Poll: 0.999... = 1

[emeraldrafael]

Snip

you're still doing what he said you cant. you're mixing a concept of infinity (no value, just a range), with a value, and you just cant. Its like mixing real and imaginary numbers. you cant, well, not really, to make one number from a combination of both.

 

[Maze1125]

why, the print out I mean? I emailed him exactly what you said, I'm just waiting for him to get back to me.

Thanks.
And don't worry, an e-mail's fine, provided it doesn't lose any of the formatting.

 

[Rubashov]

heres the thing though its not you can post all proofs you want but that is just because humans haven't fully mastered how to use infinites in math for example 1/3 does not equal .333 repeating that is just the closest approximent we can give because 1 can not be divided by 3 so no .99999 repeating is not equal to 1

No, 1/3 does in fact equal 0.333.... Here's a proof of that, which someone else already posted:

Consider the following sequences:

a_n = 0.4, 0.34, 0.334, 0.3334...
b_n = 1/3, 1/3, 1/3, 1/3...
c_n = 0.3, 0.33, 0.333, 0.3333...

The limit of a_n is obviously 0.333..., and the limit of c_n is obviously 0.333...; however, c_n < b_n < a_n for all n. Since the limit of a_n is equal to the limit of c_n, the limit of b_n must be equal to both of them by the squeeze theorem. But the limit of b_n is obviously 1/3, since all terms in b_n are 1/3. So either 0.333... = 1/3, or the squeeze theorem is incorrect. Care to disprove the squeeze theorem?

 

[emeraldrafael]

why, the print out I mean? I emailed him exactly what you said, I'm just waiting for him to get back to me.

Thanks.
And don't worry, an e-mail's fine, provided it doesn't lose any of the formatting.

Shouldnt. He may lose the sub and exponents, but he'll know what you're talking about.

 

[Athinira]

Snip

you're still doing what he said you cant. you're mixing a concept of infinity (no value, just a range), with a value, and you just cant. Its like mixing real and imaginary numbers. you cant, well, not really, to make one number from a combination of both.

Sure you can. Why shouldn't you be able to? It's right on the Wikipedia-page even (which, while it shouldn't be taken for granted as universal truth, still has a hell of alot of well-informed sources on the subject):

Students of mathematics often reject the equality of 0.999... and 1, for reasons ranging from their disparate appearance to deep misgivings over the limit concept and disagreements over the nature of infinitesimals. There are many common contributing factors to the confusion:
- Students are often "mentally committed to the notion that a number can be represented in one and only one way by a decimal." Seeing two manifestly different decimals representing the same number appears to be a paradox, which is amplified by the appearance of the seemingly well-understood number 1.

You are arguing that I'm combining two different numbers, when the entire idea is that the two numbers are in fact the same, just with different representation. I'm not making one number from a combination of two, because there is no "combination". They are the same.

It seems to me like you are, just like several other people in the thread, mixing up the difference between "infinity" and "infinite decimals". Infinity is a range, but a number with an infinite amount of decimals is a perfectly real number. Example: Pi.

No, but I've got an argument:

If I'm working in the real numbers and try to calculate the sqrt(-1), I don't just call it 0 as that is the closest real number to i, I say I can't calculate it.
If I'm working in the positive integers and try to calculate 5 - 16, I don't just call it 1 because that's the closest positive integer to -11.

According to WikiPedia, you can actually decide yourself what you want to do. Since division of integers aren't closed, you can use 4 different methods. It indicates though that the typical method in math is to convert the number system, so in your case of 5 - 16, you convert the system from Naturals to Integers (or real numbers) and conclude that it's -11.

I really wish i could find some more prominent material on the subject though. Poke me if you find something more concrete to help us out. You really have me in doubt here

 

[emeraldrafael]

Snip

you're still doing what he said you cant. you're mixing a concept of infinity (no value, just a range), with a value, and you just cant. Its like mixing real and imaginary numbers. you cant, well, not really, to make one number from a combination of both.

