[Colonol Dekker]

Gah, brain failure. . . . .
 

[CP5670]

depends on who you ask mostly, most mathematicians say yes, I say no, my position is one mostly of semantics, as I believe the other position is as well.

It's not a matter of semantics if you agree that the ... means the 9s repeat forever. As portej05 said, it's a limit of increasingly closer approximations to 1, and limits can be understood without having to consider anything infinite.

There is actually a fringe area of math called nonstandard analysis, which uses a different set of axioms that allows for the existence of infinitely small numbers. However, analysis (calculus) and basically all other mainstream math is based on the classical Zermelo-Fraenkel framework that doesn't allow such quantities.

[peterv]

Not my field but if 0.9999... = 1 would that mean that  0.099999... = 0.1 , 0.19999... = 0.2 etc. ?

[General Battuta]

I believe that's true.

[peterv]

And what about 0.99999... + 0.11111... ?

[General Battuta]

You mean 1 + .11111111111...?

I should think it equals 1 and 1/9th.

[peterv]

Here's my point: 0.999 + 0.111 = 1.110 not equal to 1 + 0.111 = 1.111
It must be semantics.

[CP5670]

Those aren't the same numbers as in your last post.

Notice that every time you tack on an extra 9 and 1 to the first two numbers, the sum gets closer and closer to 10/9. It's the limit that matters.

[peterv]

Yes, i understand this. I just think that the source of the confusion is that we symbolize a limit as a decimal.

[iamzack]

X     = 0.999...       / multiply by 10
10X = 9.999...       / -X (and since X = 0.999...)
9X   = 9                /divide by 9
X     = 1
==========

-> QED

Of course this relies on assumption that you can agree that 9.999... - 0.999... = 9
So it's still semantics, but it demonstrates nicely that the obfuscation is in your head, and not in math.

Isn't that circular reasoning?

Of course 9.999... - 0.999... = 9, but if X = 0.999... and you subtract X from 10X, you wouldn't get 9X unless X = 1.

X     = 0.999...       / multiply by 10
10X = 9.999...       / -X (and since X = 0.999...)
9.000...1X   = 9        <-- and that's false unless 0.000...1 = 0

Which it would if 0.999... = 1.

I'm not arguing that 0.999... isn't the same as 1, I'm just arguing with your argument.

[CP5670]

It's correct, although he could have written it more clearly. The idea is that since there are an infinite number of 9s, shifting all of them towards the decimal point (which is what multiplying by 10 does) will not change the fractional part. So you get 10X=9+X, which gives X=1.

[iamzack]

But with 10X = 9 + X, you still have to subtract X from 10X which is still 0.000...1.

Of course, 0.000...1 is 0 since you can't really put a one after an infinite number of zeroes.

Which is why 0.999... = 1.

Because 0.999... + 0 = 1.

[CP5670]

But with 10X = 9 + X, you still have to subtract X from 10X which is still 0.000...1.

You're just solving it as a normal linear equation at that point, without making any further assumptions on what X is. You show that the number, whatever it is, satisfies 10X=9+X, and just from that it follows that X=1.

[Uchuujinsan]

10X-X = 10*X- 1*X (1 is neutral element for multiplication, hence the identity) = (10-1)*X (distributive law) = 9*X

I dont get it, what has that to do with 0,999...? Thats simple math that is universally true for ALL numbers, as it is basically part of the definition of (common) numbers itself.

[castor]

Whatever is true, I'm satisfied with the notion that nothing in/from this world can tell the difference between 1 and 0.999...

[Col. Fishguts]

X     = 0.999...       / multiply by 10
10X = 9.999...       / -X (and since X = 0.999...)
9X   = 9                /divide by 9
X     = 1
==========

-> QED

Of course this relies on assumption that you can agree that 9.999... - 0.999... = 9
So it's still semantics, but it demonstrates nicely that the obfuscation is in your head, and not in math.

Isn't that circular reasoning?

Of course 9.999... - 0.999... = 9, but if X = 0.999... and you subtract X from 10X, you wouldn't get 9X unless X = 1.

X     = 0.999...       / multiply by 10
10X = 9.999...       / -X (and since X = 0.999...)
9.000...1X   = 9        <-- and that's false unless 0.000...1 = 0

Which it would if 0.999... = 1.

I'm not arguing that 0.999... isn't the same as 1, I'm just arguing with your argument.

Ah OK, sorry for the confusion, I was doing two things in one step. All written out it goes like this:

X     = 0.999...                / multiply by 10
10X = 9.999...                / subtract X
9X   = 9.999... -X            / substitute the X on the right side with 0.999..., as it was defined on the first line
9X   = 9.999... - 0.999... / execute the subtraction on the right side
9X   = 9                          /divide by 9
X     = 1
==========

[iamzack]

10X-X = 10*X- 1*X (1 is neutral element for multiplication, hence the identity) = (10-1)*X (distributive law) = 9*X

(10 - 1) * X = (9) * X

This works.

X     = 0.999...                / multiply by 10
10X = 9.999...                / subtract X
9X   = 9.999... -X            / substitute the X on the right side with 0.999..., as it was defined on the first line
9X   = 9.999... - 0.999... / execute the subtraction on the right side
9X   = 9                          /divide by 9
X     = 1
==========

This is still circular reasoning.

[General Battuta]

It's easier to prove with limits.

[Col. Fishguts]

This is still circular reasoning.

I fail to see why this should be circular reasoning, care to elaborate ? (I'm not arguing, I'd really want to know if there's logical flaw in my solution)

"10X - X = 9X" is always true, not matter what X is, this is basic algebra.

X can be 43782487, and yet if you subtract one X from 10X, you still get 9X.

[iamzack]

X     = 0.999...                / multiply by 10
10X = 9.999...                / subtract X
9X   = 9.999... -X            / substitute the X on the right side with 0.999..., as it was defined on the first line
9X   = 9.999... - 0.999... / execute the subtraction on the right side
9X   = 9                          /divide by 9
X     = 1
==========

When you subtract 0.999... from 10X and get 9X, you are assuming that 0.999... equals 1. It's circular reasoning because you are using the assumption that 0.999... = 1 in order to prove that 0.999... = 1.