Do you think the human notion of 'infinite'/'infinity' is actually real infinity?

[onejayhawk]

Is .99999.... = 1?

Of course not, but close enough for practical purposes. The proof is trivial. .9999... is on the unit segment open at 1, but 1 is not. QED

That said you will see proofs that they identical , silly as that is. You will even get the statement that an open segment has no endpoint. The correct statement is that an open segment has no rational endpoint. We have the concept of an infinitely small distinction, but handling it is difficult. Instead we approximate, which destroys the purpose.

J

 

[Borachio]

The equality 0.999... = 1 has long been accepted by mathematicians and is part of general mathematical education. Nonetheless, some students find it sufficiently counterintuitive that they question or reject it. Such skepticism is common enough that the difficulty of convincing them of the validity of this identity has been the subject of several studies in mathematics education.

http://en.wikipedia.org/wiki/0.999...

 

[Leoreth]

That said you will see proofs that they identical , silly as that is.

Oh no, mathematicians and their logical proofs. They are caught in a system!

Let's all just make up our own opinions instead.

You will even get the statement that an open segment has no endpoint. The correct statement is that an open segment has no rational endpoint.

It also has no real endpoint.

 

[onejayhawk]

It also has no real endpoint.

Incorrect. There is a real endpoint, but we cannot describe it directly. Like e and pi, transcendental numbers can only be approximated. The best approximation is 1, which we removed. Hence we say it does not exist. Sophistry and nonsense.

J

 

[Leoreth]

To clarify: when I say real, I mean real as in "real number". Transcendental numbers are real numbers. No number in an open segment is at the same time its bound, as long as we are talking real numbers. This is not a question of representation or approximation, but existence.

You can make up numbers that conform to rules where this would be the case, but then you have left the math you are familiar with entirely behind you.

This is the exact opposite of sophistry, this is cold hard logic, from axiom and definition to the final conclusion. You may not understand it, it may frustrate you or upset your intuition, but this doesn't change the fact that it is as true as a thing can be.

Of course I am willing to change my stance if you can show why every proof that demonstrates 0.999... = 1 is false. Or let's say just one of them.

 

[Thorgalaeg]

from wikipedia:

1/9=0,11111111...
2/9=0,22222222...
...
8/9=0,8888888...
so 9/9 should be 0,9999999... but 9/9 is also 1 so 0,999999...=1

I am not sure that is a valid mathematical demostraton.

This one is better:

1/9 = 0,11111...
9 x 1/9 = 0,9999999... = 9/9 = 1

 

[Leoreth]

It is if you actually accept that 1/9 = 0.111... .

I think it works better with 1/3 because almost everyone agrees that it is equal to 0.333... . Multiply by three and you have 1/1 on one side and 0.999... on the other.

Another popular proof goes like this: let x = 0.999..., then:

x = 0.999... | * 10
10x = 9.999... | -x
9x = 9 | / 9
x = 1

The only way x can equal both 1 and 0.999... is if they are equal themselves.

It's fascinating how people can still grasp at straws trying to deny something that can be proven in such a simple way.

 

[Giftless]

The whole question really comes down to asking if anything in mathematics is a description of something "real" that is being discovered or if it is just playing around with logic.
I don't think there will ever be a definite answer to that.

I think mathematics is a "system" which describes reality. This is fine and dandy as long as the frame works, but something to remember is that there may be exceptions to the rule at the extreme. We may be able to predict how the universe operates within a certain range; however, this doesn't mean that our interpretation of the whole is correct.

Granted, my background is in Biology where many of the molecular interactions become too difficult to understand outside the chemical pathway, which is the known portion so to speak. In terms of drug-interactions and hormones I get the impression that our doctors are still kicking the sides of tv sets in most cases.

 

[Giftless]

To clarify: when I say real, I mean real as in "real number". Transcendental numbers are real numbers. No number in an open segment is at the same time its bound, as long as we are talking real numbers. This is not a question of representation or approximation, but existence.

You can make up numbers that conform to rules where this would be the case, but then you have left the math you are familiar with entirely behind you.

