Poll: 0.999... = 1

[Maze1125]

It's not one, it's infinitely close to 1. Christ.

So, on a scale, how would you draw a line that is infinitely close to some other line? Assuming the measure has no width (as the numbers also don't), it would be on that other line. No magic necessary.

Mate, 0,99999... isn't a finite number, and as such can't be drawn. To use this sort of math you need finite numbers. By definition it's infinitely close to, not the same as 1.

No, the definition of 0.999... is lim(as n->infinity)sum(from k=1 to n) (9 * 1/10[sup]k[/sup])

And nothing in that necessarily means that it is infinitely close to 1 but not the same.
In fact, if you calculate it, you get 1 exactly.

 

[Rubashov]

It's not one, it's infinitely close to 1. Christ.

So, on a scale, how would you draw a line that is infinitely close to some other line? Assuming the measure has no width (as the numbers also don't), it would be on that other line. No magic necessary.

Mate, 0,99999... isn't a finite number, and as such can't be drawn. To use this sort of math you need finite numbers. By definition it's infinitely close to, not the same as 1.

0.999... IS a finite number. An infinitely long number and an infinite "number" are not the same thing.

People keep making the point that no matter how many 9s you add to the end of 0.9, you never actually get to 1; you just get really, really close. But this is irrelevant, because adding additional 9s to the end of 0.9 will never actually give you a number with infinite 9s at the end either. Since the number 0.999... does, in fact, have infinite nines at the end, it is equivalent to 1.

 

[emeraldrafael]

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).

Then you sir, have never written, nor read some fantastic papers on Mathematics. Some would say certain proofs were the equivalent of Shakespearean literature.

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.

Actually in certain circumstances you can divide by 0. Let R be a commutative ring and let S be an R-module. Then an element s in S is called a torsion element if there exists a non-zero element x in R such that sx = 0.

And there it is. Math has finally contradicted its self so much you can finally divide by zero.

 

[Athinira]

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.

Which is still incorrect.

The real number system actually works within the confines of the fact that all values in it's system have infinite decimals (we as humans just choose not to express them most of the time). The number 4 is actually the number 4.000... The number 12.47 is actually 12.47000... with infinite zeroes.

See where I'm going? By your argument, there actually doesn't exist any "values" in the real number system, just ranges, which is obviously false.

Also 0.999... is NOT an imaginary number.

Within the confines of the real number system, it's valid to express numbers with infinite decimals because all values in that system by definition has infinite decimals. They either have infinite 0-9's, repeat an infinite sequence or continue an infinite sequence that doesn't repeat itself (irrational numbers). But they aren't at any point imaginary, and the system not only allows us to work with the values this way, it actually expects us to.

 

[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.

Which is still incorrect.

The real number system actually works within the confines of the fact that all values in it's system have infinite decimals (we as humans just choose not to express them most of the time). The number 4 is actually the number 4.000... The number 12.47 is actually 12.47000... with infinite zeroes.

See where I'm going? By your argument, there actually doesn't exist any "values" in the real number system, just ranges, which is obviously false.

Also 0.999... is NOT an imaginary number.

Within the confines of the real number system, it's valid to express numbers with infinite decimals because all values in that system by definition has infinite decimals. They either have infinite 0-9's, repeat an infinite sequence or continue an infinite sequence that doesn't repeat itself (irrational numbers). But they aren't at any point imaginary, and the system not only allows us to work with the values this way, it actually expects us to.

well, I was actually talking about actual imaginary numbers. You know, the kinda things you get when square root a negative number? Not the .999...! and numbers like it.

And alright, but your logic, since .999...! has a such a small insignificant difference that it can not be seen, or it just doesnt exist all together, then it equals 1. Which has been said.
So, following this same logic 4.000...! would just equal 3.9 or 4.(1,01, 001, etc). So by your logic, there is no difference in any number, because the idea of infinite decimal spaces after a number creates a difference so low it cant be seen or it doesnt have a difference period.

 

[Athinira]

And alright, but your logic, since .999...! has a such a small insignificant difference that it can not be seen, or it just doesnt exist all together, then it equals 1. Which has been said.
So, following this same logic 4.000...! would just equal 3.9 or 4.(1,01, 001, etc). So by your logic, there is no difference in any number, because the idea of infinite decimal spaces after a number creates a difference so low it cant be seen or it doesnt have a difference period.

Actually no.

By my logic, 3.999... and 4.000 is the same, yes, but 4.000... wouldn't equal 4.01. Rather, 4.00999... (with the 9's being the infinitely recurring decimal) would actually equal 4.01 (or 4.01000... if you like).

If you think that by my logic, there is no difference in numbers, then you understand neither the concept, or my logic to begin with.

To help you understand, i will just point back to Linear Continuum:
For every two numbers X and Y, where X < Y, there exists a number Z so that X < Z < Y. If Z doesn't exist, then X = Y.

