Why are irrational numbers such a big deal?

Why is it such a big deal that some numbers are irrational? It means they can't be represented as integer fractions. Cool. But almost all numbers satisfy that property. So why is it that, for example on $\pi$'s wikipedia page, already in the third line it tells us that $\pi$ is irrational?

Even if it had not told me that, I would've assumed it anyways. It's like if wikipedia had a page on a certain dog breed and told me "and you know what, this breed has a tail!".

Even if it had not told me that, I would've assumed it anyways.

Really? Would you also assume that $ A=\frac{1}{\pi^2} \sum_{n=1}^\infty \frac{1}{n^2} $ is irrational? And what about $$ B=\frac{\sum_{n=1}^\infty \frac{1}{n^4}}{\left(\sum_{n=1}^\infty \frac{1}{n^2}\right)^2} \ \ ? $$ $$ C=\sum_{n=1}^\infty \frac{1}{n(n+1)} \ \ ?$$ $$ D=\sum_{n=1}^\infty \frac{1}{n(n+2)} \ \ ?$$ $$ E=\frac{\sum_{n=1}^\infty \frac{1}{n^6}}{\left(\sum_{n=1}^\infty \frac{1}{n^3}\right)^2} \ \ ? $$ $$ F= \frac{1}{\sqrt{\pi}} \int_{-\infty}^\infty e^{-x^2}\,dx \ \ ? $$ The fact is, mathematics is full of results of the form "a given number defined in some special way is rational" - for example, the number $B$ defined above is equal to $2/5$, and $F$ is equal to 1. Usually we don't describe such results in those terms however, we call them "formulas" or "identities" and write them in a way that doesn't emphasize rationality, e.g. as $$ \sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6} $$ (a different way to state the theorem "$A$ is rational and is equal to $1/6$"). So if you assume that any number with a complicated definition that you come across is irrational just because in a statistical sense almost all real numbers are irrational, you would certainly be wrong a lot of the time. And other times you'll be lucky and get it right, and yet other times (as with the number $E$ defined above) it will be an open problem whether you're right or wrong.

To summarize, it is a big deal whether an important number like $\pi$ is rational or not because rationality/irrationality is both one of the most important attributes that real numbers possess, and an attribute that is surprisingly nontrivial both to guess and to prove. The reason you think it's so obvious that $\pi$ is irrational has more to do with psychological conditioning than with mathematical obviousness: it's just that if $\pi$ were rational then the way to write it as a ratio of integers would be one of the most famous facts in mathematics, which everyone would learn in grade school, and so from the fact that you never learned such a representation of $\pi$ in grade school or heard about it anywhere, it is easy for you to guess that $\pi$ cannot in fact be represented in such a way.

It is a big deal historically. When the mathematics of numbers was developed, it was natural to start with integers and progress to rationals. Rationals are so useful that became the end of what was thought about, and eventually was like a religion: Every number is rational.

So when the first quantity was discovered that was demanded by Euclidean geometry, working with only integer lengths to start with, yet could not be a rational number, those mathematicians underwent somewhat of an existential crisis.

Another point, more relevant to the wiki page, is that it is not always easy to know that a specific definable number is irrational. As an example, consider $$\zeta(3) = \sum_{n=1}^\infty \frac1{n^3}$$

Although nobody ever expected this to turn out to be rational, or even algebraic, until 1977 nobody had proven that it is irrational. In 1978, Apéry proved it is irrational and this is important information (we still don't know whether it is transcendental). So the fact that $\pi$ has been proven to be irrational and in fact has been proven to be transcendental is indeed an important fact about $\pi$.

"It can be of no practical use to know that $\pi$ is irrational, but if we can know, it surely would be intolerable not to know."

— E.C.Titchmarsh

But almost all numbers satisfy that property.

This is the key understatement. There's a lot of irrational numbers. To get a sense of how many there are, consider the task of writing down all of the real numbers between 0 and 1. There's a lot right? Infinitely many. No way you could write them all. How about all of the rational numbers between 0 and 1. Writing the numbers at a finite rate, it would take you an infinite amount of step to write down all of the rational numbers.

But note the difference in phrasing. For the rational numbers, it would take an infinite number of steps to write down all the numbers. For real numbers, I said it was impossible. I chose a different phrasing, and for good reason.

We have this concept of counting. It goes 0, 1, 2, 3, 4... and keeps going. These are called the natural numbers. The set of natural numbers is "countably infinite." If you kept adding 1 over and over, you could eventually construct every natural number. (this is basically the definition of a "countably infinite" set)

If we look at the real numbers, and pile in $\pi$ and $\sqrt 2$ and all of their irrational friends, we have more numbers than we had natural numbers. The set of all real numbers is "uncountably infinite." There's more real numbers than there are natural numbers.

Big deal right? There's more rational numbers than natural numbers too, right? Well... not exactly.

It actually turns out that the set of rational numbers is countably infinite. There is a formal way to map all of the rational numbers onto the natural numbers. It's typically done using a diagonalizing approach:

So, as strange as it may sound there are exactly the same number of natural numbers as rational numbers, but there are more real numbers than that. This has many interesting effects deeper into mathematics, because we can use mathematical induction to prove things as long as the set of values we're proving it over is no bigger than the natural numbers. Once we move into larger sets, like the set of irrational numbers, mathematical induction is no longer a valid tool in a proof.

And yes, the ability to use mathematical induction is a big deal =)