I understand the following lines, but want to make complete sense out of them.

Question

parthkohli · Answer

Suppose that $a$ and $b$ are incommensurable; divide $b$ into $m$ equal parts each equal to $\beta$, so that $b = m \beta$, where $m$ is a positive integer.  Also suppose $\beta$ is contained in $a$ more than $n$ times and less than $n+1$ times;

Then $\dfrac{a}{b} > \dfrac{n\beta}{m\beta} ~ {\rm and } < \dfrac{(n+1)\beta}{m\beta}$.

That is, $a/b$ lies between $n/m$ and $(n+1)/m$;

So that $a/b$ differs from $n/m$ by a quantity less than $1/m$.  And since we can choose $\beta$ (our unit of measurement) as small as we please, $m$ can be made as great as we please.  Hence $1/m$ can be made as small asa we please, and two integers $n$ and $m$ can be found whose ratio will express that of $a$ and $b$ to any required degree of accuracy.

parthkohli · Answer

I think I know what is happening in the first paragraph and those two lines, but the central point is still out of my reach.

parthkohli · Answer

Probably more about the language of the writers...

amistre64 · Answer

inbetween 2 rational numbers exists at least 1 irrational number

the_fizicx99 · Answer

What level of math is this, I somewhat understand what it's saying ._. it's what's bothering me :/

amistre64 · Answer

between   2.450 and 2.451 there exists at least 1 irrational number

amistre64 · Answer

we can make the interval as small as we want and there will still exists at least 1 irrational number in the middle

parthkohli · Answer

Are you using "irrational numbers" as an example or is that exactly what is said here?

amistre64 · Answer

incommensurable means not measureable ... cannot be defined as a rational number

amistre64 · Answer

or in this case it may mean not common factor, in simplest form

amistre64 · Answer

when a and b are not divisible any further, then the gcd(a,b) = 1, in other words they are relatively prime

parthkohli · Answer

Yes, hmm - so what is meant by "required degree of accuracy" when a/b is rational?

amistre64 · Answer

take 2 measures,  Circumference of a circle, and its diameter .... C/d is some value that can be approximated to any degree of accuracy

amistre64 · Answer

a and b are 2 measures that have no common factors ... and in a ratio they form some irrational value is what im getting from it

parthkohli · Answer

A rational number is defined as the ratio of two coprime numbers, then how can a/b be irrational? :P

amistre64 · Answer

this is geometry language .. and its related to sqrt(2) and pi ... numbers that are a ratio of some measureable parts, but have no rational expression

parthkohli · Answer

ah.

amistre64 · Answer

spose we are in binary ... 1/3 is an irrational number since it takes an infinity of 1/2s to create

parthkohli · Answer

oh!

parthkohli · Answer

thanks, amistre!  that makes sense

amistre64 · Answer

youre welcome :)  with any luck i may even be right lol

parthkohli · Answer

hahahahaha