> For the complete documentation index, see [llms.txt](https://mandober.gitbook.io/math-debrief/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://mandober.gitbook.io/math-debrief/501-number-theory/615-arithmetic/arithmetic-ops2.md).

# Arithmetic operations

Operations:

* increment
* decrement
* addition
* subtraction
* multiplication
* divisions
* integer division
* modulo
* exponentiation
* square
* cube
* roots
* logarithm

Hyper ops:

* level0 Successor
* level1 Addition
* level2 Multiplication
* level3 Exponentiation
* level4 Tetration
* Pentation
* Hexation
* etc.

∞ ≈ ≠ ± × ÷ · √ ∈ ∉

increment: S(a), a+1, a+(1)

* decrement,    P(a), a-1, a+(-1)
  * addition,       a+b,  (a)+(b)
* subtraction,  a-b,  (a)+(b)

S(a), a+1, a+(1)

* inc/dec = (±a) + (±1)
* add/sub = (±a) + (±b) = (±b) + (±a)
* mul/div = (±a) × (±b) = 

## Operations and their inverses

1. increment vs decrement
2. addition vs subtraction
3. multiplication vs division
4. exponentiation vs roots, logarithms
5. tetration (hyper-4) vs super-roots and super-logarithms
6. quintation/pentation, hexation, etc.

In mathematics, the super-logarithm is one of the two inverse functions of tetration. Just as exponentiation has two inverse functions, roots and logarithms, tetration has two inverses, super-roots and super-logarithms.

$$
\begin{align}
\text{increment}:       & \quad n+1 \quad & decrement n-1 \\
\text{addition}:        & \quad n+x \\
\text{multiplication}:  & \quad n\*x \\
\text{exponentiation}:  & \quad n^x \\
\text{tetration}:       & \quad ^xn
\end{align}
$$

## Arithmetic operations

**Addition**

$$
{
\scriptstyle
\left.{
\begin{matrix}
\text{summand} + \text{summand}    \\
\scriptstyle {
\text{addend (broad sense)}
} + \text{addend (broad sense)
}    \\
\scriptstyle {\text{augend}},+,{\text{addend (strict sense)}}
\end{matrix}
}
\right
} = } \text{sum}
$$

**Subtraction**

$$
\text{minuend} - \text{subtrahend} = \text{difference}
$$

**Multiplication** (×)

$$
{\scriptstyle
\left.{
\begin{matrix}
\scriptstyle {\text{factor}},\times ,{\text{factor}}\\\scriptstyle {\text{multiplier}},\times ,{\text{multiplicand}}
\end{matrix}
}
\right
} = } \scriptstyle {\text{product}}
$$

**Division** (÷)

$$
{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\frac {\scriptstyle {\text{dividend}}}{\scriptstyle {\text{divisor}}}}\\\scriptstyle {\text{ }}\\\scriptstyle {\frac {\scriptstyle {\text{numerator}}}{\scriptstyle {\text{denominator}}}}\end{matrix}}\right},=,} {\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\frac {\scriptstyle {\text{dividend}}}{\scriptstyle {\text{divisor}}}}\\\scriptstyle {\text{ }}\\\scriptstyle {\frac {\scriptstyle {\text{numerator}}}{\scriptstyle {\text{denominator}}}}\end{matrix}}\right},=,}    {\displaystyle {\begin{matrix}\scriptstyle {\text{fraction}}\\\scriptstyle {\text{quotient}}\\\scriptstyle {\text{ratio}}\end{matrix}}} {\displaystyle {\begin{matrix}\scriptstyle {\text{fraction}}\\\scriptstyle {\text{quotient}}\\\scriptstyle {\text{ratio}}\end{matrix}}}
$$

**Exponentiation**

* base, exponent, power

$$
\text{base}^{\text{exp}} = \text{power}
$$

**n-th root**

* symbol: `√`
* latex: `\sqrt[d]{r}`
* operands:
  * degree, `d`
  * radicand, `r`
  * root, `n`
* common roots:
  * n: nth root
  * n=2: square root
  * n=3: cubic root

$$\displaystyle{ \sqrt\[d]{r} = n }$$

**Logarithm** $$log$$

$$\displaystyle{ \log\_{base}({antilogarithm}) = \text{logarithm} }$$