Imagine I draw a number line, and I took two points. What's the distribution of rational and irrational numbers between them? If I put it in a diagram where I color rational with a color and irrational with another color what pattern will I get?
[Math] Rational vs Irrational distribution
irrational-numbersrational numbersreal numbers
Related Solutions
Here's one example of where the difference between rational numbers and irrational numbers matters. Consider a circle of circumference $1$ (in any units you choose), and suppose we have an ant (of infinitesimal size, of course) on the circle that moves forward by $f$ instantaneously once per second. Then the ant will return to its starting point if and only if $f$ is a rational number.
Maybe that was a little contrived. How about this instead? Consider an infinite square lattice with a chosen point $O$. Choose another point $P$ and draw the line segment $O P$. Pick an angle $\theta$ and draw a line $L$ starting from $O$ so that the angle between $L$ and $O P$ is $\theta$. Then, the line $L$ passes through a lattice point other than $O$ if and only if $\tan \theta$ is rational.
In general the difference between rational and irrational becomes most apparent when you have some kind of periodicity in space or time, as in the examples above.
Joke (but true): The difference between a rational number and an irrational number is irrational.
Serious answer: Your question already expressed it. A rational number can be written $\frac mn$ for some integer $m$ and some positive integer $n$. An irrational number is a real number that cannot be written like that.
To show that a number is rational, the most common approach by far is to find $m$ and $n$, and prove that the number in fact equals their ratio.
To show that a number is irrational is often a good deal harder, and is usually done using some sort of proof by contradiction. For example, it took a long time for mathematicians to even prove that $\pi$ is irrational. According to https://mathoverflow.net/questions/40145/irrationality-of-pie-pipi-and-epi2, no one even knows whether $\pi^{\pi^{\pi^\pi}}$ is an integer, let alone whether it is rational (but just about anyone would bet that it's irrational).
It turns out that in several senses, almost all real numbers are irrational, and in fact even transcendental (a nastier sort of beast). There are also various techniques available for manufacturing great gobs of irrational (and even transcendental) numbers, but most of the numbers people are actually interested in are either trivially rational, trivially algebraic (not transcendental), or mysterious—no one knows for certain whether they are rational or irrational.
Part of the reason for this is that while it's very easy to put together rational numbers to get more rational numbers, you can't really put together irrational numbers to get more irrational numbers in very many ways. For example, the sum or product of two rational numbers is always rational, but the sum or product of two irrational numbers may be rational.
Best Answer
For any continuous probability distribution on $\mathbb{R}$, the probability of picking a rational number is zero. In other words, if $X$ is a random variable which has a density, $$ P(\mathbb{Q}) = 0 \text{.} $$
The reason is that for continuous probability distributions, its probability measure $\mathbb{P}$ is absolutely continuous compared to the lebesgue measure $\lambda$, meaning that for all sets $X$ with $\lambda(X) = 0$ you also have $\mathbb{P}(X) = 0$. And you have $\lambda(\mathbb{Q}) = 0$, because $$ \lambda(\mathbb{Q}) = \lambda\left(\bigcup_{k\in\mathbb{N}} \{q_k\}\right) = \sum_{k=1}^\infty \lambda(\{q_k\}) = \sum_{k=1}^\infty 0 = 0 $$ for every enumeration $(q_k)_{k\in\mathbb{N}}$ of the rational numbers. The same works, for the same reason, for any countable subset of $\mathbb{R}$.