Poll: 0.999... = 1

[emeraldrafael]

Snip

I'd be inclined to believe you, but I used (well... more went against) all twelve and college semester's worth of math knowledge and to that knowledge they proved 2+2= soemthing other then four. Like how originally someone said x = more then one number (though they misunderstood my post).

Also, its not even the words, but the ideals behind. I was always told you could NOT find the square root of a negative number. Yet if you do, the punishment inst what it is by dividing by zero, its telling you you get an imaginary number, which in essence cant be used for much of anything other then (what I'm going to guess is what mathematicians sit around and do all day) throwing it at something else and seeing if it works.

And no, I've met my fair share of those people, but its also the math... um... hm, elites? Yeah, we'll say the math elites. they put themselves on a level above me because I didnt want to be an actuary science major when I went to college and like to break down everything I do into a numbers game. Though it doesnt help that the last one that tried to do that while i was playing hockey met the blade end of my stick into his frontal lobe so i could see if he would still dare to tell me how based on his equations i only had to lift my arm this much more or hit the puck that much harder to score. But thats straying from the topic and conversation.

 

[Coldie]

I'd be inclined to believe you, but I used (well... more went against) all twelve and college semester's worth of math knowledge and to that knowledge they proved 2+2= something other then four.

Math is big. A few semesters would just scratch the surface. If you're talking about Real numbers, then 2+2 always equals 4, but if you create a different set of numbers, 2+2 will equal whatever you want. All the fields, rings, groups you want are there. Modular arithmetics (2 + 2 = 0 mod 4), Zero Divisors (2 * 2 = 0 in the Z[sub]4[/sub] ring), etc. Linear Algebra is all magic like that.

And yet, there will be no contradictions. When you create your own algebraic group, you create a new system, with new rules. Existing systems, like Real numbers will still be there, with 2+2 forever being 4.

When someone says that you cannot find a square root of a negative number, they mean it. There is no answer to x = sqrt(-1) or x = 1/0 within Real numbers. The answer exists elsewhere, in a different system. One that isn't covered by the current class. If all math classes had to say "you can do it, but that's Imaginary/Complex numbers, so don't do it in this class", that would just add unnecessary confusion. Math in the first school year forbids stuff like 2/3 and 2-3 and yet nobody complains.

 

[emeraldrafael]

I'd be inclined to believe you, but I used (well... more went against) all twelve and college semester's worth of math knowledge and to that knowledge they proved 2+2= something other then four.

Math is big. A few semesters would just scratch the surface. If you're talking about Real numbers, then 2+2 always equals 4, but if you create a different set of numbers, 2+2 will equal whatever you want. All the fields, rings, groups you want are there. Modular arithmetics (2 + 2 = 0 mod 4), Zero Divisors (2 * 2 = 0 in the Z[sub]4[/sub] ring), etc. Linear Algebra is all magic like that.

And yet, there will be no contradictions. When you create your own algebraic group, you create a new system, with new rules. Existing systems, like Real numbers will still be there, with 2+2 forever being 4.

When someone says that you cannot find a square root of a negative number, they mean it. There is no answer to x = sqrt(-1) or x = 1/0 within Real numbers. The answer exists elsewhere, in a different system. One that isn't covered by the current class. If all math classes had to say "you can do it, but that's Imaginary/Complex numbers, so don't do it in this class", that would just add unnecessary confusion. Math in the first school year forbids stuff like 2/3 and 2-3 and yet nobody complains.

Yes but for obvious reasons. I learned fractions by second grade, but they drilled additions and numbers up to 100 plus different ways to count to it (even, odd, counting).
By the first quarter of thrid grade, I knew prime numbers. Its all pacing.

But even then, I've never heard of a college that teaches a class solely on the concept of imaginary numbers.

 

[Coldie]

Yes but for obvious reasons. I learned fractions by second grade, but they drilled additions and numbers up to 100 plus different ways to count to it (even, odd, counting).
By the first quarter of third grade, I knew prime numbers. Its all pacing.

