More on 0.999~ vs 1: Comparing Numbers

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The discussion centers on the mathematical equivalence of 0.999... and 1, arguing that if two real numbers are different, a number must exist between them, which does not apply here. Participants explain that 0.999... represents the limit of an infinite series and is mathematically equal to 1, despite misconceptions about infinity and completeness. Some contributors express confusion over the concept of limits and the implications of defining numbers with infinite decimal representations. The conversation also touches on the idea of continuity and points on a graph, with some asserting that 1 is not a point in certain functions while 0.999... is. Ultimately, the thread emphasizes the importance of understanding limits and the definitions of numbers in mathematical discourse.
  • #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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