What are some curiosities or interesting things that every person interested in mathematics should know?

The story of 1729--the Ramanujan-Hardy number--is a good one.

More real numbers than natural numbers is a good one.
Dirac's delta function basically started as an abuse of notation by physicists but then mathematicians figured out what the hell it was
Every compact Riemann surface is, in fact, a complex algebraic curve.
The 5 axioms of Euclid and how breaking the parallel postulate leads to hyperbolic and elliptic geometry.
I would also suggest, if you have a chance, take a math history course. Especially if it's handled by your math dept.

My math history class was glorious. A beautiful glimpse into this living art. I think Hofstadter called mathematics "the international, trans-generational, metamind" (though I'm not finding results from altavista'ing that). It was a mix of higher-level undergrad and postgrad, and so of course there was a lot of challenging material along with the historical/cultural stuff. Immensely cool, and lots of curiosities and interesting things related to the *actual* people who *actually* crafted/discovered/invented the mathematics we often take as dropped down from on high.
If you're interested, I thoroughly recommend Simon Singh's excellent book on the subject, called either *Fermat's Last Theorem* or *Fermat's Enigma*.
I'm surprised no one has mentioned the life of Évariste Galois. While just around 19 years old, he invented a whole new branch of mathematics (group theory) and discovered a connection to which polynomial equations are solvable in radicals. This was a long-standing open problem for centuries. His personal life was also interesting, he was involved in political acts related to the French Revolution, and he died in a duel aged 20 years old.
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Ramanujan’s series of pi
There's a lot of trivia that can potentially be of interest to you. If you like algebra then perhaps take a read through the Emmy Noether's Wikipedia page. She was a female mathematician with an interesting story. I think her approach to research was very interesting as well, since she focused on developing useful results from very abstract settings. This is not something most people do, usually you start with examples and then build abstraction by looking at similar examples and connecting them, which is something she did, but she also started from very abstract objects (rings with the ascending chain condition) and found useful results that can be used in rings we actually care about. Would highly recommend looking through her work.
Galois life history
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