Poll: 0.999... = 1

[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.

 

[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".