From 9f4e42fdb4fdcb8049ab6f9487d5f19275ffb17b Mon Sep 17 00:00:00 2001
From: Andrew Lorimer <andrew@lorimer.id.au>
Date: Wed, 13 Feb 2019 12:18:47 +1100
Subject: [PATCH] properties of complex conjugates & moduli

---
 spec/complex.md | 26 +++++++++++++++++++-------
 1 file changed, 19 insertions(+), 7 deletions(-)

diff --git a/spec/complex.md b/spec/complex.md
index 374b793..c281154 100755
--- 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
 
 $z^{-1} = {1 \over z} = {{a-bi} \over {a^2+B^2}} = {\overline{z} \over {|z|^2}}$
-- 
2.43.2