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Is 1 = 0.9999......

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Originally posted by: DrPizza
If you want to really stress your brain out, how about these:

1. Inside a 1x1 grid, does there exist a curve that is completely inside the grid, has 2 endpoints, yet has infinite length?
2. Do smooth curves exists inside such a square region, such that the curve will pass through *every* point in that region?
3. If the curve in (2) has infinite lenth, does it have to pass through all the points that the curve in (1) passes through?

Answer to the first two questions is yes, answer to the 3rd question is no.
Click to expand...

interestings

pics, explaination?
 
The contents of this thread are way above my head !!!!!!!111!!!!0.999999!!!!!1!!!!!111! :Q

That's my contribution to this thread.
You know this thread is going to be archived and sent on one of those probes into space. Those aliens are gonna think we're nutzo 😛
 
Originally posted by: Stefan
1/3 * 3/3 does equal 1

When something takes the form of 1/3 it is defined as a finite number, therefore you can make the calculation.

In the form of a decimal, .333... is infinite, and as I said above, if you used the definition of infinity, you couldn't perform the calculation of .333... * 3 because in order to make the calculation you need a finite number.

.333... *3 == .999... !=1
Click to expand...

First of all, your usage of "finite" and "infinite" is hardly appropriate.

Secondly, 1/3 = .33333333.... (repeating forever)
Anyone beyond 4th grade should know that.
so,
.3333...(repeating forever) * 3 = 1/3 * 3 (substitution property of equality - if you're beyond high school algebra)

Substitution property of algebra, (just in case),
if a = b, then f(a) = f(b). In this case, f(x) = x*3

Also, .3333.... is NOT an "infinite number" (I think you mean a number with an infinite number of decimal places?? But, all numbers have an infinite number of decimal places. ex 2.00000.... )
 
Originally posted by: DoNotDisturb
that is correct.... theoretically, 0.9999 would equal it, but thats assuming that infinity ends, remember -> means "approaches"... does it reach infinity? you tell me.
Click to expand...

Yet again, the same error in reasoning. You see the word "approaches" and apply it in the wrong place. This error has been made in about 50 of the 1900 posts.

Sum = limit
limit = 1 as a dummy variable approaches infinity. It's not the limit that approaches infinity, the limit *IS* 1, as n (or any other variable used) approaches infinity.

Incidentally, I was going to toss out another one to make it more interesting here.
Take the line segment from 0 to 1 (including the points at 0 and 1)
Then, erase the middle 1/3 of the segment, leaving 2 segments each 1/3 in length.
erase the middle 1/3 of each of those segments, leaving 4 segments each 1/9 in length.
repeat forever.

The sum of the length of the "stuff" that you erased will equal 1. However, there are an infinite number of points left. Furthermore, there is an exact 1 to 1 correspondence to the points in the segment from 0 to 1 (same cardinality for the math people... "same size of infinite number of points" for everyone else.")

(I was just bored and reading an old text I have on fractals - chapter on Cantor dusts and Peano space filling curves.)
 
Originally posted by: DrPizza
Originally posted by: DoNotDisturb
that is correct.... theoretically, 0.9999 would equal it, but thats assuming that infinity ends, remember -> means "approaches"... does it reach infinity? you tell me.
Click to expand...

Yet again, the same error in reasoning. You see the word "approaches" and apply it in the wrong place. This error has been made in about 50 of the 1900 posts.

Sum = limit
limit = 1 as a dummy variable approaches infinity. It's not the limit that approaches infinity, the limit *IS* 1, as n (or any other variable used) approaches infinity.

Incidentally, I was going to toss out another one to make it more interesting here.
Take the line segment from 0 to 1 (including the points at 0 and 1)
Then, erase the middle 1/3 of the segment, leaving 2 segments each 1/3 in length.
erase the middle 1/3 of each of those segments, leaving 4 segments each 1/9 in length.
repeat forever.

