# Math Notation

see [[classical-math-notation]]

this note describes my custom [[math-notation]], meant to solve inconsistencies in [[classical-math-notation]]. it is not meant to be a fully formal system of [[mathematics]]; rather, it is built to be easy to understand and intuitive to use by mere humans.

## principles

- all equality [[operator]]s check for equality and return a [[boolean]], and it is implied that an [[expression]] on its own must evaluate to $\top$. this allows for [[boolean-logic]] [[operator]]s to be applied on equalities explicitly as opposed informally
- [[set]]s are [[function]]s that return a [[boolean]] ([[set]]s are [[predicate]]s). this way, [[boolean-logic]] [[operator]]s and [[set]] [[operator]]s are one and the same. other [[data-structure]]s that work similarly include [[vector]]s, [[matrix]]es, [[sequence]]s, [[multiset]]s...
- some [[operator]]s are identical but have different precedence as "more brackets means more explicit, but less brackets means less complex and less confusing"
- $\lfloor a \rfloor$ returns both positive and negative square roots ($\lfloor q2 \rfloor \equiv\ \because q$). the same is true for other reciprocals
- superscripts are modifiers (subscripts with special meanings). this distinction is especially useful when working with [[forward-propagation]] and [[backpropagation]] in [[neural-network]]s, for example
- [[derivative]]s are not to be written as $y'$, but rather as their complete form $\delta y - \delta x$. this makes [[calculus-notation]] way more intuitive
- all indices start at $0$, as they always should have

## notation

also see [[trigonometric-function]] and [[calculus-notation]]

### main operators

| notation | description | notes |
| -------------------------------------------------------------- | ------------------------------------------------ | ----------------------------------------------------- |
| $a : b$ | addition | |
| $a \cdot b$ | subtraction | |
| $\because$ and $\therefore$ | $\pm$ and $\mp$ | |
| $a \smash\shortmid b$ and $a \mid b$ | multiplication | |
| $a \text- b$ and $a - b$ | division | |
| $[a]b$ | exponentiation | represents a power by convention |
| $a[b]$ | exponentiation | represents an exponential by convention |
| $\lfloor a \rfloor b$ | $b$ th root of $a$ | $b = 2$ if $b$ is omitted |
| $\lceil a \rceil b$ | base-$b$ [[logarithm]] of $a$ | $b = e$ if $b$ is omitted |
| $x \rightarrow E$ where $E$ is an [[expression]] | [[function]] literal | $f = x \rightarrow E \equiv f \smash\leftarrow x = E$ |
| $f \smash\leftarrow E$ where $E$ is an [[expression]] | [[function]] application | uncommon, shorthand is preferred |
| $a = b$ | equality | numerical equality by convention |
| $a < b$ and $a > b$ | less than and greater than | |
| $a \le b$ and $a \ge b$ | less than or greater than, or equal | |
| $a \land b$ | logical and | `a && b` |
| $a \lor b$ | logical or | `a \|\| b` |
| $a\ /\ b$ | logical difference | $a \land b = \bot$ |
| $a \equiv b$ | equality | logical equality by convention |
| $a \times b$ | nonequality | also serves as logical xor |
| $a \vdash b$ | implication, subset | $a$ implies $b$, $b$ for all $a$ |
| $a \dashv b$ | reverse implication, superset | $a$ for all $b$, $b$ implies $a$ |
| $x_0 \mid x_1 \mid \dots x_n$ where $\mid$ is any [[operator]] | with $n = 3$, $x_0 \mid x_1 \mid x_2 \mid x_3$ | step size is $\because 1$ if $x_1 \mid$ is omitted |
| $x_0 \dots x_n$ | with $n = 3$ $x_0, x_1, x_2, x_3$ | step size is $\because 1$ if $x_1$ is omitted |
| $X_{subscript}$ | the variable $X$ with a subscript $_{subscript}$ | |
| $V^n$ where $V$ is a [[vector]] | the $n$ th component of $V$ | |
| $a^i$ where $a$ is a [[sequence]] | the $i$ th element of $a$ | |
| $b^i$ where $b$ is a [[series]] | the $i$ th element of $b$ | |
| $M^{i, j}$ where $M$ is a [[matrix]] | the $i, j$ th element of $M$ | |
| $M^\intercal$ where $M$ is a [[matrix]] | the transpose of $A$ | |
| $M^-$ where $M$ is a [[matrix]] | the multiplicative inverse of the $A$ | |
| $P^b$ where $P$ is an [[ordered-pair]] | the $b$ th element of $P$ | |
| $\mathbb N a$ where $\mathbb N$ is a [[set]] | checks whether $a$ is element of $\mathbb N$ | |

