Don’t be scared. Prove that shit directly.

Wake your ass up, it’s time to talk proofs.  Time to step your game up from those piddly things from high school geometry.  There are a number standard methods of proof that all y’all aspiring mathematicians need to know.  This is the first one, bitch.

Direct proof

Direct proofs are 100% fucking deductive.  If $P$ then $Q$.

Consider the following three “facts:”

1. I will listen to any hip hop not in the top 40.
2. If a song is in the top 40, then it’s getting too much radio play.
3. I have never heard a Talib Kweli track on the radio.

We can prove that I will listen to any given Talib Kweli song.  You can deduce from the fact that Kweli isn’t getting too much radio play (from fact #3), that it’s not in the top 40 (from fact #2).  Well, if it’s not the top 40, I’ll give it a listen (from fact #1).  This is the gist of direct proofs: start at something you know, use the facts at your disposal to deduce what you can, and please listen to some hip hop that’s worth a damn.

Let’s get some numbers in this bitch.  We’ll prove the following using a direct proof:

Statement

The square of an ass integer is a square, jackass.*

Proof

Oh shit!  Real math right here.

We have to start somewhere, so let’s assume that some integer–call it $n$–is odd.  If that’s true, then $n=1+2m$, where $m$ is an integer.  We know this because an even integer plus 1 is odd as hell, and $2m$ is even since it has a factor of 2.  Definition of an even number, smartass.

$$n=1+2m$$

$$n^2=(1+2m)^2$$

$$n^2=1+4m+4m^2$$

$$n^2=1+4(m+m^2)$$

Guess what, asshole?  That means $n^2$ is odd.  Why the hell? $4(m+m^2)$ is even since it has a factor of 2.  That means $1+4(m+m^2)$ is odd.

Proven, bitches.

That’s all there is to it.  Straightforward and direct.  Start with an assumption, and just roll with it until you get what you need.  Use whatever fucking theorems or properties you know to deduce your way to your conclusion..  Sherlock Holmes shit, y’all.

If you want to get real academic about this shit, we could write all this out entirely in boojie math notation:

$$n\,\mbox{odd}\implies n=1+2m\implies$$

$$n^2=1+4m+4m^2=1+4(m+m^2)\implies$$

$$n^2 \, \mbox{odd}$$

Ya dig?  This is proofs 101–mathematicians use the direct method as at least a part of pretty much every proof on the fucking earth.  Trickier methods to come later for your curious ass.

Welcome to the world of pure fucking mathematics.

---

*Statement from A Concise Introduction to Pure Mathematics, page 4.