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Prove or disprove 0.0000.....001 equals 0


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  On 1/21/2017 at 4:05 PM, bonanova said:
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One of my students used this as their proof, I am curious about what you think of it bonanova,

Assume that 0.000....1 exists then 1 - 0.000....1 = 0.999....

Since we can show 0.99999 = 1  and 0.000...1 + 0.999... = 1 then 0.000...1 must equal zero as 0+1=1.

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Well the issue is what 0,000...1 means. That question must be answered before asking whether it has a value, and if so what the value is.

The expression contains a string that is both infinite and terminating. A terminating string can be counted. Each digit occupies a numbered place. What place does the 1 occupy? This expression implies that a finite string that terminates with a 1 can contain an infinite sub-string of zeros. That fact is equivalent to saying that any explanation of the meaning of the expression must contain a contradiction.

We can construct the sequence

  • 0.1
  • 0.01
  • 0.001
  • 0.0001
  • 0.00001
  • ...

and ask whether the sequence converges. Yes. It easily can be proved to converge to zero. But the proof never mentions 0.000...1. So it does not apply to the OP.

Regarding your student's proof I question the assertion 1 - 0.000....1 = 0.999.... Why? Because it equates an infinite string to a terminating string.

I would say 1 - 0.000....1   =   0.999... - 0.000...1   =   0.999...8 whose meaning is equally problematical. Saying that it evaluates to unity carries the same difficulty as the original question.

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