fog/gof

diff --git a/spec/complex.md b/spec/complex.md
index 374b793d048fa32c484f2e1c9a314fcd73a14d4d..e7760be2d467eeee835e0783238c7cb8d0d4a891 100755 (executable)
--- a/spec/complex.md
+++ b/spec/complex.md
@@ -2,9 +2,7 @@
 
 ## Imaginary numbers
 
-$i^2 = -1$
-
-$\therefore i = \sqrt {-1}$
+$i^2 = -1 \quad \therefore i = \sqrt {-1}$
 
 ### Simplifying negative surds
 
@@ -16,9 +14,8 @@ $= \sqrt{2}i$
 
 $\mathbb{C} = \{a+bi : a, b \in \mathbb{R} \}$
 
-General form: $z=a+bi$
-- $\operatorname{Re}(z) = a$
-- $\operatorname{Im}(z) = b$
+General form: $z=a+bi$  
+$\operatorname{Re}(z) = a, \quad \operatorname{Im}(z) = b$
 
 ### Addition
 
@@ -59,10 +56,19 @@ $z_1 \times z_2 = (ac-bd)+(ad+bc)i$
 
 If $z=a+bi$, conjugate of $z$ is $\overline{z} = a-bi$ (flipped operator)
 
-Also, $z \overline{z} = (a+bi)(a-bi) = a^2+b^2$
+Also, $z \overline{z} = (a+bi)(a-bi) = a^2+b^2 = |z|^2$
 
 - Multiplication and addition are associative
 
+#### Properties
+
+- $\overline{z_1 + z_2} = \overline{z_1} + \overline{z_2}$
+- $\overline{z_1 z_2} = \overline{z_1} \cdot \overline{z_2}$
+- $\overline{kz} = k \overline{z}, \text{ for } k \in \mathbb{R}$
+- $z \overline{z} = |z|^2$
+- $z + \overline{z} = 2 \operatorname{Re}(z)$
+
+
 ### Modulus
 
 Distance from origin.
@@ -70,6 +76,12 @@ $|{z}|=\sqrt{a^2+b^2}$
 
 $\therefore z \overline{z} = |z|^2$
 
+#### Properties
+
+- $|z_1 z_2| = |z_1| |z_2|$
+- $|{z_1 \over z_2}| = {|z_1| \over |z_2|}$
+- $|z_1 + z_2| \le |z_1 + |z_2|$
+
 ### Multiplicative inverse
 
 $z^{-1} = {1 \over z} = {{a-bi} \over {a^2+B^2}} = {\overline{z} \over {|z|^2}}$
@@ -99,9 +111,12 @@ $z^2+a^2=z^2-(ai)^2=(z+ai)(z-ai)$
 
 General form:
 $z=r \operatorname{cis} \theta$
-$= r\operatorname{cos}\theta+r\operatorname{sin}\theta i$
+$= r(\operatorname{cos}\theta+i \operatorname{sin}\theta)$
+
+$z=a+bi$  
+$z=r\operatorname{cis}\theta$  
+
 
-where
 - $z=a+bi$
 - $r$ is the distance from origin, given by Pythagoras ($r=\sqrt{x^2+y^2}$)
 - $\theta$ is the argument of $z$, CCW from origin
@@ -109,12 +124,16 @@ where
 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
 
+### Conjugate in polar form
+
+$$(r \operatorname{cis} \theta)^{-1} = r\operatorname{cis} (- \theta)$$
+
 ### Multiplication and division in polar form
 
 $z_1z_2=r_1r_2\operatorname{cis}(\theta_1+\theta_2)$ (multiply moduli, add angles)
 
 ${z_1 \over z_2} = {r_1 \over r_2} \operatorname{cis}(\theta_1-\theta_2)$ (divide moduli, subtract angles)
 
-## de Moivres' Theorum
+## de Moivres' Theorem
 
-$(r\operatorname{cis}\theta)^n=r^n\operatorname{cis}(n\theta)$
+$(r\operatorname{cis}\theta)^n=r^n\operatorname{cis}(n\theta)$ where $n \in \mathbb{Z}$