0.999....=1

[JWeinCom]

Teeqoz said:

JWeinCom said:

I don't believe I said that the system is flawed.  I said it is limited which I'll stand by.  That's not to say other systems are not limited (I chose to use base 3 instead of base 2 because 1/3 and 2/3 still repeat there), but this particular example is one that exists in base 10 and not in other systems.

Still, when you say "it's only because of a limitation of the system", it has that same effect on people, ie. it makes them think "well, it is correct because of the limitations, but it's not really correct".

Also, to simply think of it as a limitation as to how our decimal system "displays" fractions isn't right, because the concept of 0.999... reccurring doesn't require you to think of it as 9 times 0.111.... reccurring, or 9*1/9. The fractions themselves are just used in (some) proofs. However you can prove that 0.999...=1 without using fractions, so they aren't the root of the problem. 

The phenomenon that one value can be written in two ways (ie. that 0.25 = 0.24999... etc. for any possible value) exist for all bases, the only thing that changes is which digit is repeated (it's the highest digit of any given base, so in base 9, the digit repeated would be 8, in base 8, the digit repeated would be 7. In base 3, fractions like 1/2 are infinite sequences of numbers (1/2 in base 3 being 0.111....)

People can think whatever they want about it.  The fact is that the system we use obviously has limitations.  Any system of symbolic representation does.