> 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/hyperops.md).

# hyperops

## Index of arithmetic operations

### The elementary arithmetic operations

* The four elementary operations
  * Addition
  * Subtraction
  * Multiplication
  * Division

```
Addition:        addend       + addend     = sum
Subtraction:     subtrahend   - minuend    = difference
Multiplication:  multiplicand × multiplier = product
Multiplication:  factor × factor = product
Division:        denominator  ÷ divisor    = (quotient, remainder)
```

## Hyperoperations

### hyper-1, zeration (successor):

$$
S(0) = 1   \\
S(n) = n \cup {n}  \\
S(n) = S \circ S^{\ \circ\ n}   \\
S^{\ \circ\ n} = S\_{n}(...(S(0))) = n    \\
\ \\
P(0) = 0  \\
P(1) = 0  \\
0 = P^{\ \circ\ n}\quad \\
S(n) = S^{\ \circ\ P(n)}
\ \\
1 = S(0) \\
2 = S^{\ \circ\ 2} = S \circ S = S(S(0)) = S(1) = 2  \\
3 = S^{\ \circ\ 3} = S \circ S \circ S = S(S(S(0))) = S(S(1)) = S(2) = 3  \\
n = S^{\ \circ\ n} = S\_{n}(...(S(0)))
$$

$$
\displaystyle{
1 + \overbrace{4+3+5}^{a-ha!} \\
1 + \underbrace{4+3+5}\_{a-ha!}
}
$$

### hyper-1, addition

$$
x =
\begin{cases}
x > 0 : & 4 \\
x = 0 : & 2 \\
x < 0 : & 5
\end{cases}
$$

$$
m + 0 = m \\
m + S(n) = S(m + n)
$$

### hyper-2, multiplication

$$
m \times 0 = 0 \\
m^S{(n)} = n \times m^n
$$

### hyper-3, exponentiation

$$
m^0 = 0 \\
m^S{(n)} = n \times m^n
$$

### hyper-4, tetration

$$
m^0 = 0 \\
m^S{(n)} = n \times m^n
$$

### hyper-i, Hyperops

$$

$$