Trying to make math cool is like trying to make "not holding your breath for five minutes" cool: It already is, and anyone thinking otherwise struggles through life with significantly reduced mental abilities. Anyone born into a world based entirely on numbers with little dollar signs in front of them, then deciding they don't need to know numbers, is volunteering to be a very small one adding up to someone else's very big one.
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"Just 10 more short years, and you'll have that degree in Klingon paid off!"
But accounting is to math what diapers are to biochemistry: dealing with the stinking mess left by some unfortunate and immature side effects of being alive while doing it. Mathematics isn't cool in the same way spacetime isn't real estate. It's much bigger and more important than the ridiculous little structures we've erected on it. And anyone who doesn't understand how truly cool it is (2.725 K) simply doesn't appreciate the sheer scale of it.
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We found reality's programming language. Physics is the operating system, but it's written in equations. E=mc^2 defines large parts of reality in fewer bytes than it takes to say "my penis" and has similar effects on local spacetime, ladies (~ 90 percent, + gentlemen ~ 10 percent). But that's physics. There are pure mathematical equations which are just as capable of blowing your mind.
#5. Infinite Pi
Pi is the ratio of a circle's diameter to its circumference, in the same way nuclear fission is a way of powering TVs to watch America's Got Talent: an appallingly simple effect of a reality-defining truth. Pi isn't a number, it's a startup constant of spacetime. Take a line in one dimension, rotate it around another, and the resulting ratio of lengths is a precise number. The existence of space has a numerical signature. It's called a transcendental number, because even attempting to think about how much it means is more mind-expanding than all the drugs.
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Seen here in the three dimensions it glues together
Calculating pi has become the computer scientist equivalent of tuning muscle cars: we don't actually need more digits for anything useful, because the basics do everything we actually need it for, but we've spent years stacking up air-cooled hardware just because. In 1985 it was calculated to 17 million digits. Srinivasa Ramanujan found the formula used. Around 1910.
It wasn't the only such formula, but was incredibly useful because it converged exponentially compared to other algorithms, making it ideal for computers. Interesting note: at the time there was no such thing as computers. Srinivasa Ramanujan had pre-empted processors by decades. In 1985 his formula was used in the world record calculation of pi to 17 million digits, and a slight variant modified by David and Gregory Chudnovsky now holds the title at 10 trillion. We're not saying he's a robot, but if a perfect calculating machine ever goes missing in a time travel experiment we already know where it ended up.
It turns out that rogue Terminators like sharp suits and counting forever.
#4. Dividing By Zero
Dividing by zero gives you undefined, infinity, or who cares, depending on whether you're talking to a mathematician, physicist, or engineer. You can tell a lot about how disconnected someone is from reality by how they contemplate infinity. For example, cult leaders often explain how infinity after nothingness means you should obey them.
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"Pay your entire life now for insurance against eternity with the King! No refunds!"
At its most basic, division means, "How many times can you take this out of that?" For example, you can take a human head from the members of One Direction five times before you get blessed silence. On the one hand you can take nothing out of something an infinite number of times, but on the other, no, you can't, because that would take forever. It's a mathematical problem so hard it once crippled a U.S. Navy cruiser. The USS Yorktown's experimental military computer control system tried it and was crippled by a buffer overrun. Making this the first ship to overflow without any water. On the upside, it's way easier than asking a computer to define love or telling it that truth is a lie, or asking if those are both the same question.
"These feelings are tearing me apart! Also the incoming torpedoes."
The solution is as brilliant as it sounds stupid. You deal with the impossible by sneaking up on it. It's nice to know calculus solves problems the same way SEAL teams do it: sneakily and permanently.
Taking the limit as X goes to zero means that you never actually get there, but you get as close as you need to be, no matter how close that is. Then the "limit" as you tend toward zero -- but never actually reach it -- gives you the answer. As X gets small, sin X is approximately equal to X, so you're always dividing something by itself and getting one. Then, with a particularly cunning flourish, you end up dividing nothingness by itself and becoming one. Which I think means mathematics is the king of zen.
His decades of calm are going to be wasted when he finds out mathematicians worked it out with a pencil.
There's a fairly easy proof using the squeeze theorem. Which doubles as a good line when you want to chat up a mathematician.
"... and you'll be with me because I give quite excellent dick, QED."
These limits as you hit zero, which is a much less depressing sentence in mathematics, are essential for calculus. And calculus is essential for everything. If math is the programming language of reality, calculus is the graphics processor working out things like explosions, lasers, and gravity, all the cool special effects. Or as we call them in physics, effects.
#3. Pick A Digit, Any Digit (of Pi)
If the Ramanujan formula was transcendent understanding of reality by the human brain, the Bailey-Borwein-Plouffe formula is outright sorcery. Developed by Simon Plouffe in 1995, this formula lets you skip straight to any digit of pi without working out the rest of the number.
You want the the 10-trillion-and-33rd digit of pi? No problem. There's a fundamental constant of reality and now, without any input apart from being in the same reality, this equation can read out any part of it. That's like pulling rabbits out of a hat where the rabbits are Star Trek energy beings and your hat is a beret.
The process of using the formula sounds like someone found a glitch in god's computer. If you want the Nth term, you split the infinite sum at the Nth term, and a bit of modulo math skims out the required digit in hexadecimal. That sounds like something you'd read on Mr Mxyzptlk's GameFAQ. Reality, mathematics, and the design of 8-bit computers lining up to accidentally output one of the universe's BIOS settings. The craziest part? It's still slower than Ramanujan for finding the whole thing. It really is just a cheat code for reality.