problem 2 and some wikipedia junk

[zenk...@gmail.com]

9 is a cool number, this is why i did question 1.3 for fun too...

anyways, #2 in the book says to prove that .999... is irrational. is
it as easy as supposing that it is and that .999...=a/b => b*.999... =
a => a is infinitely close to b but not quite b, so they can't both be
integers? i feel like i'mmissing something.

also, i'm curious what you all think of this article that i saw on a
horrible forum today (and thought it was a joke, but it's apparently
true (but not to erdos)): http://en.wikipedia.org/wiki/.999...

[Kenneth A. Ribet]

Not true that 0.999.... is irrational. It's 1. The problem states
that you will not encounter the sequence 99999... in converting a
fraction into a decimal by long division.

Ken R

[zenk...@gmail.com]

are there other repeating decimals that don't have fractional
representations? this might be high school talking but i thought that
they all did...

[zenkalia]

of course, but do they all have fractional representations other than .999...?

On 10/25/06, Christian Claiborn < brai...@gmail.com> wrote:

No, any repeating decimal is rational.  Convince yourself of this by
separating out the repeating parts and then using results about
geometric series.

[Kenneth A. Ribet]

It might be fruitful to think of the situation in terms of two
functions. Let Q be the set of rational numbers. Let D be the set
of decimal numbers that are eventually periodic. There's a function
g:D -> Q that takes each decimal to the rational number that it
represents. You calculate it by the formula for the sum of a
geometric series. One has g(1) = g(0.999....) = 1. There's a
function f:Q -> D that is defined by long division. If you have a
fraction a/b, you can divide b into a and get a decimal. It's true
that g(f(a/b)) = a/b. In other words, the value of the decimal that
a/b gives rise to is the fraction a/b. However, it is not true that f
(g(d)) = d for all d. For example, f(g(0.999...)) = 1.00000....

The function g is onto but not 1-1. The function f is 1-1 but not onto.

Hope this helps,
Ken R