A very good question is “what kinds of objects are these, anyway?” Since we have an infinite decimal they can’t be rational numbers.
This is just wrong. A rational number is a number that can be written as a fraction of two integers. Lots of infinite decimals are rational numbers. 1⁄3 = .3333333..., 1⁄9 = .1111111.… 1⁄7 = .142857142857142857… etc.
Ah, of course, my mistake. I was trying to hand-wave an argument that we should be looking at reals instead of rationals (which isn’t inherently true once you already know that 0.999...=1, but seems like it should be before you’ve determined that). I foolishly didn’t think twice about what I had written to see if it made sense.
I still think it’s true that “0.999...” compels you to look at the definition of real numbers, not rationals. Just need to figure out a plausible sounding justification for that.
I think the point is that you’re writing down “0.999...” and assuming that that must define a number at all. If you’re assuming that every decimal expression gives a number then you must be working with the reals.
I suppose you might be right for some people. For me, the fact that repeating infinite decimal expansions are rational is deeply deeply ingrained. Since your post is essentially how to square your feelings with what turns out to be mathematically true, you have a lot of room for disagreement as there is no contradiction in different people feeling different ways about the same facts.
For me the most fun thing about 0.9999.… is that 1⁄9 = .11111… and therefore 9x1/9 = 9x.111111..… and this last expression obviously = .99999...
You should also do a search on “right” in your post and edit it, you use “right” one time where you really need “write” I think it is “right down” instead of “write down” but I’ll let you do the looking.
Fixed the typo. Also changed the argument there entirely: I think that the easy reason to assume we’re talking about real numbers instead of rationals is just that that’s the default when doing math, not because 0.999… looks like a real number due to the decimal representation. Skips the problem entirely.
The OP states:
This is just wrong. A rational number is a number that can be written as a fraction of two integers. Lots of infinite decimals are rational numbers. 1⁄3 = .3333333..., 1⁄9 = .1111111.… 1⁄7 = .142857142857142857… etc.
Ah, of course, my mistake. I was trying to hand-wave an argument that we should be looking at reals instead of rationals (which isn’t inherently true once you already know that 0.999...=1, but seems like it should be before you’ve determined that). I foolishly didn’t think twice about what I had written to see if it made sense.
I still think it’s true that “0.999...” compels you to look at the definition of real numbers, not rationals. Just need to figure out a plausible sounding justification for that.
I think the point is that you’re writing down “0.999...” and assuming that that must define a number at all. If you’re assuming that every decimal expression gives a number then you must be working with the reals.
I suppose you might be right for some people. For me, the fact that repeating infinite decimal expansions are rational is deeply deeply ingrained. Since your post is essentially how to square your feelings with what turns out to be mathematically true, you have a lot of room for disagreement as there is no contradiction in different people feeling different ways about the same facts.
For me the most fun thing about 0.9999.… is that 1⁄9 = .11111… and therefore 9x1/9 = 9x.111111..… and this last expression obviously = .99999...
You should also do a search on “right” in your post and edit it, you use “right” one time where you really need “write” I think it is “right down” instead of “write down” but I’ll let you do the looking.
Fixed the typo. Also changed the argument there entirely: I think that the easy reason to assume we’re talking about real numbers instead of rationals is just that that’s the default when doing math, not because 0.999… looks like a real number due to the decimal representation. Skips the problem entirely.