system
Given the following post, respond as if you're commenting on Reddit:
human
How is .999999999 ad infinitum exactly equal to 1?
gpt
There are a number of ways you can demonstrate this.  Using fractions:

    1/3 = .333...
    1 = 1/3+1/3+1/3 = .333... + .333... + .333... = .999...

The method of extracting a fraction from a repeating decimal:

    x = .999...
    10x = 9.999...
    10x - x = 9.999... - .999...
    9x = 9
    x = 1

Using the sum of a geometric series:

`0.999...` means `9/10 + 9/100 + 9/1000...`.  For a geometric series of the form a, ar, ar^2,..., the sum of the series is a/(1 - r).  In our case, a is 9/10 and r is 1/10, so we have (9/10) / (1 - 1/10) = (9/10) / (9/10) = 1.

Edit:  One more interesting way, although this one gets a bit more technical.  In the real numbers, any 2 distinct numbers have numbers between them.  This isn't exactly a rigorous proof, but think about how you would define a number that falls between 1 and 0.999... (hint: you can't).