Poll: 0.999... = 1

[Athinira]

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

Infinity?

.oO(insert the sign yourself)

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.

All numbers are per definition infinitely long in the Real Number system. 4 is equivalent to 4.000... 3.523 is equivalent to 3.523000... I could go on.

You still don't understand what an infinite number is. 0.999... is not infinite (i can find plenty of numbers that are larger than it, like 2, 3, 4, 4.5, 4.75, 200 etc.), it's just infinitely long, just like EVERY other number in the Real Number system.

And saying that you can't draw a line on an infinitely long number is also incorrect. I could easily draw this line in a coordinate system for example:
f(x) = x * 3.333...
That line would have several passing points easily discernable:
f(2) = 6.666...
f(3) = 10
f(4) = 13.333...
f(5) = 16.666...
f(6) = 20

 

[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.

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

Assuming you're right, that would only be a problem if I was trying to find a number defined other than as a limit, by using limits.

But that is not the case.

0.999... is itself defined as a limit. So it doesn't matter if limits are just a "qualified guess", or something else. Because 0.999... is a limit, so if limits are just "qualified guesses" then 0.999... itself is just a "qualified guess".

 

[James13v]

If x = 0.999999...
Then 10x = 9.9999...
Therefore, 10x - x = 9
Which implies 9x = 9
Thus, x = 1
x also = 0.99999...

In conclusion, I have just proven 1 = 0.9999...

The thing is, infinity is relative. So ...

x = 0.9999 .... to the infinite decimal place
10x = 9.9999 .... to one less infinite place value
Therefore:
10x - x = 8.9999 .... 1, where the one is in the infinite decimal place
9x = 8.9999 .... 1
8.9999 .... 1 = 0.9999 ... to the infinite decimal place

If that doesn't make sense to you, imagine two ... let's make them space ships, travelling along the same axis in the same direction at the exact same speed which will never alter. If Rocket 1 is 10 metres in front of rocket 2, and they both start at the same time, after an infinite amount of time has passed, the distance they are from rocket 2's starting point is infinity, but rocket 1 is 10 metres in front of rocket 2. Therefore, both ships have travelled an infinite distance, but the distance between the origin and rocket 1 is a greater degree of infinity (by a distance of 10 metres) than rocket 2.

You wouldn't be able to assess their positions at infinity amount of time because they would never reach it...

Maths is a representation of our reality through the use of numbers, so my rocket ship analogy-thing was basically to give a real-world example of the case of re3lative infinities. And sure, we can't measure the distance after an infinite time, but we can state that, because of the equal velocity (and therefore constant distance between the two rockets) that one is furhter away from an infinitely far point than the other.

Actually I feel I'm explaining it badly. Basically, our understanding of reality is that, under the described rocket circumstances, they will both be an infinite distance from a particular point in space[footnote]Because the concept of infinity is something which is without limits it is basically saying "If they travel for a period of time that is without a limit, they will travel a proportionally infinite distance" (proportional to velocity and initial location that is)[/footnote], but they will both be a different distance. So they are both an infinite distance, but one is more than the other. And seeing as maths is a numerical representation of our reality, then this case of 0.99999... is an example of relative infinities as expressed by the rocket example.

So basically (in case I still worded it poorly), because the nature of our perception/reality is such that infinity can be perceived as existing to degrees (as in the 'bound by physical laws' example of rocket ships) then this 'relative infinities' quality is a propery of mathematics. So, because maths is dependent upon reality, and reality supports relative infinity, then 0.999... does not equal 1.

I get what you are trying to say, but "infinities" are not things, they do not have identities because they are not finite. You cannot comment on the ships' distances after an infinite amount of time because it is a scenario that physically/realistically can never happen (not virtually impossible, just plain impossible). I'm not extremely adept at maths, but I do consider myself pretty logical, so consider this my slightly above lay-level view of the matter.

EDIT: Nevermind, I get it now, lol.

 

[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.

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

Infinity... which isn't a number, at least not on the real number line.

0.999... cannot be an infinite number because 2 is demonstrably greater than 0.999..., and NOTHING is greater than infinity because infinity is, well, infinite.

 

[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.

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?

Infinity... which isn't a number, at least not on the real number line.

0.999... cannot be an infinite number because 2 is demonstrably greater than 0.999..., and NOTHING is greater than infinity because infinity is, well, infinite.

That's not strictly true.

The integers are infinite, the real numbers are also infinite.
But it can be proven absolutely that the real numbers are a larger set than the integers.

 

[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.

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

Infinity... which isn't a number, at least not on the real number line.

0.999... cannot be an infinite number because 2 is demonstrably greater than 0.999..., and NOTHING is greater than infinity because infinity is, well, infinite.

That's not strictly true.

The integers are infinite, the real numbers are also infinite.
But it can be proven absolutely that the real numbers are a larger set than the integers.

...Huh. That makes sense. Still, 0.999... does not have an infinite value because its value is demonstrably less than the finite value 2.

 

[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.

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.

No. The point still is that the infinity is... what it is: infinite. This value is incomprehensible and the way we use it is just to skip the evaluation to the result of it. Just see what you're saying: "it would be infitely close to 1". How does that even look? Infinity implies that whatever is in question, is of unending size. Your unending closeness to 1 is 1. Infinitely close to 1 is 1. Period.

 

[Delta342]

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.

I guess the way I have always understood is that infinity merely implied unboundedness, different ways of thinking of the same thing I guess!

Hehe, you have to give it to set Theorists; 100 pages just to prove the existence of the real numbers, now that's a rigorous proof! =P

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

By that do you mean you study for your own interest? Or a PHD?
Also; Merry belated Christmas! =D

 

[tautologico]

Quite the long thread, huh? People sure have problems with the notion of limit.