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The difference between rational and irrational numbers is always stated as: rational numbers can be written as the ratio of two integers, and irrational numbers can't. However, why do mathematicians make a distinction between these two types of numbers? Why are integers special anyway, other than being historically significant?
Is there any property that sets rational or irrational numbers apart, other than the way they are written in our number system?
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.
One thing is in how you construct them. Starting from the natural numbers (and $0$) you construct the integers by saying that $\mathbb{Z}$ is the smallest set that contains the naturals and is a group under addition. Similarly, the rationals $\mathbb{Q}$ is the smallest set containing $\mathbb{Z}$ that forms a group under multiplication (when $0$ is taken out). The reals $\mathbb{R}$ can then be constructed by defining it to be the smallest set containing $\mathbb{Q}$ in which every bounded set has a least upper bound.
Why are integers special anyway, other than being historically significant?
From a technical perspective, it is good to know that we can perform exact computations on rationals. On irrational numbers, you have to approximate unless you restrict yourself to a suitable subset of irrational numbers, like e.g. an extension like $\mathbb Q[\sqrt2]$ or the algebraic numbers $\bar{\mathbb Q}$. But even when performing computations with such fields, many operations are internally formulated on rational numbers, which in turn are formulated on integers.
So in a sense, anything you can express with rational numbers is something which you can compute in a straight-forward way (although still using rationals based on arbitrary length integers, as opposed to floating point numbers) without loosing exactness. Anything involving reals has to fail there: you cannot even enter a single irrational real number unless you do so using a formula describing how to compute it from rational numbers.
Another example where they differ. Take any polynomial $a_nx^n + a_{n-1}x^{n-1} + \ldots +a_1x + a_0$ with integer coefficients. You can easily find all rational roots of that polynomial: Any rational root $\frac{p}{q}$ (with $p$, $q$ relativly prime integers) must satisfy: $p$ divides $a_0$ and $q$ divides $a_n$. So there are finitely many possibilities which you can check by hand.
It's not easy in general to find irrational roots.
From an algebraic perspective, if you believe in the "naturalness" of $\mathbb{R}$ or $\mathbb{C}$, then $\mathbb{Q}$ sits naturally inside them as the minimal subfield of characteristic zero. Topologically it is also worth noting that $\mathbb{Q}$ is dense in $\mathbb{R}$. These are just two properties that illustrate that $\mathbb{Q}$ is really a natural and interesting set.
The Dirichlet function, which holds different values for rationals and irrationals is an example of a function defined for each real x, but continuous nowhere. This wouldn't have been true were the set of rationals not dense. This function is a straightforward example of a non Riemann integrable function.
Moreover, the fact that the rationals are countable helps in proving the Lindelöf Covering theorem.
Also, the construction of the field of fractions of an integral domain is motivated from the construction of rationals from the integers.
