@livelaughlove21 It’s good to question “obvious” steps in a proof – great thinkers do that. To @Mariah‘s GAs I would add that so long as we agree that .999… , whatever it is, has some unique real number value, then we can name it x. Once named, we use the variable as terse shorthand for the unwieldy infinite string of digits, so x may be manipulated like any other real number – an elegant mathematical trick that works in other contexts too.
Also, there’s no ambiguity in adding:
9 + 0.999… = 9.999…
It rigidly follows the rules of the decimal system.
Likewise
10 * (.999…) = 9.999…
because multiplying by 10 is accomplished by simply shifting the decimal point, no matter what the digits.
Btw there’s a wikipedia article… on this very topic, affirming that (.999…) is another name for the number 1.
@josie …windshield wipers occasionally cycle in harmony with the turn signal, then go out of phase, and then after a while go back in phase…Nature’s eternal mystery
No mystery really. They go in & out of synch periodically, at a frequency equal to the difference between individual frequencies of wiper & turn signal, akin to beat frequency heard with two slightly different tones. Nature’s eternal harmony – at least while waiting to make a turn in the rain.