TITLE: Why are irrational numbers such a big deal?
QUESTION [35 upvotes]: 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!".

REPLY [2 votes]: Why is it such a big deal that some numbers are irrational?

It isn't, always.
It is, sometimes, because classification of stuff is what we do as mathematicians.
Split groups of stuff into subgroups of stuff, and talk about how different rule changes alter the groups around.  We love classification.

So why is it that, for example on π's wikipedia page, already in the third line it tells us that π is irrational?

If you look at any well-stocked Wikipedia article, there will be plenty of information that one may find irrelevant to their given task.  For example, "I need code samples, not the history of MVC."  Or, "I want Will Smith's height, not his net worth."
"π is irrational" is just a fact about π, and "facts about subjects" is exactly what the Wikipedia pages are for.

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!".

But I'd bet you didn't know Will Smith's net worth was 250 million without being told.  Or that the patent for the cotton gin was granted on March 14, 1794.
Yet other people may have known that stuff at the top of their heads.  So why was it included on the page?
Wikipedia doesn't alter the content of their pages based on the presumed knowledge of their visitors.  It just hosts a web site full of "facts about stuff", and the editors have discussions about each page and what type of content to include.
If you're really passionate about what types of content should be included on Wikipedia pages, you should join the editor's community.  You can start with π.