When I encountered this result in school for the first time, in the context of learning the algorithm for converting a repeating decimal into a fraction, I eventually reasoned “If 1 and 0.999… are different numbers, there ought to be a number between them, but there isn’t. So it must really be true that they’re the same.”
Of all the different explanations and interpretations people have been giving in this thread this is the most satisfying to my mathematically illiterate brain. It’s troublesome for me to grasp how 0.999… isn’t always just a bit smaller than 1 because my brain wants to think that even an infinitely tiny difference is still a difference. But when you put it like that—there’s nowhere between the two where you can draw a line between them—it seems to click in. 0.999… hugs 1 so tight that you can’t meaningfully separate them.
When I encountered this result in school for the first time, in the context of learning the algorithm for converting a repeating decimal into a fraction, I eventually reasoned “If 1 and 0.999… are different numbers, there ought to be a number between them, but there isn’t. So it must really be true that they’re the same.”
Of all the different explanations and interpretations people have been giving in this thread this is the most satisfying to my mathematically illiterate brain. It’s troublesome for me to grasp how 0.999… isn’t always just a bit smaller than 1 because my brain wants to think that even an infinitely tiny difference is still a difference. But when you put it like that—there’s nowhere between the two where you can draw a line between them—it seems to click in. 0.999… hugs 1 so tight that you can’t meaningfully separate them.