This is the exact opposite of sophistry, this is cold hard logic, from axiom and definition to the final conclusion. You may not understand it, it may frustrate you or upset your intuition, but this doesn't change the fact that it is as true as a thing can be.

Of course I am willing to change my stance if you can show why every proof that demonstrates 0.999... = 1 is false. Or let's say just one of them.

Actually the 0.999 = 1 makes some sense from a significant digits standpoint; if you apply it to something like machine work (where I imagine the precision of the cut would matter) past a certain point you could just round off. Another thing is that this rounding could reflect the margin of error in one's measuring instruments

 

[warpus]

Math is based on axioms some of which mirror reality; everything in the discipline is ultimately sitting on those several assumptions. It shouldn't be a surprise then that we can use math to model reality and get results which mirror it to a degree, since the building blocks of math mirror it also.

 

[Kyriakos]

^And more crucially in science (not including math in the same way, here) people try to model what they view in the physical world, ie not the world by itself but the world sensed by humans in the first place.
The above is the main reason why Socrates/Plato are dismissive of non-math science.

 

[onejayhawk]

It is if you actually accept that 1/9 = 0.111... .

I think it works better with 1/3 because almost everyone agrees that it is equal to 0.333... . Multiply by three and you have 1/1 on one side and 0.999... on the other.

Another popular proof goes like this: let x = 0.999..., then:

x = 0.999... | * 10
10x = 9.999... | -x
9x = 9 | / 9
x = 1

The only way x can equal both 1 and 0.999... is if they are equal themselves.

It's fascinating how people can still grasp at straws trying to deny something that can be proven in such a simple way.

Your proof does not work. The first line does not subtract evenly from the second line as shown in the third line. There is an infinitessimal remainer, which you simply dropped.

As you say, the proof is trivial. 0.999... is an element of the set of points in the open unit segment. 1 is not an element, by definition.

from wikipedia:
1/9 = 0,11111...
9 x 1/9 = 0,9999999... = 9/9 = 1

If you accept that this is correct. However, it is not.

Assume 1/9 = 0.1111...

9x(1/9) = 9x0.1111... = 1 #

We have proof by contradiction that .1111... =/= 1/9.

J

 

[TheMeInTeam]

Your proof does not work. The first line does not subtract evenly from the second line as shown in the third line. There is an infinitessimal remainer, which you simply dropped.

As you say, the proof is trivial. 0.999... is an element of the set of points in the open unit segment. 1 is not an element, by definition.



If you accept that this is correct. However, it is not.

Assume 1/9 = 0.1111...

9x(1/9) = 9x0.1111... = 1 #

We have proof by contradiction that .1111... =/= 1/9.

J

It ultimately rests on how you define .3333... or .9999...


All of these representations are limited by the constraints of our definitions for them, including the usage of any numbers whatsoever. If you define 1/9 as equal to .1111... then by definition .9999... = 1.

 

[onejayhawk]

It ultimately rests on how you define .3333... or .9999...


All of these representations are limited by the constraints of our definitions for them, including the usage of any numbers whatsoever. If you define 1/9 as equal to .1111... then by definition .9999... = 1.

Exactly. If you define .333... = 1/3 then you must have conflicting definitions with a map of real numbers on a line. Godel's Theorem shows that such conflicts are inevitable. Instead we wave our hands and say, "Close enough." If you look at the foundational work of Newton and Riemann, they recognize this. Indeed the definition of limit is merely strict math speak for "close enough".

This does not mean that a line segment does not have a real end point, even if it is open. It is simply easier to say so than to say there is no rational endpoint.

J

 

[nc-1701]

Thanks

But wait, in a sphere that has an infinite periphery isn't any point its center?

(or is time non-linear and i already had that drink? )

Yep

This shouldn't even be surprising...

Is .99999.... = 1?

Of course not, but close enough for practical purposes. The proof is trivial. .9999... is on the unit segment open at 1, but 1 is not. QED

That said you will see proofs that they identical , silly as that is. You will even get the statement that an open segment has no endpoint. The correct statement is that an open segment has no rational endpoint. We have the concept of an infinitely small distinction, but handling it is difficult. Instead we approximate, which destroys the purpose.