But even then, I've never heard of a college that teaches a class solely on the concept of imaginary numbers.

Imaginary and Complex numbers are covered by Calculus and Complex Analysis. They also feature fairly prominently in some fields of physics and engineering. Also, the pretty fractals everyone loves so much? They are created via Complex calculations.

There's probably no class dedicated to C numbers, but they are still a major "player" in modern science and technology.

 

[Delta342]

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

Indeed, thus you have shown that the solution to sqrt = + sqrt(2) or -sqrt(2). Apologies it was about 2am last night and I was running on only a few hours sleep =S

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".

The reason I point you towards doing your own research is because to understand enough of non-standard analysis we would need to go through several definitions, prove some Theorems and basic properties, possibly squeeze in a few lemmas and then really look into our problem. So rather than take up huge amounts of space here and my time I referred you to material. There's no point in me proving something using say hyperreal numbers or just the symbol *R or *N without giving a good background first.

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

Another reason I know is because in my first year me and my housemates were arguing about this, we attracted the attention of a PHD student, a lecturer and a professor, it was actually quite an even split but it all really depends on what part of mathematics you hail from. It was the professor who actually said using non-standard analysis then 0.999... =/= 1 without a doubt.

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?

So you're defining infinity as infinity = (infinity, infinity)? Generally we have defined it purely as infinity. The Riemann sphere is not made up of numbers. It is generally constructed using projection and is thus points. Not quite the same. Alternatively infinity can be defined using projective Geometry. In that "points at infinity" in two dimensional projective space are elements of the equivalence class (x:y:0).

Out of interest Maze are you currently studying Mathematics?

 

[Delta342]

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.

 

[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.

 

[gl1koz3]

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.

 

[Maze1125]

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

Indeed, thus you have shown that the solution to sqrt = + sqrt(2) or -sqrt(2).

y = 2, so what you've just said there is that sqrt(2) = -sqrt(2).

Personally, I'd find it far easier, and less confusing to say that the solutions for x are sqrt(2) and -sqrt(2). Which are, of course, also sqrt and -sqrt.

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".

The reason I point you towards doing your own research is because to understand enough of non-standard analysis we would need to go through several definitions, prove some Theorems and basic properties, possibly squeeze in a few lemmas and then really look into our problem. So rather than take up huge amounts of space here and my time I referred you to material. There's no point in me proving something using say hyperreal numbers or just the symbol *R or *N without giving a good background first.

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

Another reason I know is because in my first year me and my housemates were arguing about this, we attracted the attention of a PHD student, a lecturer and a professor, it was actually quite an even split but it all really depends on what part of mathematics you hail from. It was the professor who actually said using non-standard analysis then 0.999... =/= 1 without a doubt.

Well I have done some research into this specific problem and from what I've found there isn't even an agreed definition of what "0.999..." means when you try to extend it to the hyperreals. And it's quite possible to come up with perfectly good definitions that go either way. Where 0.999... = 1 and where 0.999... =/= 1.

So, for the moment, it seems that it would be far more sensible to keep this issue confined to standard analysis, where 0.999... is well-defined.

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?

So you're defining infinity as infinity = (infinity, infinity)? Generally we have defined it purely as infinity. The Riemann sphere is not made up of numbers. It is generally constructed using projection and is thus points. Not quite the same. Alternatively infinity can be defined using projective Geometry. In that "points at infinity" in two dimensional projective space are elements of the equivalence class (x:y:0).

You're twisting everything just to sound right. Yes, the Riemann Sphere can be considered as such a set of points on a sphere, but it can also be perfectly well considered as a set of numbers.

And it's hardly the case that that is the only situation where infinity is considered to be a number. Set theory is founded on numbers that have cardinality of infinity, several different ones no less, and infinity has to be a value for that to be true.
Yes, they're all given different names to avoid confusion, but the concept is the same.

Out of interest Maze are you currently studying Mathematics?

I do study mathematics, yes, but I've finished what most people would call "studies".

 

[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.