Every rational number has two decimal expansions, induced by the metric on the real numbers.

Every terminating decimal has two decimal expansions. But how is this induced by the metric on the real numbers (absolute value)?

Absolute value of a number would be the metric of that number with zero, the metric d(1,0.9...) = 0 which follows pretty soon after the axioms.

]]>Every rational number has two decimal expansions, induced by the metric on the real numbers.

Every terminating decimal has two decimal expansions. But how is this induced by the metric on the real numbers (absolute value)?

]]>Sometimes during our discussions we have been confusing "infinite repetition" with infinity. 0.333... has an infinite repetition, but it is simply 1/3 which is not "infinite"

]]>In this same way, I suppose that 0.0000000000...001=0 if there are an infinite number of zeros before the 1, although anything with an infinite number is mostly theoretical (with the possible exception of pi).

Couldn't be further from the truth. First off, many rational numbers have an infinite decimal expansion... it's just that this decimal expansion repeats. But for irrationals, there are tons of constants known to be irrational, some which are believed to be, which are used every day for practical purposes.

]]>I may be contradicting things I've said earlier, but I don't mind this thread. The posts are thoughtful, and we haven't descended into flaming, like in the one you linked to.

I think Identity's inequality is correct. It's obvious that no real number x exists such that 0.999... < x < 1, so if 0.999... < 1 then that means that the reals aren't dense. But they are.

Therefore, 0.999... = 1.

Please discuss in this thread, in which there is a fair amount of controversy.

For extra information on the subject, click here, here, here, here, here, here and/or here.

]]>you have to remember that the 9... indicates there are an 'infinite' amount of 9's, in otherwords, multiplying by 10, does not make it have 'one less' 9 at the end that you would normally assume when multiplying a number by 10, the number of 9's after it, is still infinate, so when you subtract, they completely cancel out, leaving you with an integer value.

normally you would write the above proof as:

x = 0.999...

10x = 9.999...

9x = 9.999... - 0.999... = 9

x = 9/9 = 1

0.999... = 1

if you find it hard to think about that, think about it this way, if they were *not* the same number, then there would be a number that goes inbetween them, like 1 != 2, because there is for example, the number 1.5 that fits between them, but that isn't the case with 0.999... and 1, there ** cannot ** exist a number that goes between them, so they ** must ** be the same

This doesnt make sense.

0.9999....*10=9.99999....

9.9999-0.99999=9

9/9=1

1=0.99999....