Try to play around with some of the numbers in your own post and this, assign them to X and Y and see if you can find Z. When you start to understand where Z exists and where it doesn't without having to do the math first, then you should understand the logic

 

[Maze1125]

And there it is. Math has finally contradicted its self so much you can finally divide by zero.

Will you please stop saying that mathematics contradicts itself.
Just because you don't understand why two seemingly contradictory things can both be true, doesn't mean there is an actual contradiction.

 

[emeraldrafael]

And there it is. Math has finally contradicted its self so much you can finally divide by zero.

Will you please stop saying that mathematics contradicts itself.
Just because you don't understand why two seemingly contradictory things can both be true, doesn't mean there is an actual contradiction.

BUt its breaking the cardinal rule of math. you odnt divide by zero.

 

[Maze1125]

And there it is. Math has finally contradicted its self so much you can finally divide by zero.

Will you please stop saying that mathematics contradicts itself.
Just because you don't understand why two seemingly contradictory things can both be true, doesn't mean there is an actual contradiction.

BUt its breaking the cardinal rule of math. you odnt divide by zero.

There's nothing wrong with dividing by zero, provided you only do it in a system that has been constructed in such a way that it is allowed.

The standard numbers systems do not have such a construction, so dividing by zero isn't allowed there and, as most students would never come across a system where it is allowed, they are just simply told "Never divide by zero."

That doesn't mean that it can't ever be done, just that your teacher doesn't have time to qualify every single mathematical rule they tell you.
Edit: And even if they did have the time, it wouldn't even necessarily be a good thing for them to tell you, as then your head would be full of qualifications you didn't need to know quite possibly resulting in enough confusion to reduce your mark in the exams.

 

[emeraldrafael]

Snip

*sigh* I give up. Just flat out give up. Math, like anything else, should have practical use, and there is no practical use to dividing by zero. Maybe thats just me, BUt I dont know. I just give up. I'll never use it. Doesnt mean its wrong, doesnt mean others cant have their fun with this stuff, but to me, Its just not for me.

So I give up/

 

[Maze1125]

there is no practical use to dividing by zero.

How would you know?
You're claiming there that in the entirety of everything ever, there could never be a time where dividing my zero could be in any way useful. That's one hell of claim to make.

A large proportion of the mathematics we use in the sciences today was originally discovered just for the hell of doing mathematics. There was absolutely no use for it at the time, it was made up just because someone wanted to, and then a use was found for it later.
If people had only ever researched mathematics that had obvious immediate practical use then many discoveries would never have been made, and science would have been held back because of it.

There is no gain in having a rule that "It must be practical or you can't do it." No gain at all.

 

[emeraldrafael]

there is no practical use to dividing by zero.

How would you know?
You're claiming there that in the entirety of everything ever, there could never be a time where dividing my zero could be in any way useful. That's one hell of claim to make.

A large proportion of the mathematics we use in the sciences today was originally discovered just for the hell of doing mathematics. There was absolutely no use for it at the time, it was made up just because someone wanted to, and then a use was found for it later.
If people had only ever researched mathematics that had obvious immediate practical use then many discoveries would never have been made, and science would have been held back because of it.

There is no gain in having a rule that "It must be practical or you can't do it." No gain at all.

Like I said, I give up. And yes, there is no reason to divide by nothing. What in any purpose does that serve, ever? the only wnat i could even think of it to mean is to just show how poor you are.

 

[Maze1125]

Like I said, I give up. And yes, there is no reason to divide by nothing. What in any purpose does that serve, ever? the only wnat i could even think of it to mean is to just show how poor you are.

The use isn't in dividing by zero, but in the number systems where the phenomenon exists.

And, really, you're being extremely arrogant here. Just because you can't think of a way for it to be useful doesn't mean there isn't a way, it only means that you have failed to think of it.
Or do you think you're omniscient or something?

 

[emeraldrafael]

Like I said, I give up. And yes, there is no reason to divide by nothing. What in any purpose does that serve, ever? the only wnat i could even think of it to mean is to just show how poor you are.

The use isn't in dividing by zero, but in the number systems where the phenomenon exists.

And, really, you're being extremely arrogant here. Just because you can't think of a way for it to be useful doesn't mean there isn't a way, it only means that you have failed to think of it.
Or do you think you're omniscient or something?

No, I'm just saying, for me personally, (i.e. My OPINION, on this OPINION thread), I dont see any use in math if its not practical and applicable to my life. if I dont use it, there's not much reason to learn it. But thats just me. And I'm just stating that I dont see the use, but I dont now everything. I dont devote myself to math, cause I find other things fun and more fulfilling.

 

[Athinira]

No, I'm just saying, for me personally, (i.e. My OPINION, on this OPINION thread), I dont see any use in math if its not practical and applicable to my life. if I dont use it, there's not much reason to learn it. But thats just me. And I'm just stating that I dont see the use, but I dont now everything. I dont devote myself to math, cause I find other things fun and more fulfilling.