Sure you can. Why shouldn't you be able to? It's right on the Wikipedia-page even (which, while it shouldn't be taken for granted as universal truth, still has a hell of alot of well-informed sources on the subject):

Students of mathematics often reject the equality of 0.999... and 1, for reasons ranging from their disparate appearance to deep misgivings over the limit concept and disagreements over the nature of infinitesimals. There are many common contributing factors to the confusion:
- Students are often "mentally committed to the notion that a number can be represented in one and only one way by a decimal." Seeing two manifestly different decimals representing the same number appears to be a paradox, which is amplified by the appearance of the seemingly well-understood number 1.

You are arguing that I'm combining two different numbers, when the entire idea is that the two numbers are in fact the same, just with different representation. I'm not making one number from a combination of two, because there is no "combination". They are the same.

No, no not with that example. What I'm saying is taht you're taking two different plains of mathematical reality and trying to mix it. You're trying to mix the idea of imaginary with real.

With the .999....! you're mixing the idea of a range (infinity), with a value, 1. Its the RANGE v. VALUE part that I'm focusing on of why you cant.

 

[Delta342]

Yes, I did say that, because it is true.

It is useful to have the concept of a square root be a function, which can't happen if it can be both negative and positive. Because it would be then one to many.

Occasionally it is useful to use square root as a function (I.e. yes we do require it to be injective) but not particularly often, or if we do then we do it by proving that the negative case results in say a contradiction in our initial assumptions. One thing I invite you to look at is determining the number of intersects of the line y=2 and the curve y=x^2 in 2 dimensional euclidean space.

0.999... is defined as a limit. So if the limit is 1, then it is 1, as they are the same thing.

Only in a standard sense. As I say, do some research into non-standard analysis, it's becoming popular again in a few applications I believe.. Mainly physics though I think. As for proving this just look up some basic definitions it all hinges on the idea of an infinitesimal.

Yeah, I'm going to have to disagree with you there too.

Then I suggest you look into vector bundles, K-Theory and a few other areas of very abstract pure Mathematics and you will see that equality isn't necessarily the strongest relation =)You could even just look at some elementary group theory. For instance, an isomorphism is much stronger than equality. Right, back to proving the Mordein-Weil Theorem for me. Amazing result!

 

[Delta342]

Ooh also, food for thought: if you define 1 := x/x then there is actually a singularity at x = 0, hehe.

 

[Delta342]

Not quite, if you're working in the integers, then 30/8 doesn't have a solution, you don't just round.

Got a source for that, because my old math teacher disagress

I must agree with Maze here. On the integers, division is not a binary operation. It is not closed as, like in the example given by Maze 30/8 is not an integer.

Oh and sorry Maze:

Infinity is not a number.

Actually, in some number systems, such as the Riemann Sphere [http://en.wikipedia.org/wiki/Riemann_sphere] infinity is a number. But that's fairly irrelevant here.

But even on the Riemann sphere infinity is not defined as a NUMBER. It is defined as a point (both top and bottom of the sphere) hence, point at infinity.

 

[Maze1125]

Occasionally it is useful to use square root as a function (I.e. yes we do require it to be injective) but not particularly often, or if we do then we do it by proving that the negative case results in say a contradiction in our initial assumptions. One thing I invite you to look at is determining the number of intersects of the line y=2 and the curve y=x^2 in 2 dimensional euclidean space.

I believe the solutions would be ±sqrt(2). Agreed?

0.999... is defined as a limit. So if the limit is 1, then it is 1, as they are the same thing.

Only in a standard sense. As I say, do some research into non-standard analysis, it's becoming popular again in a few applications I believe.. Mainly physics though I think. As for proving this just look up some basic definitions it all hinges on the idea of an infinitesimal.

I understand the basics of non-standard analysis, but you cannot come in and claim that it suddenly makes a difference and if anyone questions you just go "research it yourself".

I admit that I don't know a huge amount of non-standard, but nothing in what I do know has given me much reason to think that 0.999... =/= 1 or that 0.999... would have a different definition.

If you want to claim that it does make a difference, then you're the one who has to provide the definitions and proofs.

Yeah, I'm going to have to disagree with you there too.

Then I suggest you look into vector bundles, K-Theory and a few other areas of very abstract pure Mathematics and you will see that equality isn't necessarily the strongest relation =)You could even just look at some elementary group theory. For instance, an isomorphism is much stronger than equality. Right, back to proving the Mordein-Weil Theorem for me. Amazing result!

