[spec] clean up

diff --git a/spec/complex.md b/spec/complex.md
index e72225e3b7bbf2d5a170930723ed6a9fa50e03c3..bb98525c3eb4456c32a9a79f461a7b60d7fff475 100755 (executable)
--- a/spec/complex.md
+++ b/spec/complex.md
@@ -1,5 +1,5 @@
 ---
-geometry: margin=2cm
+geometry: margin=1.9cm
 <!-- columns: 2 -->
 graphics: yes
 tables: yes
@@ -66,9 +66,7 @@ $$z_1 \times z_2 = (ac-bd)+(ad+bc)i$$
 
 ### Conjugates
 
-If $z=a+bi$, conjugate is
-
-$$\overline{z} = a-bi$$
+$$\overline{z} = a \mp bi$$
 
 ##### Properties
 
@@ -108,19 +106,21 @@ $${z_1 \over z_2} = {{(a+bi)(c-di)} \over {c^2+d^2}}$$
 - horizontal $= \operatorname{Re}(z)$; vertical $= \operatorname{Im}(z)$
 - Multiplication by $i$ results in an anticlockwise rotation of $\pi \over 2$
 
-## Solving complex polynomials
+\vfil \break
 
-**Include $\pm$ for all solutions, including imaginary**
+## Complex polynomials
 
-## Solving complex quadratics
+**Include $\pm$ for all solutions, including imaginary**
 
-To solve $z^2+a^2=0$ (sum of two squares):
+### Sum of two squares (quadratics)
 
 $$z^2+a^2=z^2-(ai)^2=(z+ai)(z-ai)$$
 
+Complete the square to get to this point.
+
 #### Dividing complex polynomials
 
-Dividing $P(z)$ by $D(z)$ gives quotient $Q(z)$ and remainder $R(z)$ such that:
+$P(z) \div D(z)$ gives quotient $Q(z)$ and remainder $R(z)$:
 
 $$P(z) = D(z)Q(z) + R(z)$$
 
@@ -129,14 +129,19 @@ $$P(z) = D(z)Q(z) + R(z)$$
 Let $\alpha \in \mathbb{C}$. Remainder of $P(z) \div (z - \alpha)$ is $P(\alpha)$
 
 #### Factor theorem
+
 If $a+bi$ is a solution to $P(z)=0$, then:
 
 - $P(a+bi)=0$
 - $z-(a+bi)$ is a factor of $P(z)$
 
+#### Sum of two cubes
+
+$$a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2)$$
+
 ## Conjugate root theorem
 
-If $a+bi$ is a solution to $P(z)=0$, with $a, b \in \mathbb{R}$, then the conjugate $\overline{z}=a-bi$ is also a solution.
+If $a+bi$ is a solution to $P(z)=0$, then the conjugate $\overline{z}=a-bi$ is also a solution.
 
 ## Polar form
 
@@ -146,8 +151,8 @@ If $a+bi$ is a solution to $P(z)=0$, with $a, b \in \mathbb{R}$, then the conjug
 - $\theta=\operatorname{arg}(z)$ (on CAS: `arg(a+bi)`)
 - **principal argument** is $\operatorname{Arg}(z) \in (-\pi, \pi]$ (note capital $\operatorname{Arg}$)
 
-Note each complex number has multiple polar representations:
-$z=r \operatorname{cis} \theta = r \operatorname{cis} (\theta+2 n\pi$) where $n$ is integer number of revolutions
+Each complex number has multiple polar representations:  
+$z=r \operatorname{cis} \theta = r \operatorname{cis} (\theta+2 n\pi$) with $n \in \mathbb{Z}$ revolutions
 
 ### Conjugate in polar form
 
@@ -157,9 +162,9 @@ Reflection of $z$ across horizontal axis.
 
 ### Multiplication and division in polar form
 
-$z_1z_2=r_1r_2\operatorname{cis}(\theta_1+\theta_2)$ (multiply moduli, add angles)
+$$z_1z_2=r_1r_2\operatorname{cis}(\theta_1+\theta_2)$$
 
-${z_1 \over z_2} = {r_1 \over r_2} \operatorname{cis}(\theta_1-\theta_2)$ (divide moduli, subtract angles)
+$${z_1 \over z_2} = {r_1 \over r_2} \operatorname{cis}(\theta_1-\theta_2)$$
 
 ## de Moivres' Theorem
 
@@ -179,6 +184,17 @@ $$x^2 + y^2 = (|a|^{1 \over n})^2$$
 
 ## Sketching complex graphs
 
-- **Straight line:** $\operatorname{Re}(z) = c$ or $\operatorname{Im}(z) = c$ (perpendicular bisector) or $\operatorname{Arg}(z) = \theta$
-- **Circle:** $|z-z_1|^2 = c^2 |z_2+2|^2$ or $|z-(a + bi)| = c$
-- **Locus:** $\operatorname{Arg}(z) < \theta$
+### Straight line
+
+- $\operatorname{Re}(z) = c$ or $\operatorname{Im}(z) = c$ (perpendicular bisector)
+- $\operatorname{Arg}(z) = \theta$
+- $|z+a|=|z+bi|$ where $m={a \over b}$
+- $|z+a|=|z+b| \longrightarrow 2(a-b)x=b^2-a^2$
+
+### Circle
+
+$|z-z_1|^2 = c^2 |z_2+2|^2$ or $|z-(a + bi)| = c$
+
+### Locus
+
+$\operatorname{Arg}(z) < \theta$