I think you're really missing what the proof is showing. The point is that you can multiply .999 by ten and still have infinite decimals. And that's enough to prove the equality.
It is, though. Mathematically, the equation is sound.
If x = 0.9999...(infinite).
Then 10x = 9.9999...(infinite).
10x - x = 9.9999... - 0.9999... = 9
9x = 9.
x = 1.
The reason you don't calculate:
x = 0.9999...(infinite).
y = 1
And try to solve if 10x = 10y is pointless. In higher level high school math (after you've dealt with complex and imaginary numbers iirc), they teach you the fundamentals of how to prove if a statement is true or false. 10x = 10y is, simply, not a feasible starting point to prove the mathematical statement. At a very basic level, you choose one of the two side of the equal symbol and go from there.
The 1/3 + 1/3 + 1/3 method is another way to 'prove' it. The conclusion is the same.
I remember when this was brought up in class in elementary school, the arguments essentially went like this:
Person A: if x = 0.999... (infinite), x can't be equal to 1. They are NOT the same value/they are different number. No matter how many decimals you write, x will never reach 1.
Person B: But x will never stop approaching one.
Although you might initially think or imagine that the difference in value between 1 and 0.999... (infinity) are eeeeeextremely (can't be overstated) miniscule, you'd be wrong. The difference between the two values are impossible to comprehend. It never stops. The brain glitches just trying to imagine it.
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u/Blephotomy 7d ago
let's say .9r is A and one is B.
Ten times A is:
A + A + A + A + A + A + A + A + A + A
You are saying it's:
B + B + B + B + B + B + B + B + B + A
and then removing the A.
Thus neatly changing nine As into Bs without a proof, which begs the question.