update planner, create complex reference

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diff --git a/spec/complex.md b/spec/complex.md
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--- a/spec/complex.md
+++ b/spec/complex.md
@@ -1,81 +1,90 @@
+---
+geometry: margin=2cm
+<!-- columns: 2 -->
+graphics: yes
+tables: yes
+author: Andrew Lorimer
+classoption: twocolumn
+header-includes:
+- \usepackage{harpoon}
+- \usepackage{amsmath}
+- \pagenumbering{gobble}
+
+---
+
+

 # Complex & Imaginary Numbers
 
 ## Imaginary numbers
 
 # Complex & Imaginary Numbers
 
 ## Imaginary numbers
 

-$i^2 = -1 \quad \therefore i = \sqrt {-1}$
+$$i^2 = -1 \quad \therefore i = \sqrt {-1}$$

 
 ### Simplifying negative surds
 
 
 ### Simplifying negative surds
 

-$\sqrt{-2} = \sqrt{-1 \times 2}$  
-$= \sqrt{2}i$
+\begin{equation}\begin{split}\sqrt{-2} & = \sqrt{-1 \times 2} \\ & = \sqrt{2}i\end{split}\end{equation}

 
 
 ## Complex numbers
 
 
 
 ## Complex numbers
 

-$\mathbb{C} = \{a+bi : a, b \in \mathbb{R} \}$
+$$\mathbb{C} = \{a+bi : a, b \in \mathbb{R} \}$$

 
 General form: $z=a+bi$  
 $\operatorname{Re}(z) = a, \quad \operatorname{Im}(z) = b$
 
 ### Addition
 
 
 General form: $z=a+bi$  
 $\operatorname{Re}(z) = a, \quad \operatorname{Im}(z) = b$
 
 ### Addition
 

-If $z_1 = a+bi$ and $z_2=c+di$, then  
-$z_1+z_2 = (a+c)+(b+d)i$
+If $z_1 = a+bi$ and $z_2=c+di$, then
+
+$$z_1+z_2 = (a+c)+(b+d)i$$

 
 ### Subtraction
 
 
 ### Subtraction
 

-If $z_1=a+bi$ and $z_2=c+di$, then $z_1−z_2=(a−c)+(b−d)i$
+If $z_1=a+bi$ and $z_2=c+di$, then
+
+$$z_1−z_2=(a−c)+(b−d)i$$

 
 ### Multiplication by a real constant
 
 
 ### Multiplication by a real constant
 

-If $z=a+bi$ and $k \in \mathbb{R}$, then $kz=ka+kbi$
+If $z=a+bi$ and $k \in \mathbb{R}$, then

 
 

-### Powers of $i$
-$i^0=1$
-$i^1=i$
-$i^2=-1$
-$i^3=-i$
-$i^4=1$
-$\dots$
+$$kz=ka+kbi$$

 
 

-Therefore..
+### Powers of $i$

 
 - $i^{4n} = 1$
 - $i^{4n+1} = i$
 - $i^{4n+2} = -1$
 - $i^{4n+3} = -i$
 
 
 - $i^{4n} = 1$
 - $i^{4n+1} = i$
 - $i^{4n+2} = -1$
 - $i^{4n+3} = -i$
 

-For $i^n$, divide $n$ by 4 and let remainder $= r$. Then $i^n = i^r$.
+For $i^n$, find remainder $r$ when $n \div 4$. Then $i^n = i^r$.

