README.md initial LaTeX cleanup

README.md

@@ -115,27 +115,35 @@ If we want to correct $e$ byte-errors, we will need $n = 2e$ fixed
 points. We can construct a generator polynomial $P(x)$ with $n$ fixed
 points at $g^i$ where $i < n$ like so:
 
-$$
+$$ \\
 P(x) = \prod_{i=0}^n \left(x - X_i\right)
 $$
 
 We could choose any arbitrary set of fixed points, but usually we choose
 $g^i$ where $g$ is a [generator][generator] in GF(256), since it provides
 a convenient mapping of integers to unique non-zero elements in GF(256).
 
-Note that for any fixed point $g^i$, $x - g^i = g^i - g^i = 0$. And
-since multiplying anything by zero is zero, this will make our entire
+Note that for any fixed point $g^i$:
+
+$$ \\
+\begin{aligned}
+x - g^i &= g^i - g^i \\
+        &= 0
+\end{aligned}
+$$
+
+And since multiplying anything by zero is zero, this will make our entire
 product zero. So for any fixed point $g^i$, $P(g^i)$ should also evaluate
 to zero:
 
-$$
+$$ \\
 P(g^i) = 0
 $$
 
 This gets real nifty when you look at the definition of our Reed-Solomon
 code for codeword $C(x)$ given a message $M(x)$:
 
-$$
+$$ \\
 C(x) = M(x) x^n - (M(x) x^n \bmod P(x))
 $$
 
@@ -144,7 +152,7 @@ gives us a polynomial that is a multiple of $P(x)$. And since multiplying
 anything by zero is zero, for any fixed point $g^i$, $C(g^i)$ should also
 evaluate to zero:
 
-$$
+$$ \\
 C(g^i) = 0
 $$
 
@@ -163,9 +171,11 @@ $$
 Check out what happens if we plug in our fixed point, $g^i$:
 
 $$
-C'(g^i) = C(g^i) + E(g^i)
-        = 0 + E(g^i)
-        = E(g^i)
+\begin{aligned}
+C'(g^i) &= C(g^i) + E(g^i) \\
+        &= 0 + E(g^i) \\
+        &= E(g^i)
+\end{aligned}
 $$
 
 The original codeword drops out! Leaving us with an equation defined only