Let's discuss Mathematics

The closure of a set is always closed, no?

 

The closure of a set is defined as the set joined with its limit points. Since you simply added the point 1 to the set [0,1) you assumed the conclusion. However, I did rephrase to make my objection clearer.

Once again, the fact that lim (0.9+0.09+0.009+...) = 1 is insufficient to show that 0.999... = 1.

J

 

I've seen this one before.

1 divided by 3 is 0.333....
Multiply by 3 and the answer is 0.999...

1 / 3 * 3 = 0.999... = 1

So as long as you believe that multiplication and division are inverse operations, 0.999... = 1.



The solution is obviously to abolish the use of decimals and switch entirely to fractions.
When asked what each decimal place means, the teacher always uses fractions like 1/10th, 1/100th, 1/1000th etc. to explain anyway.
Fractions are far superior to decimals!

https://abcnews.go.com/Technology/story?id=4194473&page=1

Pistols at dawn sirrah
Math should always be precise.

 

That's one approach. The problem is that 1/3 is not equal to 0.333... for the same reason that 1 is not equal to 0.999.... It works as a practical matter because Rational numbers are dense on the Real numbers.

J

 

1 is between [0,1) and (1,2] right?

If 0.999... is not equal to 1, and is in fact less than 1, then it must be part of [0,1)
By that logic, an infinitely small value must exist, and 1 - 0.999... is equal to the infinitely small value.
1 plus the infinitely small value must exist too, and would be written as 1.000...
This 1.000... with an infinite amount of zeros is part of (1,2]

I find it hard to imagine the 1.000... with infinite zeros is somehow greater than 1, but it must be if the infinitely small value exists since it was added to 1 to create 1.000...
In addition, the infinitely small value must exist, because how do we count from 0 to 1 without it?


I think I see your objection to 0.999... = 1
https://en.wikipedia.org/wiki/0.999...

It is something like this maybe?
https://en.wikipedia.org/wiki/Infinitesimal

They tried to make math work where 0.999... did not equal 1 with Non-Standard Analysis I think?
https://en.wikipedia.org/wiki/Non-standard_analysis

Here is some talk about it I found.
http://math.coe.uga.edu/tme/Issues/v21n2/5-21.2_Norton & Baldwin.pdf

Shouldn't we use the current math system with limits where 0.999... = 1 if the math is easier to understand and gives more useful results?
The other math system with infinitesimals where 0.999...≠ 1 requires learning the current math system first I think?

I know there should only be one truth for 0.999... = 1 or 0.999... ≠ 1, but non-math people should be spared too much pain!

 

You lost it here. For this to work the distance between [0,1) and 1 must be nonzero. It is not even defined.

You are correct that 0.9999... is in [0,1) and that 1 is not. This provides the contradiction of the assumption that 0.999... = 1.

J