Reforming Math

Here are my suggestions for making math notation better:

• Use \(\tau\) instead of \(\pi\) for the circle constant
• Instead of writing \(a_{12}\) to mean the edge between points \(1\) and \(2\) in graph theory, write \(a_{1\leftrightarrow2}\). Otherwise, it looks like indexing \(12\) of \(a\). When indexing a 2d-array, write \(a_{1,2}\) instead of \(a_{12}\). In a 100x100 array, \(a_{12}\) should return a 1x100 vector (analogous to a[12]), not a number (analogous to a[1][2])
• A big source of confusion in math is not knowing what letter variables stand for. The problem with saying \(\text{charge}=\text{capacitance}\times \text{voltage}\) is that it can be interpreted as \( c\times h\times a\times r\times g\times e=c\times a\times p\times a\times c\times i\times t\times a\times n\times c\times e\times v\times o\times l\times t\times a\times g\times e \). The solution is to write a line over multiletter variables (inspired by the usage of the vinculum as a grouping operator).
  Write \(\overline{\text{charge}}=\overline{\text{capacitance}}\ \overline{\text{voltage}}\)
  If you want start the equation with a multiletter variable and use single-letter variables later, you can use \(\rm as\), like so:
  \[\overline{\text{charge as Q}}=\overline{\text{capacitance as C}}\times\overline{\text{voltage as V}}\]\[V=\frac{Q}C\]
• \(\sin^2(x)\) should mean \(\sin(\sin(x))\) and \(sin^{-1}(x)\) should mean \(arcsin(x)\) whereas
  \(\sin(x)^2\) should mean \(\sin(x)\cdot \sin(x)\) and \(\sin(x)^{-1}\) should mean \(\frac1{\sin(x)}\)
• Problem: \((1, π)\) can be a cartesian or polar coordinate.
  Solution: make polar coordinates explicitly polar with a \(^p\) superscript, such as \((1, π)^p\). You only have to do this if it is unclear.
• Use == when writing a proof, so that you remember that you cannot manipulate both sides like an equation. For example, prove \(\frac{\cos 2x-1}{\sin 2x}==-\tan x\)

Reforming Physics

• Make an electron positive charge
• Use moles of e as a unit of charge instead of coulombs C because that way it is easier to visualize as opposed to being a completely arbitrary unit.
• Do not use \frac{1}{4\pi\epsilon_0}