 
 ### Multiplying complex expressions
 
 
 ### Multiplying complex expressions
 

-If $z_1 = a+bi$ and $z_2=c+di$, then  
-$z_1 \times z_2 = (ac-bd)+(ad+bc)i$
+If $z_1 = a+bi$ and $z_2=c+di$, then

 
 

-### Conjugates
+$$z_1 \times z_2 = (ac-bd)+(ad+bc)i$$

 
 

-If $z=a+bi$, conjugate of $z$ is $\overline{z} = a-bi$ (flipped operator)
+### Conjugates

 
 

-Also, $z \overline{z} = (a+bi)(a-bi) = a^2+b^2 = |z|^2$
+If $z=a+bi$, conjugate is

 
 

-- Multiplication and addition are associative
+$$\overline{z} = a-bi$$

 
 

-#### Properties
+##### Properties

 
 - $\overline{z_1 + z_2} = \overline{z_1} + \overline{z_2}$
 - $\overline{z_1 z_2} = \overline{z_1} \cdot \overline{z_2}$
 - $\overline{kz} = k \overline{z}, \text{ for } k \in \mathbb{R}$
 
 - $\overline{z_1 + z_2} = \overline{z_1} + \overline{z_2}$
 - $\overline{z_1 z_2} = \overline{z_1} \cdot \overline{z_2}$
 - $\overline{kz} = k \overline{z}, \text{ for } k \in \mathbb{R}$

-- $z \overline{z} = |z|^2$
+- $z \overline{z} = = (a+bi)(a-bi) = a^2+b^2 = |z|^2$

 - $z + \overline{z} = 2 \operatorname{Re}(z)$
 
 ### Modulus
 
 Distance from origin.
 - $z + \overline{z} = 2 \operatorname{Re}(z)$
 
 ### Modulus
 
 Distance from origin.

-$|{z}|=\sqrt{a^2+b^2}$

 
 

-$\therefore z \overline{z} = |z|^2$
+$$|{z}|=\sqrt{a^2+b^2} \quad  \therefore z \overline{z} = |z|^2$$

 
 

-#### Properties
+###### Properties

 
 - $|z_1 z_2| = |z_1| |z_2|$
 - $|{z_1 \over z_2}| = {|z_1| \over |z_2|}$
 
 - $|z_1 z_2| = |z_1| |z_2|$
 - $|{z_1 \over z_2}| = {|z_1| \over |z_2|}$


@@ -83,11 +92,11 @@ $\therefore z \overline{z} = |z|^2$
 
 ### Multiplicative inverse
 
 
 ### Multiplicative inverse
 

-$z^{-1} = {1 \over z} = {{a-bi} \over {a^2+B^2}} = {\overline{z} \over {|z|^2}}$
+\begin{equation}\begin{split}z^{-1} & = {1 \over z} \\ & = {{a-bi} \over {a^2+B^2}} \\ & = {\overline{z} \over {|z|^2}}\end{split}\end{equation}

 
 ### Dividing complex numbers
 
 
 ### Dividing complex numbers
 

-${{z_1} \over {z_2}} = {{z_1\ {z_2}^{-1}}} = {{z_1 \overline{z_2}} \over {{|z_2|}^2}}$
+$${{z_1} \over {z_2}} = {{z_1\ {z_2}^{-1}}} = {{z_1 \overline{z_2}} \over {{|z_2|}^2}} \quad \text{multiplicative inverse}$$

 
 (using multiplicative inverse)
 
 
 (using multiplicative inverse)
 


@@ -97,20 +106,18 @@ ${z_1 \over z_2} = {{(a+bi)(c-di)} \over {c^2+d^2}}$
 ## Argand planes
 
 - Geometric representation of $\mathbb{C}$
 ## Argand planes
 
 - Geometric representation of $\mathbb{C}$

-- Horizontal $= \operatorname{Re}(z)$; vertical $= \operatorname{Im}(z)$
+- horizontal $= \operatorname{Re}(z)$; vertical $= \operatorname{Im}(z)$

 - Multiplication by $i$ results in an anticlockwise rotation of $\pi \over 2$
 
 - Multiplication by $i$ results in an anticlockwise rotation of $\pi \over 2$
 

-## Solving complex quadratics
-
-To solve $z^2+a^2=0$ (sum of two squares):
+## Solving complex polynomials

 
 