J

Eh... It actually isn't, that's the rub.

This isn't a rabbit hole that I should be wandering down, but 0.999...=1 can be prove rigorously, without a limit hand wave that occurs in the commonly shown "proofs".

Spoiler :

Define S[n] = {the sum of 9*(1/10^i) for i=1...n} = 0.999...9 (n times).

Then for any epsilon > 0 consider 1 - epsilon, we can now find a N such that s[N]>=1-epsilon, but we already know that for all S[n]=<1 for all n trivially, but since epsilon was chosen arbitrarily we can conclude that S[infinity]=0.999...=1

Once the initial shock wears off and we remember that numbers are defined inherently and decimal writings of numbers are just ways to describe them it shouldn't be surprising that decimal writings of numbers are not unique.

Finally (0,1) does not have a rational endpoint or an irrational endpoint. As can be shown by a proof very similar to the one above, it's not like e or pi. Intervals with irrational endpoints are still very much closed. Open/Closed is a topological concept which cares very little about rather your number is rational/irrational.



To answer Kyriakos, I think that a notion of "real" non-mathematical infinity is not useful. Unboundedness is I think an absolutely sufficient notion of infinity for looking at the Universe.

 

[nc-1701]

Oh... And if we're going to disparage limit points we need to remember you can't construct the real line without them

 

[onejayhawk]

Define S[n] = {the sum of 9*(1/10^i) for i=1...n} = 0.999...9 (n times).

Then for any epsilon > 0 consider 1 - epsilon, we can now find a N such that s[N]>=1-epsilon, but we already know that for all S[n]=<1 for all n trivially, but since epsilon was chosen arbitrarily we can conclude that S[infinity]=0.999...=1

This is exactly what I was referring to when I stated, "strict math speak for 'close enough'." Anything involving an epsilon is essentially the same hand waving. I have already proven that 0.999... =/= 1. You merely demonstrate that there is no measurable distance. Since the real numbers are a measure space, this is significant, reflecting that the Rational numbers are dense on the real numbers

You make one good point. Saying an open segment does not have a rational endpoint is insufficient. There are an uncountable number of irrational numbers that are not the end point, for any epsilon around 1.

J

 

[Borachio]

^ Nice one. I take it, then, that calculus as a whole is just so much hooey. And that Zeno was right after all: no arrow has ever hit anything.

The equality 0.999... = 1 has long been accepted by mathematicians and is part of general mathematical education. Nonetheless, some students find it sufficiently counterintuitive that they question or reject it. Such skepticism is common enough that the difficulty of convincing them of the validity of this identity has been the subject of several studies in mathematics education.

 

[nc-1701]

This is exactly what I was referring to when I stated, "strict math speak for 'close enough'." Anything involving an epsilon is essentially the same hand waving. I have already proven that 0.999... =/= 1. You merely demonstrate that there is no measurable distance. Since the real numbers are a measure space, this is significant, reflecting that the Rational numbers are dense on the real numbers

Supposing we do take your position that limit points aren't meaningfully defined as numbers how do we go about constructing the real line?

Spoiler :

Finally I'll offer a second argument for the equality, this is less mathematically rigorous, but I think more intuitive.
1) Given two distinct real numbers we can always find a third real number that lies between them.

2) There is no number which lies between 0.999... and 1.

=> they are not distinct real numbers.

In the end it depends on whether you accept limit points as being well defined numbers, but if you don't it's impossible to construct the real line.

 

[onejayhawk]

^ Nice one. I take it, then, that calculus as a whole is just so much hooey. And that Zeno was right after all: no arrow has ever hit anything.

Good grief. Where did you get that? I am stating that calculus is the art of "close enough".

Supposing we do take your position that limit points aren't meaningfully defined as numbers how do we go about constructing the real line?

You don't. Like "1" and "+" it is undefined. If you prefer, you could call it axiomatic. Some things are irreducible.

BTW We have had all this discussion, but no one has ever challenged, or even addressed my proof. Instead you offer flawed proofs that depend on an epsilon. That is a you-first argument. I went first. Your turn.

J