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!
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!