 1. Also, it is not a matter of belief, it is a matter of proof. And no, it is not reasonable to state that 0.0(...)1 is equal to 1 - 0.9(...); you are stating that in the concept of infinity, there are numbers beyond it. Since infinite sets includes all numbers within that set (in this case, to +infinity and -infinity), it is essentially a violation and a grave error.

Quote:

We believe that pi extends forever, don't we? And doesn't pi change numbers along the way? Absolutely. Think of 0.0(...)1 as pi. The pattern will never repeat itself, and we'll never find the end of it.

0.0(...)1 is not like pi. You are committing a mathematical fallacy by stating that 0.0(...)1 exists. Pi does not commit such an error. There's no need to find any end as 0.0(...)1 is non-existent.

Quote:

As was stated in my post, several posts up in about the middle of this page, you can either accept it or you can't. And it's because you have the choice to do so, that makes this such a topic worthy of discussion.

Again, this is not a matter of belief nor acceptance, it is a matter of proof.

zoozoc · Feb 05, 2006, 02:51 AM // 02:51

pie just got owned

Rayndeon · Feb 05, 2006, 03:28 AM // 03:28

In any case, PieXags, I believe I understand what you are talking about. Mathematics can get quite confusing these areas (usually those dealing with zero and infinity); it can become an intellectual confusion in respects.

Eclair · Feb 05, 2006, 03:32 AM // 03:32

Quote:

Originally Posted by PieXags

A lot of math is absurd, personally I find the reasoning behind something like "imaginary numbers" to be just about the stupidest thing I've ever seen in my life. I find the IDEA of infinite to be entirely absurd. I never said you would enjoy what you get with 1 - 0.9(...) = 0.0(...)1, I'm just saying that's what it is. To comprehend infinite takes about as much reasoning as it does to believe that there is something after infinite.

Math is probably one of the few things that we can almost firmly cement as a fundamental truth. Pretty much everything we've achieved today in terms of the development of technology and the understanding of the natural sciences all derive from some form of math. Even those very obscure, seemingly useless mathematical concepts are probably used as the basis for some important technology today. For example, in the 1800's Bool developed a type of math called Boolean algebra. At that time, even mathmaticians thought his contributions were utterly useless and completely not applicable to anything tangible. Flash forward to around the World War II era. Engineers who developed the first computers found Bool's theorems and created the fundamental structure of how computers processors would be designed and implemented.

So don't hate on math XD

calamitykell · Feb 05, 2006, 03:34 AM // 03:34

Quote:

Originally Posted by PieXags

...I don't know about you, but 9/9 is one, buddy. 9/9 is not 0.999999(...) Just because it follows a pretty pattern doesn't mean it makes a damn bit of sense.

if A = B and B = C, then A = C, correct?

.33(...) + .33(...) + .33(...) = 1/3 + 1/3 + 1/3

1/3 + 1/3 + 1/3 = 3/3 = 1

.33(...) + .33(...) + .33(...) = .99(...)

And since we stated that .33(...) was equal to 1/3, then we also see that .99(...) = 3/3. Which is 1.

.99(...) = 3/3.

3/3 = 1.

A = B.
B = C

A = C

Mr.Style · Feb 05, 2006, 04:08 AM // 04:08

I agree that for all practical purposes in completing a math problem etc. that 0.999...=1. But, you have to remember that we are dealing with mathematics, and in mathematics, rather than dealing with measured quantities, we are dealing with exact numbers. Therefore, it can be concluded that, in mathematics, 0.999... does not equal to one, simply because it is an exact number that we know is less than one. In order to demonstrate this we may take the two exact numbers, one and one, to get zero, because 1-1=0, and all numbers in the equation are exact, it is safe to say that in the equation 1-0.999... which contains all exact numbers that 1-0.999...=0.000...001 and not 0. This shows that although the number 0.999... is determined to be 1 for all practical purposes it in fact does not equal to one.

PieXags · Feb 05, 2006, 04:27 AM // 04:27

*sigh* I don't expect anyone to accept what I've said, people have a bad habit of sticking only to what they've been taught. You can "prove" such theories only because you've been taught how to do so within the given existing realm of numbers that make up our mathematics. I can guarantee that what I've said about 0.0(...)1 existing is in fact true in a certain light, just not by the means you people are looking at it. My explaination earlier about pi. We assume that it never ends, don't we? But can we really prove that it is without an end so far as we know? Can we prove that 0.9(...) even exists? If you can prove to me that infinite exists at all, then by all means try to do so and I'll say alright. Infinite is an IDEA, and only an idea. It is not a fact, you cannot show me in any sense that 0.9(...) even exists. We only know of the idea, of 0.9(...) and can only REPRESENT 0.9 recurring by labeling it as such---but we can't prove it at all, can we?