If that's all you meant, then yes, you're right.

Infinity is not a number.

Actually, in some number systems, such as the Riemann Sphere [http://en.wikipedia.org/wiki/Riemann_sphere] infinity is a number. But that's fairly irrelevant here.

But even on the Riemann sphere infinity is not defined as a NUMBER. It is defined as a point (both top and bottom of the sphere) hence, point at infinity.

The definition on "number" that I'm using is that a number is an element of a number set.
Riemann Sphere is a number set and infinity is an element of it, and is therefore a number with-in that context.

What definition are you using?

 

[havass]

Therefore, 10x - x = 9

I sense a disturbance in the Force...

The Force does not comprehend algebra, it would seem.

 

[Coldie]

No, no not with that example. What I'm saying is taht you're taking two different plains of mathematical reality and trying to mix it. You're trying to mix the idea of imaginary with real.

With the .999....! you're mixing the idea of a range (infinity), with a value, 1. Its the RANGE v. VALUE part that I'm focusing on of why you cant.

0.999... is a number (yes, this representation of the number has an infinite number of digits, numbers can do that). 1 is a number (approximately 1 digit). Where's this "range" thing you're talking about? Don't try to create an unfounded semantic argument.

P.S. The word you're looking for is probably "plane", not "plain", although it doesn't make all that much more sense, either... On the other hand, "Rolling hills of mathematical reality" would sound kinda good, actually.
P.P.S. Oh, and if you mix Imaginary with Real, you get Complex. That's a big part of math, too. The more you know...

 

[emeraldrafael]

No, no not with that example. What I'm saying is taht you're taking two different plains of mathematical reality and trying to mix it. You're trying to mix the idea of imaginary with real.

With the .999....! you're mixing the idea of a range (infinity), with a value, 1. Its the RANGE v. VALUE part that I'm focusing on of why you cant.

0.999... is a number (yes, this representation of the number has an infinite number of digits, numbers can do that). 1 is a number (approximately 1 digit). Where's this "range" thing you're talking about? Don't try to create an unfounded semantic argument.

P.S. The word you're looking for is probably "plane", not "plain", although it doesn't make all that much more sense, either... On the other hand, "Rolling hills of mathematical reality" would sound kinda good, actually.
P.P.S. Oh, and if you mix Imaginary with Real, you get Complex. That's a big part of math, too. The more you know...

I know about a complex number, but its till basically something like 5i (imaginary i symbol should be there but I dont know how to make it). Which is just taping the two together, since they can be easily broken apart again. Thats just separation, not elimination like 2-1=1 where you're phsycially deconstructing the number to make it something else.

And eh, I've seen it either way, though as stated before, grammar doesnt work in math or doesnt have much point to it since one is numbers and the other is letters and the combination of the two never yielded and epic like Romeo & Juliet (Speaking of plains v planes).

the range is infinity. it doesnt have value. Again, like I siad before, give me infinity somethings that i can hold,and if not hold see physically like you can give me one (1) dollar, apple, hooker, gun, rubber ducky, etc. To say it has infinite of them means yes, thats great, but as anyone will (usually) tell you when you do something like that in numbers you follow a trend. Hence that being you're just going back infinitely with nines, adding another somethingth to the number (tenth, hundredth, thousandth, zettath). Its sequencing in numbers, and allows someone to do soemthing like this:

1, 2, 3, 4, ... , 97, 98, 99, 100
or
2, 4, 6, 8, ... , 94, 96, 98, 100
or
-100, -99, -98, -97, ... , 97, 98, 99, 100

And save space and time by not writing them out. By saying .999...! you're saying you're creating a never ending sequence of nines that will continue to go back till someone either just gets bored and leaves or says stop (since nothing in the human mind is infinite [save the soul if you're religious]), and so everyhting has an end to to say something is infinite is just impractical. And that the reason it equals one is because there is a difference so minute it doesnt exist hence the logical action is to just make it one (even though when you do you deny what math, science, and all that good stuff was made to, which is to find everyhting and prove its existence. Math even has imaginary numbers while saying these dont exist in the real world, but someone needed to know what happened when you square rooted a negative number.

