Is there a "positive" definition for irrational numbers?
Solution 1:
An irrational number is a real number that can be expressed as an infinite simple continued fraction.
Solution 2:
A irrational number is a real number $x$ such that for any integer $q'$ there exists a rational number $p/q$ with $q > q'$ and
$$0 < \left|x - \frac pq \right| < \frac{1}{q^2}.$$
Solution 3:
If you want simple definition that's not based on Dirichlet's or Hurwitz's theorem, try this:
A real number $x$ is irrational if and only if for all positive integers $n$ there exists an integer $m$ such that $0\lt nx-m\lt1$.
The underlying theorem here is that for all real numbers $x$ and all positive integers $n$ there is a unique integer $m$ (namely, $m=\lfloor nx\rfloor$) such that $0\le nx-m\lt1$. If $0=nx-m$, then $x=m/n$ is rational, and vice versa.
Solution 4:
You can give one such characterization using continued fractions. An irrational number is a real number that can be expressed as an infinite continued fraction $$a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3 + {_\ddots} }}}$$ for a sequence $(a_n)_{n\in\mathbb{N}}$ where $a_0\in\mathbb{Z}$ and $a_n\in\mathbb{Z}_+$ for all $n>0$. (In fact, the sequence $(a_n)$ is then unique, and this correspondence is actually a homeomorphism between the space of irrational numbers and the space of such sequences in the product topology.)
Solution 5:
In addition to Dirichlet's approximation theorem as given by @Eric M. Schmidt, you have Hurwitz's theorem, saying that an irrational number $\alpha$ is a real number such that there exist infinitely many relatively prime integers $p$, $q$ such that $$\displaystyle \left|\alpha -{\frac {p}{q}}\right|<{\frac {1}{{\sqrt {5}}\,q^{2}}}.$$
Here, $\sqrt {5}$ is the best possible constant. It is attained for instance by the irrational $\phi = \frac{1+\sqrt{5}}{2}$ (the golden mean or golden ratio). The reason is explained in Why is there a $\sqrt {5}$ for instance in Hurwitz's Theorem? See for instance Introduction to number theory, M. Klazar.
As a side note, what happens to mere rationals: if $\alpha = \frac{r}{s}$, ($r$ and $s$ relatively prime):
$$ \left| \frac{r}{s} - \frac{p}{q}\right| = \left| \frac{rq-ps}{sq} \right| \geq \frac{\left| \frac{q}{s}\right|}{q^2}$$
and
$$\frac{\left| \frac{q}{s}\right|}{q^2}> \frac{k}{q^2}$$ except for a finite number of $q$ such that $|q|\leq |ks|$. Nice graphic versions with circles of Hurwitz's theorem (and some more on fractions) can be found in Ford, Fractions, 1938, American Mathematical Monthly. You can check that out, for instance, at Ford circles:
