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

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