If 0.9(...) equals one, then how do we know infinite exists? Since 0.9(...) extends forever, and according to you, is also equal to one---doesn't that mean there is no infinite? Afterall, one has an end, doesn't it? One is just one, it has an end and there's no recurring at all. To say that an infinite number is also equal to a number that has a one is to say that 0.9(...) might not as well mean anything at all. Also, keep in mind that just because you can't prove something doesn't by any means mean it isn't (or can't be) true. If we go back to my example of imaginary numbers---what we really have is a simple representation of something that can't exist in any other form. Would it be so wrong to consider 0.0(...)1 an "imaginary number"? We know it can't really exist, but what it represents is something that would by all means make enough sense to represent a difference between what would then be two entirely different numbers.

It's easy to prove when you declare everything else to be impossible. Of course, the "proof" we have is based on the ACCEPTANCE that "infinite" can actually exist. Of course if it couldn't people wouldn't give a damn anyways, they'd just use it to represent something that would exist otherwise. (Which might be what we're doing right now.)

...this is a really confusing thing to read I'm sure, but do you understand what I'm saying? Lets pretend for a moment that math actually does make a bit of sense: We've got imaginary numbers, representing the square root of negative one in the simplest of senses. We know you can't have a square root of negative one, but apparently mathematicians didn't care to stop there so they decided to just tack on an i to represent what would OTHERWISE be impossible. You accept that, even though it's absolute "bs". Probably because it's in textbooks.

Now lets say for example (assuming we've accepted the possiblity of infinite) you've got 0.0(...)1---the difference between one and 0.9(...). We know that in any reasonable sense, 0.0(...)1 can't exist ---but lets be like mathematicians of old and not let that stop us. I would label 0.9(...), as something like...^1. Which would represent 0.0(...)1 away from one. Do you see what I'm getting at? So you'd have 0.9(...) = ^1, not 1. It doesn't make any sense at all and represents something we know you can't have, but that's EXACTLY what something like imaginary numbers does. Represents something that doesn't exist outside the little symbol, that doesn't exist in the real world. (Hence we call them imaginary.) In the given example, a ^ might indicate a number tacked on with infinite. You could then right the difference as 0.0^1.

...now granted, that was just an example I tossed out of my mind, it might or might not make much sense but that's alright. Think of it as another axiom, you can't prove or disprove it, it's a matter of acceptance. (Of course you could tell me that by other means it would have conflicting ideas, but then you'd have to do a little bit of rethinking, wouldn't you?) You guys are going by the book, PieXags never liked the book because it had way too much shit that didn't even exist in real life that was represented by symbols. You people accept those only because they're in those books and because people before you accepted them as truths. What made me so angry about things like that were simply that you can throw out ANY bit of information, declare it an axiom, and base a system around it. (What do you think fiction books are?) I see absolutely no problem with an alternative solution to expressing the difference between 1 and 0.9(...), because that's all ALL of it is, is just ways to express things, that may or may not exist. Math is full of both.

Ah, now I'm going to go do something a bit more useful, like get a drink. Drinks do exist, and have nothing to do with trying to prove infinite.

Edit: If it WASN'T a matter of acceptance and how you percieve things, do you really think this topic would've been created?

Mathias Deathwater · Feb 05, 2006, 04:31 AM // 04:31

The fact that depending on how you look at it .999... may or may not equal 1 just proves that we haven't really "figured out" math completely yet, and we maybe never will. If you look at the problem as a geometric sequence, then yes, .999... does equal 1. But if you take the idea that x-x=0, then no, .999 does not equal 1, because you would end up with .000...9 (gah, i really wish I had that line you put over partially repeating decimals, just imagine it's there)

PieXags · Feb 05, 2006, 04:36 AM // 04:36

There we go, Deathwater! That's how people need to start thinking. So far we've seen mostly "proof" using only what we know about math, but what they fail to understand is that math is so many different things in so many different senses most of it IS how you accept things. If you could prove, without flaw, that one equals the other there would be absolutely no discussion over it. The fact that we can't prove infinite (we can only represent it) is enough in itself to throw the ENTIRE thing far beyond any sort of "proof" we have right now, because you have to accept many, many different things as true before you can accept the "proof", which is in fact only one sense of looking at it. Math is just a great big bunch of things that exist only within itself. You can't even use half of it to figure out real life situations, only more math! This gives people like PieXags every right to run away from "inside the box" thinking, and create new possiblities and representations for people to look at.

Mr.Style · Feb 05, 2006, 04:40 AM // 04:40

Ummm...yeah, so Deathwater just pretty much repeated what I had already said...

Mathias Deathwater · Feb 05, 2006, 04:45 AM // 04:45

Quote:

Originally Posted by Mr.Style

Ummm...yeah, so Deathwater just pretty much repeated what I had already said...

Yeah, I just expounded a bit. And since it was a good point, I figured it warranted repeating, especially because it was kind of lost in the shuffle.

Mr.Style · Feb 05, 2006, 04:55 AM // 04:55

Yeah, no problem, thanks man.