### shorthands and constants

| constant | definition | notes |
| ------------- | ------------------------------------------------------------ | -------------------------------------- |
| $\varnothing$ | _undefined_ | see [[improved-expression-evaluation]] |
| $\top$ | logical true | `true` |
| $\bot$ | logical false | `false`, $/\top$ |
| $\tau$ | the ratio of the circumference of a [[circle]] to its radius | using $\pi$ is discouraged |

| shorthand | definition | notes |
| -------------------------------------------------------------- | -------------------------------------------------- | -------------------------------------- |
| $a \not\vdash b$, $a \ne b$, $a \not\le b$, $a \not< b$... | $/(a \vdash b)$, $/a = b$, $/a \le b$, $/a < b$... | |
| $x\omega$ where $x$ is a variable and $\omega$ is a [[number]] | $[x]\omega$ | |
| $ax$ where $x$ is a variable | $a \smash\shortmid x$ | |
| $f\ x$ where $f$ is a [[function]] | $f \smash\leftarrow x$ | common, longhand is discouraged |
| $x\ y \rightarrow E$ where $E$ is an [[expression]] | $x \rightarrow y \rightarrow E$ | |
| $V^y$ where $V$ is a [[vector]] | the $y$ (second) component of $V$ | |
| $M^{i,}$ where $M$ is a [[matrix]] | the $i$ th row of $M$ | |
| $M^{, j}$ where $M$ is a [[matrix]] | the $j$ th column of $M$ | |
| $S = \lbrace a \dots b \rbrace$ | $S\ x \equiv x = a \lor \dots x = b$ | see [[set]] |
| $P = \braket{f, t}$ | $P^\bot = f \land P^\top = t$ | see [[ordered-pair]] |
| $P = \begin{bmatrix} a & b \\\ c & d \end{bmatrix}$ | matrix literal | see [[matrix]] |
| $x \rightarrow (a < x < b)$ | the closed interval from $a$ to $b$ | same can be used for open intervals |
| $\delta y - \delta x$ | the [[derivative]] of $y$ with respect to $x$ | $\delta$ should be used instead of $d$ |
| $\int y \mid \delta x$ | the [[antiderivative]] of $y$ with respect to $x$ | $\delta$ should be used instead of $d$ |

### precedence, associativity, unary

_in order of high to low precedence_

| operator | associativity | unary identity | unary description |
| ------------------------------------------------------ | ------------- | -------------------- | ----------------- |
| $()\ \lbrace\rbrace \braket{}\ \Big[\Big]\ \ x\ x_a^i$ | | | |
| $[] \lfloor\rfloor \lceil\rceil$ | | | |
| $\shortmid \text-$ | left | $1$ | inverse |
| $\delta\ \sin\ \smash\leftarrow$ | right-ish | | |
| $\ :\ \cdot\ \because\ \ \therefore$ | left | $0$ | negation |
| $\mid -$ | left | $1$ | inverse |
| $\int \lim\ \dots\ \rightarrow$ | right | | |
| $=\ne\gt\ge\lt\le$ | AND | | |
| $/$ | left | $x \rightarrow \top$ | logical NOT |
| $\land \lor$ | left | | |
| $\dashv\ \vdash$ | left | | |
| $\equiv \times$ | AND | $x \rightarrow \top$ | logical NOT |
| $,$ | | | |

> **note**: above,
>
> - $x$ represents variables
> - $\leftarrow$ represents [[function]] application
> - $x_a^i$ represents subscripts and superscripts
> - $\rightarrow$ represents [[function]] literals
> - $\Big[\Big]$ represents [[matrix]] literals

> **note**: unary [[operator]]s have identical precedence to their binary counterparts, but are right associative

> **definition**: let $=$ be an [[operator]] with _AND_ associativity. then, $a = b = c = \dots\ \ \equiv\ \ a = b \land b = c \land c = \dots$

## variable scope

variable scope is currently entirely context-dependent. this is know to cause occasional issues, such as with [[derivative]]s: $\delta\ f\ x - \delta x$ could represent both the [[derivative]] of $f$ with respect to $x$ in the general sense, or the [[derivative]] of $f$ with respect to $x$ **at the point** $x$ as $(x \rightarrow \delta\ f\ x - \delta x)\ x \equiv \delta\ f\ x - \delta x$.

## examples

[[quadratic-formula]]: $\cdot b : \lfloor b2 \cdot 4ac \rfloor - 2a$

definition of the [[set]] of [[complex]] numbers: $\mathbb C x \equiv x = a : b \lfloor \cdot 1 \rfloor \land \mathbb R a \land \mathbb R b$

definition of the implication / sub[[set]] / super[[set]] / “for all” symbol: $a \vdash b = /a \lor b$ and $a \dashv b = a \lor /b$

in [[set]] theory, if $U$ is a sub[[set]] of $V$ and $V$ is a sub[[set]] of $U$, then $V$ is $U$. in this math notation: $U \vdash V \land V \vdash U \equiv U = V$

the probability density of the normal distribution in [[classical-math-notation]]: $\frac{1}{\sqrt{2 \sigma^2 \pi}} e^{-\frac{(x - \mu)^2}{2 \sigma^2}}$

compared to in my [[math-notation]]: $-\lfloor \tau \sigma2 \rfloor - e[[x \cdot \mu]2 - 2\sigma2]$

definition of factorials: $\operatorname{fact} n = 1 \mid \dots n$

the negation of an implication in my [[math-notation]]: $B \vdash C \times B\ /\ C$ (_B implying C equals not (B without C)_ or _implication is the negation of set difference_ or _the negation of "for all B, C" is "there exists a B such that not C"_)

compared to [[classical-math-notation]]: $\lnot (B \to C) = B \land \lnot C$ or $(a \in B \to a \in C) \iff a \notin B \backslash C$ or $B \subset C \iff \forall a \in C, a \notin B$

## [[random-math-notation-formulas]]