To say that there is nothing there is to undermine math. Its like dividing by zero. You dont, because if you did, you get nothing, and the point of math is that there is always something.

Its as my Math Professor says, "Infinity is just a tired math teacher's answer for when someone says, 'and what comes after that?' Its the same as when I a parent says, 'because I said so,' 'because it just does,' and 'there just is.' There's no need for it in math or any other form of education save perhaps philosophy or theological speculation because it is just speculation. You'll never use it in your life because we live in a world of tangible values, and there will never be found and infinite something for simple fact if there were it would destroy us all by taking up all the living space in the this known universe and any other universe. Now stop bothering me with this infinity thing and do the same equation, its not going to get you out of it."

 

[Maze1125]

And save space and time by not writing them out. By saying .999...! you're saying you're creating a never ending sequence of nines that will continue to go back till someone either just gets bored and leaves or says stop (since nothing in the human mind is infinite [save the soul if you're religious]), and so everyhting has an end to to say something is infinite is just impractical.

What you're basically saying there is that 0.999... doesn't exist at all, and neither do any other infinite decimals, because if the digits carry on for that long, then they're just too impractical to consider.

 

[Coldie]

Words

Don't mix apples and oranges, there are no ranges used anywhere in the original equation or its proofs. When a number is endless, it's not a range, it just is an endless number. It doesn't have an end. Infinity never stops.

There's no "real world", there's no "physical" component, there's no "human mind", there's no such thing as "impractical", nothing "has to end", there is only math. Infinity never stops and is used very, very often (look up Set Theory, Cardinals/Ordinals are all about infinity). Math has no foundation in the real world, math is pure abstraction and infinity is awesome. The point of Math is not "there's always something", there's a lot of places where there is nothing at all and nobody is eager to fill them in, as that would violate the integrity of the current system.

Do not pollute math with thoughts of "real world" applications and values that end "because someone got bored". Math is not physics or philosophy, it does not allow speculation or guesswork. Math is pure and absolute abstraction.

 

[emeraldrafael]

Snip

Math is hardly that. Its a giant contradiction, as to say yeah, this is proof, but give us three days, we'll think if a way is completely right in an opposite way. I'd hardly say its pure abstraction. Math (to me, with all its proofs like this) is the lazy man's way of wanting to sound intelligent and confuse people enough they dont argue back while saying why yes, this is the answer, but its completely and total opposite is the answer as well. which any logically sane person looks at and thinks no, but then again you're going to through more numbers at me and i have far more important things in my life to sit and look at numbers all day. Like games. Or girls. Or eating.

But then again, I dont get my jimmies jumped by seeing a complex problem written on the board, I get them jumped by seeing a single dot or seeing ten dots that are alternatively coloured when they touch and trying to decipher it (i.e. I'm more artistic then logical and more right brained then left).

And yes there is range in infinity. Do you know what range means? Range is when you go from something, to something. This is infinity at its core since infinity isnt a value and is just someone saying hey, this doesnt begin or end, so its obviously not one single thing but many things.

Also I never said infinity isnt awesome. The idea of it in theory is awesome cause its everything and anything all in one. If you want to look at it from the math perspective its every number that is, was, and ever will be into one conitunously growing stream that in all reality equates to zero since every new number found with all have its opposite and negative. its that same idea that makes me htink when i get bored in math class that I, Jesus, the boy beside me, the girl three rows and two columns over, the teacher, Stalin and every yet to be born being are in fact that same. thats about the only time Math gets abstract.

And save space and time by not writing them out. By saying .999...! you're saying you're creating a never ending sequence of nines that will continue to go back till someone either just gets bored and leaves or says stop (since nothing in the human mind is infinite [save the soul if you're religious]), and so everyhting has an end to to say something is infinite is just impractical.

What you're basically saying there is that 0.999... doesn't exist at all, and neither do any other infinite decimals, because if the digits carry on for that long, then they're just too impractical to consider.

I suppose so. But like I said, this is subjective really to how you think and what you're ideas of "fun" are. though more I say that the because the difference between .999...! and 1 doesnt exist it shouldnt make them equal.

