How to find how many irrational numbers are between $1$and $2$? I know that the  open interval $(1,2)$ is uncountable. So there are an infinite amount of real numbers in the interval. Does there exist a ratio of rational numbers to irrational numbers? 
 A: We can put the set of rational numbers and natural numbers in one-to-one correspondence, and hence show that the number of natural numbers is equal to the number of rational numbers.
Hence, we have $\mathbb{R}$ (Real numbers) an uncountable infinity. 
$\mathbb{Q}$ (Rational numbers) a countable infinity, and, $\mathbb{R}-\mathbb{Q}$ (Irrational numbers) an uncountable infinity.
Hence, the number of irrational numbers is strictly greater than irrational numbers.
Hence, the ratio of rational and irrational numbers is $0$.
A: The function
$$f(x):=\frac1{x-1},\quad x\in(1,2)$$
is a bijection from $(1,2)$ to $(1,\infty)$ with the property that map rational numbers to rational numbers and irrational numbers to irrational numbers.
If you knows that $(1,\infty)$ have the same number of rational and irrational than $\Bbb R$ you are done.
A: The ratio is zero, the set of rationals is denumerable (countable) where as irrationals are uncountable. For a proof of cardinality of $\mathbb Q$ see Produce an explicit bijection between rationals and naturals?. Then note that $((1,2)\cap\mathbb Q)\cup((1,2)\cap(\mathbb R - \mathbb Q))=(1,2)$ where $\mathbb R - \mathbb Q$ denotes the irrationals. Since $(1,2)\cap\mathbb Q$ is countable and $(1,2)$ is uncountable it must be the case that $(1,2)\cap(\mathbb R - \mathbb Q)$ is uncountable as the finite union of countable sets is countable.
A: In other words, you are asking what is the Lebesgue integral of the Dirichlet function (the characteristic function of the rationals):
$$
D(x) =1_{\mathbb Q}(x)=\left\{\begin{matrix} 
1 & \mbox{if } x\in\mathbb Q  \\ 
0 & \mbox{if } x \notin \mathbb Q \end{matrix}\right.
$$
Because the rationals are countable, the Lebesgue measure of the rational numbers is $0$. Therefore the Dirichlet function equals $0$ almost everywhere and the answer is $\int D(x)=0$.
