I was able to make sense of this argument through the (rather unsophisticated) reasoning that 0.333...= 1⁄3, and multiplying both sides by three gives 0.999...=1.
(I’m not sure if this actually adds anything, but it was how I made myself believe the validity of the argument)
Thank you for the reply!
I get that 1/3= 0.333… from just dividing 1 by 3, but I do have a lack of understanding for what it means to multiply an infinite decimal by some integer. I appreciate the explanation with the decimals to the left!
I was able to make sense of this argument through the (rather unsophisticated) reasoning that 0.333...= 1⁄3, and multiplying both sides by three gives 0.999...=1. (I’m not sure if this actually adds anything, but it was how I made myself believe the validity of the argument)
But how do you know that 0.333… = 1/3? (And that multiplying an infinite decimal by 3 corresponds to multiplying each of its digits by 3?)
In the spirit of my comment, consider the analogous argument for infinite decimals to the left. Let x = …333. Then
3x = …999 = −1 (we established this earlier)
so x = −1/3. Are you satisfied with that?
Thank you for the reply! I get that 1/3= 0.333… from just dividing 1 by 3, but I do have a lack of understanding for what it means to multiply an infinite decimal by some integer. I appreciate the explanation with the decimals to the left!