More on 0.999~ vs 1: Comparing Numbers

  • Context: Undergrad 
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SUMMARY

The discussion centers on the mathematical equivalence of 0.999... and 1, with participants debating the implications of infinite decimal representations. Key arguments include the definition of limits in sequences, specifically that 0.999... is the limit of the series 0.9, 0.99, 0.999, and so forth, which converges to 1. Participants also explore the concept of completeness in numbers, asserting that 0.999... is not incomplete but rather a valid representation of the real number 1. The conversation highlights the confusion surrounding infinite sequences and their limits, ultimately affirming that 0.999... equals 1.

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  • #61
d_leet said:
It definitely does not show that 0.9999=1, .9999 = 9999/10000 which is not equal to one, but if we take the notation that .999... represents the infinite geometric series with .9 as the first term and 1/10 as the common ratio then this is equal to one .999... = 1.

I wanted to say 0.9999...
I always talked about 0.9999... but i forgot the ... because that's not the notation I learned.
I learned that if you want to write...let's say 0.6767 repeating you write
0.(67) in parenthesys like that. But you all use the ... instead so I tried to use yours but at times I forgot it.

0.9999 is a banality to convert...also an irrelevance.
However 0.(9) using the method I did with the long number as example would look like this:

(the whole number - the non-repeating number)/as many 9s as different repeating decimals followed by as many zeroes as the decimals not repeating.

So...

(09-0)/9 is just 9/9 or 1.
 

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