Thursday, September 25, 2014

Divergent Movie: Division by zero and Calculus of infinities

1. Abnegation and Additive Inverses

The people in the Abnegation Faction try their best to be selfless. Suppose $a$ represents the selfish tendencies of man, which we may theologically refer to as concupiscence. The aim of an Abnegation is to make their self, which we shall represent as $x$, approach the $-a$, the additive inverse of $a$, so that the $\lim_{x\rightarrow -a} (x + a)\rightarrow 0$. 

2. Divergence and division by zero 

Divergents become dangerous when they are paired with non-divergent people, because their Divergence shows up. Mathematically, we say that when a real number such as 3 is divided by an Abnegation $(x+2)$, the result can result to a Divergence: \begin{equation} \lim_{x\rightarrow -2^+}{\frac{3}{x+2}} = \frac{3}{0^+} = +\infty. \end{equation} To prevent Divergences from showing up, two people who can be Abnegations (remember that Divergents can fit any faction) must be paired to each other. For example, if two Abnegations are both modeled by $x+a$, then their pairing can result to a real number which is not divergent: \begin{equation} \lim_{x \rightarrow -a^+}\frac{x+a}{x+a} = \lim_{x\rightarrow a^+}\frac{1}{1} = +1, \end{equation} by L'Hopital's Rule. This is why Four and Tris can control the potential Divergence that may result from their pairing: they can both behave as if they are normal members of Dauntless. 

3. Factions and Fractions 

In the faction system, society is divided into five groups or factions based on their personalities: Abnegation (selfless), Amity (peaceful), Candor (truthful), Erudite (intelligent) and Dauntless (brave). Thus, each faction is a fractional part of the whole society. But divergents destroy this faction system. For example, if you have an ordinary fraction $2/3$ and add to it the fraction $4/(x+3)$, the sum cannot be controlled if you take the limit as $x\rightarrow -3$ and becomes divergent: \begin{equation} \lim_{x\rightarrow 3^ +}\left(\frac{2}{3} + \frac{4}{x+3}\right) = \frac{2}{3} + \frac{4}{0^+} = +\infty. \end {equation} This is why Divergents are dangerous and must be eliminated. For example, when the Dauntless soldiers were injected with a mind-conditioning serum, one Divergent went out of the line and questioned why they were lining up. He was shot dead.

 4. Simulations and Imaginary Numbers

Even if we multiply divergents by the imaginary number $i$, the result is still divergent and would have an infinite norm or length. For example, \begin{equation} \left |2 + \frac{3}{0^+}i\right | = \left |2 + \infty i\right | = +\infty. \end{equation} If we liken simulations to multiplying by the imaginary number $i$, then no amount of simulations can defeat a Divergent. That is why, a divergent like Tris can break the glass of her fears: she took the norm of the complex imaginary world and went back to reality. But even in reality, she is still divergent.

5. Divergences and computer programs

Computer programmers are always wary of divergences in their computer codes. If there is a possibility of division by zero, they try to tame this problem by making them undergo several tests, because there is a big difference whether $x\rightarrow 0$ from the right ($0^+$) or from the left $0^-$. The NaN (not a number) was introduced in 1985 by the IEEE 754 (IEEE Standard for Floating Point Arithmetic) in order to make distinctions between several types of Divergents and handle them accordingly. In the movie, these tests are done by making each person face their fears: fire, crows, drowning, closure, and heights. But Divergents can easily pass through these tests. And in the end the Divergents Four and Tris destroyed the computer program of the Erudite faction that aims to destroy all members of the Abnegation.
Divergent (Plus Bonus Features)
Divergent (Plus Bonus Features)
Divergent Series Ultimate Four-Book Collection: Divergent; Insurgent; Allegiant; Four
Divergent Series Ultimate Four-Book Collection: Divergent; Insurgent; Allegiant; Four
Divergent: From Book to Screen [HD]
Divergent: From Book to Screen [HD]
Inside Divergent: The Initiate's World
Inside Divergent: The Initiate's World