Actually thats not what you said. You originally said that there was no use at all in dividing by zero, you didn't state "for me personally", although it's a good thing you changed it.

All i think Maze is asking of you is to open your mind and say "Okay, i don't see the practicality in dividing by zero, but I'm open to the fact that it may, some day, actually be a useful tool in some branch of (mathematical) science, so i understand why some other people might find it interesting to check out. I just don't see the benefit personally, but don't let me stop you guys from trying."

Simply outright denying that it has any use isn't very constructive. I can't think of a use for it either, but if someone who is smarter than me someday wants to tell me that he found a use for it, then I'll of course be listening.

 

[emeraldrafael]

No, I'm just saying, for me personally, (i.e. My OPINION, on this OPINION thread), I dont see any use in math if its not practical and applicable to my life. if I dont use it, there's not much reason to learn it. But thats just me. And I'm just stating that I dont see the use, but I dont now everything. I dont devote myself to math, cause I find other things fun and more fulfilling.

Actually thats not what you said. You originally said that there was no use at all in dividing by zero, you didn't state "for me personally", although it's a good thing you changed it.

All i think Maze is asking of you is to open your mind and say "Okay, i don't see the practicality in dividing by zero, but I'm open to the fact that it may, some day, actually be a useful tool in some branch of (mathematical) science, so i understand why some other people might find it interesting to check out. I just don't see the benefit personally, but don't let me stop you guys from trying."

Simply outright denying that it has any use isn't very constructive. I can't think of a use for it either, but if someone who is smarter than me someday wants to tell me that he found a use for it, then I'll of course be listening.

I'd like to be open minded, but everytime i am and form an Idea, i get bitched and screamed at for the idea I had, so i just keep myself close minded and practical. Its been alot easier on my life, though my imagination is being stifled.

 

[zoulza]

It's not one, it's infinitely close to 1. Christ.

So, on a scale, how would you draw a line that is infinitely close to some other line? Assuming the measure has no width (as the numbers also don't), it would be on that other line. No magic necessary.

Mate, 0,99999... isn't a finite number, and as such can't be drawn. To use this sort of math you need finite numbers. By definition it's infinitely close to, not the same as 1.

Christ, it's amazing how many times you can point out that the reals do not permit infinitesimals, and yet people keep on using that same dumb argument.

 

[Houmand]

It's not one, it's infinitely close to 1. Christ.

So, on a scale, how would you draw a line that is infinitely close to some other line? Assuming the measure has no width (as the numbers also don't), it would be on that other line. No magic necessary.

Mate, 0,99999... isn't a finite number, and as such can't be drawn. To use this sort of math you need finite numbers. By definition it's infinitely close to, not the same as 1.

0.999... IS a finite number. An infinitely long number and an infinite "number" are not the same thing.

People keep making the point that no matter how many 9s you add to the end of 0.9, you never actually get to 1; you just get really, really close. But this is irrelevant, because adding additional 9s to the end of 0.9 will never actually give you a number with infinite 9s at the end either. Since the number 0.999... does, in fact, have infinite nines at the end, it is equivalent to 1.

Well, if an infinitely long number is not an infinite number, then what is an infinite number?

 

[Houmand]

It's not one, it's infinitely close to 1. Christ.

So, on a scale, how would you draw a line that is infinitely close to some other line? Assuming the measure has no width (as the numbers also don't), it would be on that other line. No magic necessary.

Mate, 0,99999... isn't a finite number, and as such can't be drawn. To use this sort of math you need finite numbers. By definition it's infinitely close to, not the same as 1.

No, the definition of 0.999... is lim(as n->infinity)sum(from k=1 to n) (9 * 1/10[sup]k[/sup])

And nothing in that necessarily means that it is infinitely close to 1 but not the same.
In fact, if you calculate it, you get 1 exactly.

Dude, the mathematical limit is only a "qualified guess", it's not the exact number. Google it, Einstein.

 

[Houmand]

It's not one, it's infinitely close to 1. Christ.

So, on a scale, how would you draw a line that is infinitely close to some other line? Assuming the measure has no width (as the numbers also don't), it would be on that other line. No magic necessary.

Mate, 0,99999... isn't a finite number, and as such can't be drawn. To use this sort of math you need finite numbers. By definition it's infinitely close to, not the same as 1.

What? The point is that in order to accomplish the process of drawing that, you'd need to draw it at 1. This is the whole point of it. Otherwise you keep sitting in the corner and singing "but it is not 1" and nothing ever happens.

Mate, you CAN'T draw such a line, because it doesn't exist. To draw a line, you need a FINITE number, and since 0,999... is INfinite, the line cannot be drawn. And if one did draw it, it would never be 1, it would be infitely close to 1.
1 - 0.999.... isn't 0, it's 0,00...001.