0.999..., does or does not equal 1 thread

[ParadigmShifter]

I'll open this thread to try and deflect spam from the serious maths thread

By the way, 0.999.... is exactly 1

 

[Perfection]

DEPENDS ON THE BASE.

 

[Stylesjl]

It can be proven that 0.999 = 1 by this

1/3 = 0.3333333
0.33333*3 = 0.999999999 or 1

If 0.999 didn't equal 1 it would by something new in mathematics to find an exactly reversed equation that didn't come back to the original number I.e 3/3 = 1 and 1*3 = 3

 

[dutchfire]

We're talking about real numbers here, which technically are Cauchy sequences in Q. To prove the equality of two such sequences (let's call them a_n and b_n), we need to show that for every e>0, there is an N with for all i>N |a_n-b_n|<e. In our case a_n=1 for all n and b_n=1-10^-n (you could also define a_n and b_n in other equivalent ways). That makes |a_n-b_n|=10^-n.

Now let e>0, there exists a number 0<e'<e with e'=10^-K for some K in N. In our previous definition, this K is exactly the N we're looking for.

 

[lovett]

It can be proven that 0.999 = 1 by this

1/3 = 0.3333333
0.33333*3 = 0.999999999 or 1

This is a very superficial line of argument. If you deny 0.999... = 1 then you would reasonably deny that 0.333... = 1/3.

 

[wer]

Before answering the question, two things must be cleared up:

1) What is the set of numbers we are dealing with? (What construction of numbers do we use?)
2) What definition of decimal expansion do we use? (Notice we can define it just for &#8474;.)

We're talking about real numbers here&#8230;

Are we? The question is meaningful within &#8474;, that is even without defining &#8477;.

&#8230;which technically are Cauchy sequences in Q.

No, they are not. One Cauchy sequence could be technically one particular representation of a real number, not the real number itself. To define a real number using the Cauchy sequences, you have to define it as an equivalence class of Cauchy sequences.

 

[dutchfire]

Before answering the question, two things must be cleared up:

1) What is the set of numbers we are dealing with? (What construction of numbers do we use?)
2) What definition of decimal expansion do we use? (Notice we can define it just for &#8474;.)


Are we? The question is meaningful within &#8474;, that is even without defining &#8477;.

Well, for 0,99999... to be an element of Q, there should be two integers p,q with 0,99999....=p/q. Those integers don't exist (you'll only ever get a finite expansion)

No, they are not. One Cauchy sequence could be technically one particular representation of a real number, not the real number itself. To define a real number using the Cauchy sequences, you have to define it as an equivalence class of Cauchy sequences.

Quite right of course. You then have to prove that 0,999999... and 1 are equivalent Cauchy-sequences, which is exactly what I did in my post.

 

[wer]

Well, for 0,99999... to be an element of Q, there should be two integers p,q with 0,99999....=p/q.

If number 1 is rational and your definition of decimal expansion maps

to the number 1, then the number represented by

is indeed rational.

Those integers don't exist (you'll only ever get a finite expansion)

For every rational (real) number there exists at least one infinite decimal expansion.



For any non-negative rational number x you can define decimal expansion of the number x as any expression of the form

where

is a non-negative integer and

for i=1,&#8230;,n are elements of {0,1,&#8230;,9}, such as

Using this definition you can obtain the result directly

or, if you want it so, this way

Quite right of course. You then have to prove that 0,999999... and 1 are equivalent Cauchy-sequences, which is exactly what I did in my post.

No, decimal expansion is not the Cauchy sequence itself, but you can map every expansion to a Cauchy sequence.

Using the construction of real numbers from Cauchy sequences, you can prove the equivalency of decimal expansions

and

by checking the equivalency of their respective Cauchy sequences. The equvalency of

and 1 is, once again, matter of definition.

 

[holy king]

depends on whether it's apples or pears.

 

[Mirc]

This is a very superficial line of argument. If you deny 0.999... = 1 then you would reasonably deny that 0.333... = 1/3.

It might be superficial, but it's very intuitive - you don't need advanced math knowledge to understand it, and everyone I know who denies that 0.(9)=1 is not exactly a math genius. That line of thought is exactly what I used to convince myself of the truth of this statement when I was in 4th grade, and my father explained this to me. I remember being quite proud about "discovering" this argument myself.

DEPENDS ON THE BASE.

That was my first thought too, actually, upon reading the title of the thread.

 

[dutchfire]

No time to reply to everything.

No, decimal expansion is not the Cauchy sequence itself, but you can map every expansion to a Cauchy sequence.

A decimal expansion is a way to construct the Cauchy-sequence you want. Because of the nice properties of equivalent Cauchy-sequences, it suffices to prove something like this for a specific sequence that applies to the entire class of equivalent cauchysequences.

 

[Quackers]

cansomeone write out the explanation

 

[ParadigmShifter]

Post #8 pretty much nails it.

Or try this:

What is

1 - 0.9999.....

?

 

[Hehehe]

This one is propably the easiest to understand for a layperson

X=0,999...
10X=9,999...
10X-X=9,999...-0,999...
9X=9
X=1

 

[Quackers]

I got a B at GCSE mathematics yeah I know I'm a genius :wink:

but i still don't get that hehehhehe one and PS's explanation.

my anser to PS is .111111111...... makes sense, eh!


In line 5 of hehehe's 9 x 0.9999.... = 9 thats what i want to know why thats considered correct! cuz then it would mean o.999999..=1! but doesn't prove anything cuz i already know that i want to know y!

 

[madviking]

1/9 = .11111...
2/9 = .22222...
3/9 = .33333...
9/9 = .99999... = 1

 

[nonconformist]

Simplest explanation is:

1/3 represents something that cannot be physically represented on paper

0.333....is a way of noting the same thing

0.333333 (no mater how many .3's you put) are a visual approximation of thing
but is not the same as it is not an irrational number

so 3*0.3333.... is 1

but 0.333*1 is not

but we assume it is for conventions sake (unless you do a mathematics based vocation, in which case the concequences are disasterous)

 

[ParadigmShifter]

I got a B at GCSE mathematics yeah I know I'm a genius :wink:

but i still don't get that hehehhehe one and PS's explanation.

my anser to PS is .111111111...... makes sense, eh!


In line 5 of hehehe's 9 x 0.9999.... = 9 thats what i want to know why thats considered correct! cuz then it would mean o.999999..=1! but doesn't prove anything cuz i already know that i want to know y!

LOL at 0.1111....

0.000... would be a better answer.

9*0.999... is indeed 9 but that isn't part of the proof. (hey! eyes are handy when reading!)
The proof says 9.9...... - 0.9999...... = 9

 

[Eastwinn]

Simplest explanation is:

1/3 represents something that cannot be physically represented on paper

0.333....is a way of noting the same thing

0.333333 (no mater how many .3's you put) are a visual approximation of thing
but is not the same as it is not an irrational number

so 3*0.3333.... is 1

but 0.333*1 is not

but we assume it is for conventions sake (unless you do a mathematics based vocation, in which case the concequences are disasterous)

0.(3) is not an irrational number, it has a pattern.

 

[Huayna Capac357]

Of course it does. There's a wicked easy proof for it. A second-grader could figure it out.