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[ZenGum]

Originally Posted by Happy Monkey

1/9 = .¯1
9/9 = .¯9
1 = .¯9

This has been messing with my head for some time.

(i'm having trouble with the notation. I'll write r to mean recurring last digit.)

okay, the argument is:

1/9 = .1r
2/9 = .2r
3/9 = .3r
etc
8/9 = .8r
so
9/9 = .9r
But since any x/x = 1
9/9 = 1
hence .9r = 1

Hmm. Troubling.

Intuitive reply.
No it bloody doesn't. See, this is 1. It starts with a 1. That over there is 0.9r. It starts with either a zero or a nine, depending, but either way, it is different from 1. Any fool can see that.

Think of the number line. Zero in the middle, negatives off to the left, positives off to the right. .9r would be immediately to the left of 1. We zoom in, closer and closer; .9r is always abutting 1 on 1's left, but never quite in the same place. Keep zooming, it is always there.

Okay, that's not very convincing if you weren't already convinced.

Here's a better counter argument.

0.9 < 1
0.99 < 1
0.999 < 1
0.9999 < 1

etc

0.999999999999999999999999999999999999999999999 < 1

Observation: adding more nines does not change the fact that it is less than one, no matter how many you add.

so,

0.9r < 1.

Hence, 0.9r =/= 1.

So, apparently, 0.9 both is and is not equal to one.

Man, I think we just accidentally the whole mathematics. Or my head.