Neat, overlines.

Okay, now we're dividing into an infinite number so this is a lost cause. --Flying Mullet

No, you can divide an infinite series by a number, or multiply it by a number, etc. I did so in one of my proofs.

(1/10 + 1/100 + 1/1000 +...) / 10 = (1/100 + 1/1000 + 1/10000+...)


1) In mathematical theory (which I believe is what were are dealing with), simplest terms are always important. You don't say 2A=2(pi)r^2. You don't say .77e=.77mc^2. You don't say a^2 + b^2 + 10=c^2 +10. You can, but you don't. Simplest terms are always used in mathematical theory. --SeedRankLou

I don't see any reason why you have to simplify everything. I've never heard this. Could you explain why this is true?

I'm fairly certain that the definition is specifying a ratio of two integers that is not also an integer itself, or the definition could also go: any real number that cannot be expressed as an integer (making fractions irrational number).

The definition of rational number is a ratio of two integers, where the denominator isn't 0. That's the only restriction. 1/1 is a rational number. 10/2 is a rational number. It doesn't matter if the rational number can be written in another form as an integer. Just a different way of representing the same value.

The list of rational numbers is like this (pretty sure):

...

... -1/2  -1/1  1/1  1/2  1/3  1/4  1/5 ...

... -2/2  -2/1  2/1  2/2  2/3  2/4  2/5 ...

... -3/2  -3/1  3/1  3/2  3/3  3/4   3/5 ...

...


If you want a set where no element is repeated, you can go through and eliminate all the repeats. (2/1, 3/1 for example)

1/1 = 2/1 = 3/1 = 1 = 1.0 = 9/9 = .9, all just different ways of representing the same rational number.

Is .999 repeating a rational number? Well, a number is rational if it can be written as A/B (A over B): .3 = 3/10 and .55555..... = 5/9, so these are both rational numbers. Now look at .99999999..... which is equal to 9/9 = 1. We have just written down 1 and .9999999 in the form A/B where A and B are both 9, so 1 and .9999999 are both rational numbers. In fact all repeating decimals like .575757575757... , all integers like 46, and all finite decimals like .472 are rational. -- http://mathforum.org/dr.math/faq/faq.integers.html