fossilesqueI just cited myself.
Valthornx=.9999… 10x=9.9999… Subtract x from both sides 9x=9 x=1
There it is, folks.
Divide 1 by 3: 1÷3=0.3333…
Multiply the result by 3 reverting the operation: 0.3333… x 3 = 0.9999… or just 1
0.9999… = 1
In this context, yes, because of the cancellation on the fractions when you recover.
1/3 x 3 = 1
I would say without the context, there is an infinitesimal difference. The approximation solution above essentially ignores the problem which is more of a functional flaw in base 10 than a real number theory issue
The context doesn’t make a difference
In base 10 --> 1/3 is 0.333…
In base 12 --> 1/3 is 3
But they’re both the same number.
Base 10 simply is not capable of displaying it in a concise format. We could say that this is a notation issue. No notation is perfect. Base 10 has some confusing implications
They’re different numbers. Base 10 isn’t perfect and can’t do everything just right, so you end up with irrational numbers that go on forever, sometimes.
chaonautThis seems to be conflating
0.333…3 with 0.333… One is infinitesimally close to 1/3, the other is a decimal representation of 1/3. Indeed, if 1-0.999… resulted in anything other than 0, that would necessarily be a number with more significant digits than 0.999… which would mean that the … failed to be an infinite repetition.