0.999...= 1 - Hackdot

Re:0.999...= 1 by nybble41 on Wednesday May 06, 2020 @02:15PM (#23094441)

No, they start at the same element (they're both 0.999… and thus start with 9/10) and have the same (infinite) "number" of elements. If you lined them up digit by digit there is never a point where one digit is a 9 and the other is a 0.

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Re:0.999...= 1 by nybble41 on Wednesday May 06, 2020 @06:47PM (#23097314)

Incidentally, this confusion is part of the reason why actual infinite sequences are written with a trailing ellipsis (9/10 + 9/10² + 9/10³ + …) and not a final term involving infinity. There is no final term—not even 9÷10^∞. So the correct way to write the separation of the leading term in your formula is:

    x = 9÷10 + 9÷10² + 9÷10³ + …
    x = 9÷10 + (9÷10 + 9÷10² + …) ÷ 10

Note that the number of elements in the sequence is the same no matter how many leading terms you write, so long as the pattern doesn't change. The notation { 1, 3, 5, 7, 9, … } and { 1, 3, 5, … } both refer to exactly the same set; the first notation is merely a bit more verbose. Similarly, the parenthesized portion of the second formula above is exactly equal to x, despite being written with two explicit leading terms rather than three.

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