It just means you broke the world and logical way of looking at things. You divided by zero. you placed a value where there was supposed to be a hole.

 

[Maze1125]

Math is hardly that. Its a giant contradiction, as to say yeah, this is proof, but give us three days, we'll think if a way is completely right in an opposite way. I'd hardly say its pure abstraction. Math (to me, with all its proofs like this) is the lazy man's way of wanting to sound intelligent and confuse people enough they dont argue back while saying why yes, this is the answer, but its completely and total opposite is the answer as well. which any logically sane person looks at and thinks no, but then again you're going to through more numbers at me and i have far more important things in my life to sit and look at numbers all day. Like games. Or girls. Or eating.

None of that is true at all.

Mathematical proofs are meant for mathematicians. And they all know what the terms mean and don't have anything better to do than sit and look at numbers all day. If a proof is wrong, then it will be shown to be wrong.

There are no contradictions, as mathematicians have spend the last 400 years eliminating them all, if one thing is true, then the opposite is not true.

And, dear lord, to call mathematics the "lazy man's" anything is just flabbergasting. The amount of work needed to get anywhere in maths is huge. Sure, I can real off a limit proof very quickly nowadays, but that's only because those were the most basic things I was taught first at university years and years ago.

 

[emeraldrafael]

Math is hardly that. Its a giant contradiction, as to say yeah, this is proof, but give us three days, we'll think if a way is completely right in an opposite way. I'd hardly say its pure abstraction. Math (to me, with all its proofs like this) is the lazy man's way of wanting to sound intelligent and confuse people enough they dont argue back while saying why yes, this is the answer, but its completely and total opposite is the answer as well. which any logically sane person looks at and thinks no, but then again you're going to through more numbers at me and i have far more important things in my life to sit and look at numbers all day. Like games. Or girls. Or eating.

None of that is true at all.

Mathematical proofs are meant for mathematicians. And they all know what the terms mean and don't have anything better to do than sit and look at numbers all day. If a proof is wrong, then it will be shown to be wrong.

There are no contradictions, as mathematicians have spend the last 400 years eliminating them all, if one thing is true, then the opposite is not true.

And, dear lord, to call mathematics the "lazy man's" anything is just flabbergasting. The amount of work needed to get anywhere in maths is huge. Sure, I can real off a limit proof very quickly nowadays, but that's only because those were the most basic things I was taught first at university years and years ago.

If math isnt contradictory, then how can you have a REAL and IMAGINARY number exist in it at the same time, put them together, and make a complex number? When the definition of both the are completely opposite.

I've seen people disprove 2+2=4. they make it equal something else, how is that not contradictory?

EDIT: really, the only opposite I've ever seen that hasnt had an equal and opposite statement is return is that you cant divide by zero. But I'm sure someday someone will come up with something as to say you can, so to me, its just time.

 

[Liam Moriarty]

2003 called, it wants its math meme back.

This has nothing to do with any meme. I'm not some 4chan idiot. I want to see how many people reject the concept.

Rule 50

OT: I guess this makes sense, but I'm going to show my geometry teacher this just to see what she says.

What is rule 50?

Rule 50 of the internet: Anything can be a meme

 

[Maze1125]

If math isnt contradictory, then how can you have a REAL and IMAGINARY number exist in it at the same time, put them together, and make a complex number? When the definition of both the are completely opposite.

Those are just names.
You can't claim a contradiction based on names that were picked hundreds of years ago. That's just nonsense.

I've seen people disprove 2+2=4. they make it equal something else, how is that not contradictory?

It is. Which is how you can know those proofs are false. Of course, showing where they went wrong is a different matter. But we know that 2 + 2 = 4, therefore any statement that claims that 2 + 2 =/= 4, must be false.

EDIT: really, the only opposite I've ever seen that hasnt had an equal and opposite statement is return is that you cant divide by zero. But I'm sure someday someone will come up with something as to say you can, so to me, its just time.

It rather seems your problem is that you've run into a load of people who've taken one mathematics class and used what they've heard to try and sound clever by spouting off things that are contradictory.

That doesn't mean that mathematics itself is contradictory, just that you've met a load of pseudo-mathematicians who wanted to sound clever.