User:ConMan/Proof that 0.999... does not equal 1

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This is an attempt to provide those who dispute the accuracy of Proof that 0.999... equals 1 with a space in which to give their own careful, meticulous proofs to the contrary.

[edit] Definitions:

  • 0.999\ldots = 0.\dot{9}, in other words it is the recurring decimal represented by an infinite number of nines following the decimal point.

[edit] Rules:

  • By contributing to this page, you agree to abide by the rules.
  • I (Confusing Manifestation) may introduce additional rules or definitions as necessary. If you do not agree with a new definition, then give your alternative definition in your proof along with justification why it is a valid definition. If you do not agree with a rule, take it up on the talk page. I will attempt to make the rules fair, with the aim of letting you provide your arguments.
  • Your proof must hold up to the same scrutiny that you attempt to apply to the proofs on the original page. Thus, the following apply:
  • You cannot claim anything as "trivial" or "obvious" unless you are prepared to provide further explanation when questioned.
  • You must give careful definitions of anything not defined in the above Definitions section. You must be ready to justify these definitions as well.
  • Wikipedia policy such as no personal attacks applies as usual. This is a page for cogent proofs, not insults and jibes.

[edit] Algebra proof

(Moved from above to ease editing)

Another kind of proof adapts to any repeating decimal. When a fraction in decimal notation is multiplied by 10, the digits do not change but the decimal separator moves one place to the right. Thus 10 × 0.9999… equals 9.9999…, which is 9 more than the original number. To see this, consider that subtracting 0.9999... from 9.9999… can proceed digit by digit; the result is 9 − 9, which is 0, in each of the digits after the decimal separator. But trailing zeros do not change a number, so the difference is exactly 9. The final step uses algebra. Let the decimal number in question, 0.9999…, be called c. Then 10c − c = 9. This is the same as 9c = 9. Dividing both sides by 9 completes the proof: c = 1."

As promising as this theory may seem, I have spotted what I believe to be a flaw in the calculations. 10c - c should be equal to 9 following the evidence given above. However, since 10c is c multiplied by 10, there will be one less decimal place than c. This means that 10c - c is not equal to 9, but 8.999.......1 and so 9c is not equal to 9 and c is not equal to 1. —The preceding unsigned comment was added by 166.87.255.133 ([[User talk:|talk]] • contribs) 06:33, 1 August 2006 (UTC)

A good try, but it falls flat because there are an infinite number of decimal places and one less than an infinite number is still infinite. No matter how far you go, you can never reach the end of that decimal expansion to include the extra 1 that you think should be there. This is one of the simpler proofs that, by necessity, has some flaws, but they are flaws that can be fixed by application of more rigorous definitions and calculations than the simplicity of the proof allows. (In fact, if you tighten the proof, you end up with something resembling the limit / infinite series proofs.) Confusing Manifestation 10:54, 1 August 2006 (UTC)

Yeah, I agree with ConMan on that one, whether the infinite number is 8.9999999..1 or 9.99999.., the two sides of the algebraic equation remain equal, but c still doesn't equal 1. My algebra is very rusty, that said, the only problem I see is with the algebraic equation as provided by ConMan, where Step 3 is written as 9c = 9. This is incorrect. It should be 9 = 9. Step 1 is: (10 * c) - c = 9.999.. - c; So we process (10 * c), resulting in 9.999... and then removed c from both sides of the equation, leaving 9 = 9, not 9c = 9. Well, you can do this with any variable.
10x = 40.
10x - x = 40 - x.
36 = 36.
36/36 = 36/36.
x doesn't equal 1, x = 4.
Nor is c equal to 1, c = .999...
lets say that x = 0.999...

multiply both by 10 10x = 9.999...

take the first from the second 9x = 9

and divide by 9! x = 1

There fore, 1 = x = 0.999... If you want further proof that .999... != 1, try getting 1 to equal .999... hah = )
This was great fun though, thanks = )

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