The sum of the length of the "stuff" that you erased will equal 1. However, there are an infinite number of points left. Furthermore, there is an exact 1 to 1 correspondence to the points in the segment from 0 to 1 (same cardinality for the math people... "same size of infinite number of points" for everyone else.")

(I was just bored and reading an old text I have on fractals - chapter on Cantor dusts and Peano space filling curves.)
Click to expand...


The home work set that really drove home the fact that 1 = .999... involved starting with the above mentioned Cantor set and deriving the binary number system. In this process it was made very clear that .999... was a representation of 1. Unfortunately I have been unable to find that paper. ...

Come on, give me a break....Its been 20 yrs!
 
Ross,

Once you change to binary numbering sets then you change the rules. Thats like saying 1/3 = .333.... when it really does not translate well in base 10. If the base numbering systems possible are limitless then we should be able to use infinity to define the limit of numbering sets. Using an infinite base number set then .999... should be easy enough to define.
 
Originally posted by: MadRat
Ross,

Once you change to binary numbering sets then you change the rules. Thats like saying 1/3 = .333.... when it really does not translate well in base 10. If the base numbering systems possible are limitless then we should be able to use infinity to define the limit of numbering sets. Using an infinite base number set then .999... should be easy enough to define.
Click to expand...

Ordinarly I would post something here about how Madrat is still using a different definition of infinity then the rest of us ("infinite base number" makes NO sense with the definition everyone else is uing) but I just realized that his avatar is supposed to be a snake and not a weird shaped head with a really big eye. And I've been on anandtech for how long . . . .

-Chu
 
Originally posted by: Chu
Originally posted by: MadRat
Ross,

Once you change to binary numbering sets then you change the rules. Thats like saying 1/3 = .333.... when it really does not translate well in base 10. If the base numbering systems possible are limitless then we should be able to use infinity to define the limit of numbering sets. Using an infinite base number set then .999... should be easy enough to define.
Click to expand...

Ordinarly I would post something here about how Madrat is still using a different definition of infinity then the rest of us ("infinite base number" makes NO sense with the definition everyone else is uing) but I just realized that his avatar is supposed to be a snake and not a weird shaped head with a really big eye. And I've been on anandtech for how long . . . .

-Chu
Click to expand...

Hmmm, I guess an infinite based number system would be one where you just incremented the one digit forever. Something like 0, 1, 2, 3,... 9, a, b, c, d, e, f, g, .... x, y, z, A, B, C, ... X, Y, Z, !, @, #, $, %... obviously I'm gonna run out of ascii characters, but I guess if you could go on forever with one digit you could make an infinite base number system.
 
The 0.999999... is an incorrect notation, based on what I know about the infinite cyclic decimals number (or whatever, as IANAA, I am not an american). One can write 0.(3) for 0.333333... or 1/3, and so on. But 0.(9) or 0.9999... is just incorrect notation...

Calin
 
Originally posted by: Calin
The 0.999999... is an incorrect notation, based on what I know about the infinite cyclic decimals number (or whatever, as IANAA, I am not an American). One can write 0.(3) for 0.333333... or 1/3, and so on. But 0.(9) or 0.9999... is just incorrect notation...

Calin
Click to expand...

How so?

I prefer .999...

To the best of my knowledge that is accepted notation. You are first I am aware of to make this claim. What do you consider correct notation?
 
Originally posted by: RossGr
Originally posted by: Calin
The 0.999999... is an incorrect notation, based on what I know about the infinite cyclic decimals number (or whatever, as IANAA, I am not an American). One can write 0.(3) for 0.333333... or 1/3, and so on. But 0.(9) or 0.9999... is just incorrect notation...

Calin
Click to expand...

How so?

I prefer .999...

To the best of my knowledge that is accepted notation. You are first I am aware of to make this claim. What do you consider correct notation?
Click to expand...

Where is your reference. As with any good scholar, you should always put references (author, book, page numbers, etc)... I've never come across anyone not accepting 0.9999.... as correct notion and correct understanding besides you.
 
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