A Mathematical Paradox?

Originally posted by twhitehead
I am sure you are more knowledgeable than me in this area, so I won't dispute that. But do you have any references along those lines?

I did come across this:
http://en.wikipedia.org/wiki/Point_at_infinity

But its not quite the same thing.

They use it in very specific formal contexts:
The extended real number line
Non-standard analysis
In complex analysis you have the concept of riemann surfaces that really only work well with infinity
etc.

In an informal way mathematicians say "at infinity" a lot but what they usually mean to say is that we are considering the behavior of mathematical entities at larger and larger values...

Originally posted by Great King Rat
Yes. In statistical analysis there can be a very clear difference between 1.0 and 1.00.

There is no difference between 007 and 7.

http://www.purplemath.com/modules/rounding2.htm

Well without having looked at your link I'd imagine that what they are referring to is precision. So the trailing zero shows that the number is only known to that precision and the next (unknown) digit could be non-zero, so that in this case 1.0 means any number between 0.95 and 1.05. But the number 1 when treated as an exact entity on the real line is no different to 1.0 or 1.00···

I used to worry about this but stopped, basically through acclimatization, unfortunately there wasn't a single "aha" moment which I could relate to you. The only way I can express it is that if 0.9999··· were not 1.0 then there would be a number ŋ > 0 such that 0.999··· < 1 - ŋ < 1, since we cannot find such a number 0.999··· = 1.

Originally posted by DeepThought
Well without having looked at your link I'd imagine that what they are referring to is precision. So the trailing zero shows that the number is only known to that precision and the next (unknown) digit could be non-zero, so that in this case 1.0 means any number between 0.95 and 1.05. But the number 1 when treated as an exact entity on the real line is ...[text shortened]... a number ŋ > 0 such that 0.999··· < 1 - ŋ < 1, since we cannot find such a number 0.999··· = 1.

Well without having looked at your link I'd imagine that what they are referring to is precision.

That is correct. I just made a quick look at that link and straight away saw its title which says;

“Rounding and Significant Digits”.


(you will obviously already know all of what I am about to say below but I say it for the benefit of a few other readers here who know a bit less about maths notation than we do)

But when we write 1.0 out of context of precision i.e. when we are not trying to say anything about its precision in particular such as when we write it in this thread, then “1.0” really IS just another way of writing “1”.

In addition, “0.9999...”, not to be confused with just “0.9999 ”, simply does equal 1.
The “...” part sort of means “and so on for infinitum” or words of that effect. So it isn't just only merely a case that 0.9999 “tends towards” 1 as you keep adding more and more finite number of digit 9s to the end of 0.9999 (although it does tend to 1 as you do so) but, rather, you can rationally talk about it exactly equaling 1 if you had (hypothetically ) literally an infinite number of digit 9s after that 0.9999. So;

0.9999... = 1

is literally correct.

Originally posted by Great King Rat
Yes. In statistical analysis there can be a very clear difference between 1.0 and 1.00.

There is no difference between 007 and 7.

http://www.purplemath.com/modules/rounding2.htm

So if you substract one from the other (1.0 - 1.00) then you actually think that you get something other than zero?

Funny...

What says the calculator?

Originally posted by Great King Rat
Calm down Fabian, this is not a debate. I'll be the first to admit that I'm in the minority. I've already made this crystal clear. But I'm not going to say "yeah, I guess it's true" when every fiber in my body says it isn't so. I'm don't care that that makes me look stupid or whatever.

I'm calm. Are you?

Math is not about your opinion. It is about one truth deduced from another truth. So when you say there is another number between 0.999... and 1.000... then it is your opinion. Math says otherwise.

Originally posted by FabianFnas
So if you substract one from the other (1.0 - 1.00) then you actually think that you get something other than zero?

Funny...

What says the calculator?

If I'm not mistaken:

1.0 - 1.00 = 0.0
1.00 - 1.00 = 0.00
1 - 1 = 0
0.999... - 0.999... = 0.000...

Don't blame me, I didn't make the rules.

Originally posted by Great King Rat
If I'm not mistaken:

1.0 - 1.00 = 0.0
1.00 - 1.00 = 0.00
1 - 1 = 0
0.999... - 0.999... = 0.000...

Don't blame me, I didn't make the rules.

0 = 0.0 = 0.00 = 0.000...

Now I know that you've learned something new.

Does this mean that you understand the 'paradox' now?

Originally posted by FabianFnas
So if you substract one from the other (1.0 - 1.00) then you actually think that you get something other than zero?

Funny...

What says the calculator?

Fabian in the context data analysis it is different to say that something measures:
1 cm
1.0 cm

basically when you say that something measures 1 cm it means that you used a ruler that is scaled in centimeters and centimeters only. For all you know the real length of the body can be 1.9 cm
On the other hand if you say that something measures 1.0 cm than your ruler can go until the fist decimal place and in terms of accuracy we're talking about almost an 100% gain.

Of course that like someone said before me in mathematics of the real line 1=1.0 but in experimental physics for instance 1 is very different to 1.0.

Originally posted by adam warlock
Fabian in the context data analysis it is different to say that something measures:
1 cm
1.0 cm

basically when you say that something measures 1 cm it means that you used a ruler that is scaled in centimeters and centimeters only. For all you know the real length of the body can be 1.9 cm
On the other hand if you say that something measure ...[text shortened]... tics of the real line 1=1.0 but in experimental physics for instance 1 is very different to 1.0.

I've never mentioned anything about centimeters, nor measurements, nor experimental physics.

Originally posted by FabianFnas
0 = 0.0 = 0.00 = 0.000...

Now I know that you've learned something new.

Does this mean that you understand the 'paradox' now?

Yeah... um... apparently you feel the need to be arrogant and condescending to me.

Based on the posts I've read of you in the past I really see no reason why you would think you've earned the right to be arrogant, but to be fair I also don't really care.

You'll forgive for not paying any more attention to it.

Originally posted by FabianFnas
I've never mentioned anything about centimeters, nor measurements, nor experimental physics.

I know that Fabian, but that was the context that people used when saying that 1 is not the same as 1.0. Which is different than the initial context of the discussion.

But I won't sidetrack on this issue ay longer. Maybe this discussion deserves a thread of its own?