Primes, they aren't just for Transformers

It has to be said.
There is just something magical about primes.

I mean they are numbers that represent something basic. Multiplicatively, they can't be broken down and they end up becoming the special cases upon which every mathematical theory must be tried. And yet once you throw them into the world of math theory you get all sorts of weird math facts, that are just undeniably cool.

Like the fact that (p-1)! + 1 is divisible by p, if and only if, p is a prime.

Dude, like awesome.

And yet primes remain mysterious. It was only a few years ago people learned how to test if a number was prime within polynomial time (see here), people still can't prove that there are an infinite number of primes where p and p+2 are prime. And there's the fact that as you go to infinity, the number of primes approaches the function x/ln(x). Zuh?!

I say zuh not because prime numbers are hard to understand, although sometimes they are, but because their wonder and bounty are just mind-boggling.

So let it be understood then, primes are awesome.

And since primes are awesome, math is awesome.

Of course, this is but one of the proofs of math's awesomeness, which are as numerous as prime numbers themselves, and thus proven to be infinite.

Happy St. Valentine's Day

Ah, St.Valentine's Day, a day of love, a day of romance, a day of... MATH!!!

Since I have spent much energy elsewhere, I cannot explain the full dimensions of St.Valentine's Day mathematical importance, but I can share with you a Valentine's Day treat.

Want to make your sweetie's heart swoon, give here the formula for Cardioids, vaguely heart shaped graphs using polar coordinates!

(just to refresh you, polar coordinates use distance from the origin (the center of the graph) as r, combined with an angle Θ (it's hard to represent it on a computer, but its basically an O with a line in the middle, or sometimes just a cursive-ish O) to form a location for a point)



or if you're feeling more sine-y



(formulas courtesy of Paul's Online Calc. II Notes, which is itself courtesy of and property of Professor Paul Dawkins of Lamar University)

But if you've messed up former Valentine's Day gifts, perhaps you need something a little bit more spectacular, well then I direct you to this fantastic site:

All About Heart Curves! (not it's actual name)

From the mind of Professor Jürgen Köller.

Well, I think that's a heart healthy start to Valentine's Day. But you can't slack off, after all, you still need to give Valentines to people. Just don't forget to give a Valentine to one very special girl, Math!!!

Five Def. of Pi's

Ah Pi, that elusive impossible goal. It has driven many to madness (especially the guy in this movie), but many also to great feats.

And so unsurprisingly it has a special place in the hearts of many mathematicians. The 16th century German mathematician Ludoph van Ceulen even had his tombstone engraved with his 32 decimal place estimations of pi.

Ah, Pi!

Fiercely irrational, blissfully transcendent, and so essential to the universe and how it works...

But it's so tricky. Oh course, I know how to write Pi's true form, but instead of revealing that truth to you and thus blowing your minds, I'll just share with you some other def.s of pi, beyond just the ratio of the perimeter of a circle and its diameter (meter! meter!) (all better than the old stand-by 3.14).

1. Good old Archimedes came up with a rough estimate: π is between 223/71 and 22/7

He figured this by drawing a circle, drawing a polygram inside the circle, touching all the sides, and then drawing a polygram outside the circle where the circle touched all of its sides. Then he compared the two polygrams, and by making more and more sides... and this is how you get the rough estimate 22/7 which I used through much of elementary school.

2. Here's one an Indian did write up in the 15th century AD:
and so on... FOREVER!!!


Pretty damn awesome. INFINITE SERIES RULE!!!
Good work Madhava of Sangamagrama! It would take until the 17th century for the series to be rediscovered through the hearty work of James Gregory and Gottfried Leibniz.

3. And if you like your arctangents (and who doesn't?), how about this def. of pi:


Kudos John Machin!

And if you don't like your arctans, well,
and so on, still FOREVER!!!
At least according to a nice little Taylor expansion of arctangent.

4. Well, to take Pi to the next level, another Indian had to get into the game. And so in the early 19th century another Indian did (actually Indians had been working all along, but this 5 def.s not a total history). I'm talking about the one, the only Srinivāsa Rāmānujan!!!
Check out his method for finding Pi, derived from the highest halls of Number Theory:


While that method didn't hit the big time till 1985, in that year William Gosper used it to calculate Pi to 17 million digits. Dude, sweet.

5. But if you want to go a little ways by the abstract route. Well, remember that Euler with some Taylor formulization of e^(ix), sin x and cos x, came up with

which when x = pi leads to a nice little identity involving pi:


Ah, Euler, you may have lived in the 18th century and pronounced your name like a greaser, but you're still one of the best.

So in conclusion let me give you one more def. of pi:

Pi = Awesome

Exactly