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off topic: 0.999 repeating Equals 1, proof inside!

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Hon1nbo

Sun Dec 06 2009, 06:59PM

Registered Member #902 Joined: Sun Jul 15 2007, 08:17PM
Location: North Texas
Posts: 1040

here is a proof that I find accurate, yet possibly irritating some people's understanding of math

given the number 0.999 repeating, (all decimals shown below are repeating FYI)

0.999 / 3 = 0.333 = 1/3

1/3 * 3 = 1

by transitive property, 1 = 0.999 repeating

any comments? - I personally think this is just ridiculous (even though it is technically true), because for example in a function that has an asymptote you can get infinitely close but never actually get there.

here is the wikipedia page devoted to the thought: wikipedia page

Edit 1: come to think of it, is there any sign of misuse of operations in this proof? such as in the classic example of using x=1 and derivatives to make 1=0?

EDIT: I decided to post another method after thoughts about legitimacy of using 1/3 in such a proof, again all numbers with a decimal place are repeating

let x = 0.999

10x = 9.999

10x - x = 9.999 -0.999

9x = 9

x = 1

Steve Conner

Sun Dec 06 2009, 07:08PM

Registered Member #30 Joined: Fri Feb 03 2006, 10:52AM
Location: Glasgow, Scotland
Posts: 6706

Are you saying that there's a difference between "infinitely close" and "there?"

If you look, the Wikipedia article for Zeno's paradoxes links the one for 0.999...
You might also enjoy reading Cantor's theories about "different kinds of infinity".

Z28Fistergod

Sun Dec 06 2009, 07:48PM

Registered Member #2040 Joined: Fri Mar 20 2009, 10:13PM
Location: Fairfax VA
Posts: 180

DaJJHman wrote ...

here is a proof that I find accurate, yet possibly irritating some people's understanding of math

given the number 0.999 repeating, (all decimals shown below are repeating FYI)

0.999 / 3 = 0.333 = 1/3

1/3 * 3 = 1

by transitive property, 1 = 0.999 repeating

any comments? - I personally think this is just ridiculous (even though it is technically true), because for example in a function that has an asymptote you can get infinitely close but never actually get there

I don't think that is right. 0.3 repeating is only an approximation of 1/3, so I don't think you can use that in your proof. For that to work you would need to prove that 0.3 repeating equals exactly 1/3. That is where we run into infinitesimal differences. For example, the limit of 1/x as x becomes infinite is 0, but the value of 1/x never actually equals zero, no matter what value of x we choose. The value of 1/x merely approaches 0. Where did you find this proof?

StevenCaton

Sun Dec 06 2009, 07:56PM

Registered Member #1845 Joined: Fri Dec 05 2008, 05:38AM
Location: California
Posts: 211

I don't think that is right. 0.3 repeating is only an approximation of 1/3 so I don't think you can use that in your proof. Where did you find that?

It is not an approximation if it is *infinitely* long. They are the same thing

The first time I came across the proof I was somewhat baffled too.
I posted the "monty hall" problem a while back. It is another strange math brain teaser.

Z28Fistergod

Sun Dec 06 2009, 07:59PM

Registered Member #2040 Joined: Fri Mar 20 2009, 10:13PM
Location: Fairfax VA
Posts: 180

SteveC wrote ...

It is not an approximation if it is *infinitely* long. They are the same thing

The first time I came across the proof I was somewhat baffled too.
I posted the "monty hall" problem a while back. It is another strange math brain teaser.

Well if you want to use that logic 0.9 repeating is infinitely long and is the same as 1, so the proof is superfluous.

I disagree with both and maintain that it is only an approximation.

Prove that 1/3 = 0.3 repeating and your other proof will be valid.

Otherwise I think it is wrong.

Hon1nbo

Sun Dec 06 2009, 08:25PM

Registered Member #902 Joined: Sun Jul 15 2007, 08:17PM
Location: North Texas
Posts: 1040

Steve McConner wrote ...

Are you saying that there's a difference between "infinitely close" and "there?"

If you look, the Wikipedia article for Zeno's paradoxes links the one for 0.999...
You might also enjoy reading Cantor's theories about "different kinds of infinity".

I state this difference after the first few months of Calculus class in which limits were shoved into our heads and to never think of being infinitely close and there as the same thing

Z28Fistergod wrote ...

Prove that 1/3 = 0.3 repeating and your other proof will be valid.

Otherwise I think that is wrong.

I guess this is possibly a problem, but similarly there are lots of problems in math that are just assumed true: for example, the only reason what the Triangle Congruences (Side Angle Side, Angle Side Side, Side Side Side, etc) can not all be proved without assuming one is true is because when proving the Pythagorean theorem you cannot assume that there are triangles in existence that are congruent to the triangle at hand unless you start with them and make the proof unusable in most cases, so they assume that one triangle congruency is true to prove the pythagorean theorem to prove the rest.

if the above were not the case them all of the triangle congruences could be proven with about a page each with no loops or paradoxes formed from using one to prove the other

in every proof, something must be assumed as true from either obviousness or logic otherwise nothing would be able to be proven.

klugesmith

Sun Dec 06 2009, 08:32PM

Registered Member #2099 Joined: Wed Apr 29 2009, 12:22AM
Location: Los Altos, California
Posts: 1716

How about a numerical solution using decimal representation of real numbers?

For any x greater than zero, 0.999... + x will clearly be greater than 1.

Z28Fistergod

Sun Dec 06 2009, 08:50PM

Registered Member #2040 Joined: Fri Mar 20 2009, 10:13PM
Location: Fairfax VA
Posts: 180

DaJJHman wrote ...

...in every proof, something must be assumed as true from either obviousness or logic otherwise nothing would be able to be proven.

Ok, but like I said above, if we assume 0.3 repeating = 1/3 then we might as well assume that 0.9 repeating = 1. I don't see why either assumption is valid.

Hon1nbo

Sun Dec 06 2009, 10:22PM

Registered Member #902 Joined: Sun Jul 15 2007, 08:17PM
Location: North Texas
Posts: 1040

I decided to put in the first post another method not involving fractions or any other infinite decimal besides 0.999 and what can be manipulated from it, and also I included the link to the wikipedia page

StevenCaton

Sun Dec 06 2009, 11:33PM

Registered Member #1845 Joined: Fri Dec 05 2008, 05:38AM
Location: California
Posts: 211

There shouldn't be any debate here about the fact that .9999 (repeating) equals 1. It is a well established fact of mathematics.

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