-$z^2+a^2=z^2-(ai)^2=(z+ai)(z-ai)$
+**Include $\pm$ for all solutions, including imaginary**

 
 

-*Must include $\pm$ in solutions*
+## Solving complex quadratics

 
 

-## Solving complex polynomials
+To solve $z^2+a^2=0$ (sum of two squares):

 
 

-Include $\pm$ for all solutions, including imaginary.
+$$z^2+a^2=z^2-(ai)^2=(z+ai)(z-ai)$$

 
 #### Dividing complex polynomials
 
 
 #### Dividing complex polynomials
 


@@ -124,13 +131,13 @@ Let $\alpha \in \mathbb{C}$. Remainder of $P(z) \div (z - \alpha)$ is $P(\alpha)
 
 ## Conjugate root theorem
 
 
 ## Conjugate root theorem
 

-If $a+bi$ is a solution to $P(z)=0$, with $a, b \in \mathbb{R}$, the the conjugate $a-bi$ is also a solution.
+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.

 
 ## Polar form
 
 
 ## Polar form
 

-$$\begin{equation}\begin{split}z & =r \operatorname{cis} \theta \\ & = r(\operatorname{cos}\theta+i \operatorname{sin}\theta) \\ & = a + bi \end{split}\end{equation}$$
+\begin{equation}\begin{split}z & =r \operatorname{cis} \theta \\ & = r(\operatorname{cos}\theta+i \operatorname{sin}\theta) \\ & = a + bi \end{split}\end{equation}

 
 

-- $r=|z|$, given by Pythagoras ($r=\sqrt{\operatorname{Re}(z)^2 + \operatorname{Im}(z)^2}$)
+- $r=|z|=\sqrt{\operatorname{Re}(z)^2 + \operatorname{Im}(z)^2}$

 - $\theta=\operatorname{arg}(z)$ (on CAS: `arg(a+bi)`)
 - **principal argument** is $\operatorname{Arg}(z) \in (-\pi, \pi]$ (note capital $\operatorname{Arg}$)
 
 - $\theta=\operatorname{arg}(z)$ (on CAS: `arg(a+bi)`)
 - **principal argument** is $\operatorname{Arg}(z) \in (-\pi, \pi]$ (note capital $\operatorname{Arg}$)
 


@@ -151,17 +158,22 @@ ${z_1 \over z_2} = {r_1 \over r_2} \operatorname{cis}(\theta_1-\theta_2)$ (divid
 
 ## de Moivres' Theorem
 
 
 ## de Moivres' Theorem
 

-$(r\operatorname{cis}\theta)^n=r^n\operatorname{cis}(n\theta)$ where $n \in \mathbb{Z}$
+$$(r\operatorname{cis}\theta)^n=r^n\operatorname{cis}(n\theta) \text{ where } n \in \mathbb{Z}$$

 
 ## Roots of complex numbers
 
 
 ## Roots of complex numbers
 

-$n$th roots of $r \operatorname{cis} \theta$ are:  
-$z={r^{1 \over n}} \cdot (\cos ({{\theta + 2k \pi} \over n}) + i \sin ({{\theta + 2 k \pi} \over n}))$
+$n$th roots of $z = r \operatorname{cis} \theta$ are
+
+$$z={r^{1 \over n}} \operatorname{cis}({{\theta + 2 k \pi} \over n})$$

 
 Same modulus for all solutions. Arguments are separated by ${2 \pi} \over n$
 
 
 Same modulus for all solutions. Arguments are separated by ${2 \pi} \over n$
 

+The solutions of $z^n=a \text{ where } a \in \mathbb{C}$ lie on circle
+
+$$x^2 + y^2 = (|a|^{1 \over n})^2$$
+

 ## 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 $|z-(a + bi)| = c$
 ## 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:** $\operatorname{Arg}(z) \lt \theta$
+- **Locus:** $\operatorname{Arg}(z) < \theta$