Prove that 0.999.... = 1 is a contradiction

[mustang19]

Homework Statement

Prove that 0.999... = 1 is a contradiction

Homework Equations

None

The Attempt at a Solution

Assumptions :

A. Let g(x) be a function of the form
G(1) = 0.999^2
G(2) = 0.9999^2

Etc for all integer x > 0

B. 0.999... = 1

Argument:

1. G(1) ends in decimal 1 or 8
2. G(2) ends in decimal 1 or 8
3. By linear relation all g(n) ends in decimal 1 or 8
4. G(infinity) = 0.999... Squared
5. G(infinity) = 1 ^2 by assumption (b)
6. By (3), G(infinity) ends in decimal 1 or 8
7. 1^2 ends in a decimal value of zero
8. By (6), (7), and assumption (b), contradiction
9. QED 0.999... = 1 is a contradiction.

 

[Mark44]

Homework Statement

Prove that 0.999... = 1 is a contradiction

I'm not sure what you're trying to do here. You can't prove that this is a contradiction, because it's true. Are you supposed to prove that 0.999... = 1 using a proof by contradiction?

Homework Equations

None

The Attempt at a Solution

Assumptions :

A. Let g(x) be a function of the form
G(1) = 0.999^2
G(2) = 0.9999^2

Etc for all integer x > 0

B. 0.999... = 1

Argument:

1. G(1) ends in decimal 1 or 8

The right-most digit of G(1) is 1, not 8

2. G(2) ends in decimal 1 or 8

Same here. The right-most digit of G(2) is 1, not 8.

3. By linear relation all g(n) ends in decimal 1 or 8

What do you mean, "by linear relation"?

4. G(infinity) = 0.999... Squared

This makes no sense. Infinity is not a number, the expression G(∞) is meaningless.

5. G(infinity) = 1 ^2 by assumption (b)

You can't use assumption B as part of your proof, since that's precisely what you're trying to prove.

6. By (3), G(infinity) ends in decimal 1 or 8
7. 1^2 ends in a decimal value of zero
8. By (6), (7), and assumption (b), contradiction
9. QED 0.999... = 1 is a contradiction.

 

[SammyS]

...
What is assumption b?.

Mark, I suppose that OP used a word-processor which was doing Auto-capitalization.

Likely that assumption b is:

B. 0.999... = 1

.

 

[Mark44]

Mark, I suppose that OP used a word-processor which was doing Auto-capitalization.

Yes I noticed that just before you posted. I also noted that he said he has to "Prove that 0.999... = 1 is a contradiction."

He's going to have a hard time doing that...

 

[SammyS]

As Mark indicated above, you will have a hard time completing your proof.

The Attempt at a Solution

Assumptions :

A. Let g(x) be a function of the form
G(1) = 0.999^2
G(2) = 0.9999^2

Etc for all integer x > 0

Firstly:
You should be more consistent. Is the function g(x), or is it G(x) .

1. G(1) ends in decimal 1 or 8
2. G(2) ends in decimal 1 or 8

(I'll use g.)

There is no mystery here:

g(1) = 0.999² = 0.998001
g(2) = 0.9999² = 0.99980001​

.
This does not get you anywhere, but ...
the square of a 0 followed by a decimal point followed by N 9s gives

a 0 followed by
a decimal point followed by
(N− 1) 9s followed by
an 8 followed by
(N− 1) 0s followed by
a 1 .​

 

[mustang19]

Okay so I don't see any criticism of the proof. Closing question.

 

[SammyS]

Okay so I don't see any criticism of the proof. Closing question.

... other than @Mark44 's comment that G(infinity) makes no sense?

Also, he asks you to define what you mean when you say "by linear relation".

Let me ask this:

Can you give me a positive number that's less than (1 − 0.9999... ) ?​

.

 

[Mark44]

Okay so I don't see any criticism of the proof.

See post #2.
What you have isn't a proof, nor is it clear what you're trying to prove.

 

[FactChecker]

Okay so I don't see any criticism of the proof. Closing question.

Ok, then to be more clear -- The "proof" is fatally flawed and is trying to prove something that is false. Almost every step of it is wrong.

 

[mustang19]

... other than @Mark44 's comment that G(infinity) makes no sense?

Also, he asks you to define what you mean when you say "by linear relation".

Let me ask this:

Can you give me a positive number that's less than (1 − 0.9999... ) ?​

.

That's another contradiction

By linear relation

Is there a value of g(n) where that does not hold?

 

[SammyS]

That's another contradiction

Is there a value of g(n) where that does not hold?

I suspect that your difficulty with accepting the fact that 0.999... = 1 is that you have a lack of understanding of the concept of infinity.

 

[mustang19]

I suspect that your difficulty with accepting the fact that 0.999... = 1 is that you have a lack of understanding of the concept of infinity.

Okay

 

[FactChecker]

Using terms like "linear" incorrectly does not convince me of anything. So here is a simple and direct way that you can convince me -- answer this question:

If the numbers 1 and 0.9999... are not equal, you must be able to give me a number in between the two. What is that number?

 

[mustang19]

Using terms like "linear" incorrectly does not convince me of anything. So here is a simple and direct way that you can convince me -- answer this question:

If the numbers 1 and 0.9999... are not equal, you must be able to give me a number in between the two. What is that number?

That's a contradiction

 

[berkeman]

Thread closed for Moderation...

 

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