the ultimate question: does 0.9999... equal 1

[SetGaming]

Okay, so is this
0.999... (infinite 9s here) ...999
or is it
0.999... (infinite 9s here, with no literal end)

literal no end

[Ad Bot]

[Blockomaniac]

nerds

[Shift Kitty]

Here's an idea...

Let's get rid of the decimal system.

[Ipquarx]

In the set of real numbers, 0.9999... = 1.
In the set of hyperreals where you can have numbers infinitely close to but not equal to 1, 0.999... is not a valid representation of a number as far as I'm aware.

So yes, it's equal.
If you want some ideas of why, here are a couple proofs:
x = 0.9999...
10x = 9.9999...
10x - x = 9.0
9x = 9
x = 1

1/9 = 0.1111...
2/9 = 0.2222...
3/9 = 0.3333...
...
7/9 = 0.7777...
8/9 = 0.8888...
So 9/9 = 0.9999..., and 9/9 = 1.

[Texan101]

Easy solution.
.999... - 1 = What?
0.000....
Never has a 1 or higher, so it gets simplified to 0.