Come see me dream math.

So for some reason I dreamed of my highschool math teacher!ⁱ

In my dream, I was playing with a classmate's decolletageⁱⁱ when Mr. D called me out. With his very particular eye glimmer, of course - sorta like a big happy tomcat's "o I see you there, buster!" glee. And so obviously I was going to have to say something quite momentuous and important, because what! This is a school, right ? And we're smart people and doing maths over here! Superharddifficult maths of arcana, no less.ⁱⁱⁱ

So I went to the blackboard and I wrote out the cannonical polynome, equated it to its count to the power of X and woke up. Here it is, graphically :

Obviously, i and n are natural numbers (positive integers and 0) ; the a_i series is however rational, not real. And X can be anything. So you'd have to prove that given a certain n (the thing's "class" so to speak), the equality holds for any X, or doesn't (in which case find the X or explain how the X'd be found).

I honestly don't remember if this "exists"^iv, but it occurs to me it is in fact a pretty cool approach to inquiring into the nature of irrational numbers.

For one thing, the general degenerate case where n = 1 obviously holds true : one to any arbitrary power stays one, and so all one needs is a₀ = 1.

If n = 2, things become quite complicated. Should X be an integer (ie, the particular degenerate case), the equality also obviously holds (just make a₀ equal n^X and you're done). Should X be something more interesting, like say e, the equation suddenly reads

2^e = a ∙ e ² + b ∙ e.

If we log the thing (in 2) we get

e = log (a ∙ e ²) + log (b/a ∙ e ^-1), or
e = log (a) + 2 / ln (2) + log (b/a) - 1 / ln (2)

unless I fucked it up somewhere. I don't readily remember if e minus the natural logarithm of two can always be written as the sum of two arbitrary logs. I wish I could say something intelligent including pi at this point, but nothing comes to mind.

Anyway. Welcome to the world of my dreams.

1. Hi Mr. Diaconu!

   Check out the comments there btw,

   DOMNUL PROFESOR ESTE FOARTE BUN IN CEEA CE PRIVESTE PREDAREA MATEMATICII INSA NU ESTE UN BUN PEDAGOG CU CEI CARE NU SUNT ATAT DE STRALUCITI LA MATEMATICA SI PUR SI SIMPLU II INHIBA.

   ie, "very good math teacher, but he doesn't cater to the less gifted, which is inhibiting to them". Doh, no wonder I still remember the guy. Exactly what a teacher should be doing : help those who can learn learn, and those who can't learn find something else to do with their time. Math is not for everyone, exactly like the world it stands for. Which also isn't for everyone. [↩]

2. This is quite true to form : that's exactly what I'd mostly be doing in those 30% of cases where I actually showed up for class (well, to be perfectly fair : sometimes I'd play ritzi pitzi with my genius male classmates instead). [↩]
3. This very peculiar brand of transactional hypocrisy, vaguely reminescent of cavalry officers in peacetime, where it's okay to be doing whatever just as long as you can deploy full strength if called upon with a moment's notice is not only the exact practical opposite to the puritan stolidity, but also was by far the best part of Romania of the time. Meanwhile it's all gone the way of NATO, and the corporation. Perhaps a few sysadmins scattered here and there maintain the flame alive, but otherwise, it got swallowed up in a cellulotic inferno of three ring binders churned by really boring idiots.

   Choices matter, of course, and if you choose "no kids left behind" instead of Mr. D's quite very feminine "mock them to all hell - most it discourages, some it'll encourage" you will get a world populated by Danika and the twerking dweebs instead of the world populated by me and the sort of girls I'd fuck. Which I suppose is no great loss, if you're the sort of mentally limited fucktard that'd even consider the tradeoff. Because you don't understand very much, and so the necessary result of your choice, being three steps removed, is also necessarily safe from your prying dullard eyes. [↩]

4. Obviously it exists, being math. Whether we know about it or not does not in any way impinge upon its existence, much like in the case of... I dunno, say Bitcoin ?

   What I mean is merely it existing in the literature, having a name, a long trail of petrified shit produced by the brains trying to engulf it, ingest it over the endless years and so on. Because you know... let's make an aside. Years ago when my age was single digit, we had a Chinese chess set, which I dearly loved, because those shiny acrylic beads with a whisp of colored nylon inside! I ate most of them.

   No kidding, I swallowed those delicious beads like there was no tomorrow - in spite of my parents' best efforts by the time I was old enough to grow pubic hair I had eaten most of the set (my sister also had some). Obviously, being indigestible, they came out the same way they went in, and continue their existence unmolested by human agency - as they can never be molested by human agency.

   That, roughly speaking, is your relationship to math. [↩]