${\ displaystyle a + (b + 1) = 1 + \ left (a + b \ right)}$

${\ displaystyle a \ cdot (b + 1) = a + \ left (a \ cdot b \ right)}$

${\ displaystyle a ^ {(b + 1)} = a \ cdot \ left (a ^ {b} \ right)}$

define a three-digit operator recursively (with )
${\ displaystyle a, b, n \ geq 0}$

${\ displaystyle a ^ {(n)} b: = {\ begin {cases} b + 1, & {\ text {if}} n = 0 \\ a, & {\ text {if}} n = 1, b = 0 \\ 0, & {\ text {if}} n = 2, b = 0 \\ 1, & {\ text {if}} n> 2, b = 0 \\ a ^ {(n-1 )} \ left (a ^ {(n)} (b-1) \ right) & {\ text {otherwise}} \ end {cases}}}$

and introduces the following terms:

${\ displaystyle \ operatorname {hyper} {\ mathit {n}} (a, b) = \ operatorname {hyper} (a, n, b) = a ^ {(n)} b.}$

(It should be noted with this notation that the spelling of and does not represent a multiplication, i.e. every actually occurring multiplication with the explicit operator must be noted. Likewise, there is no exponentiation. The use of the notation , on the other hand, rules out such possible confusion.
${\ displaystyle a ^ {(n)}}$${\ displaystyle b}$${\ displaystyle \ cdot}$${\ displaystyle a ^ {(n)}}$${\ displaystyle \ operatorname {hyper} (a, n, b)}$

Thus hyper1 is the addition , hyper2 the multiplication and hyper3 the exponentiation . hyper4 is also known as tetration or superpotency and can be noted as follows:

More generally understandable one could also say: Write the number - times in a row and insert the operator one step lower in between.
${\ displaystyle a}$${\ displaystyle b}$

The family has been expanded for not for real numbers because there are several “obvious” ways to do this, but they are not associative .
${\ displaystyle n> 3}$

Another notation uses the symbol instead of the arrow . With the definition, the following applies
${\ displaystyle \ uparrow}$${\ displaystyle {\ hat {\ hbox {}}}}$

${\ displaystyle a \ underbrace {\ uparrow \ dotsb \ uparrow} _ {n {\ mbox {mal}}} b = a \ underbrace {{\ hat {\ hbox {}}} \ dotsb {\ hat {\ hbox { }}}} _ {n {\ mbox {mal}}} b = \ operatorname {hyper} (a, n + 2, b) = a ^ {(n + 2)} b}$.

This notation is used to represent very large numbers such as Graham's number .

Another extension

There is another way to get a more general definition of the link from the specifications, because it also applies

${\ displaystyle \, a + b = (a + (b-1)) + 1}$

${\ displaystyle a \ cdot b = (a \ cdot (b-1)) + a}$

${\ displaystyle a ^ {b} = \ left (a ^ {(b-1)} \ right) \ cdot a}$,

because the links are + and commutative . This gives the definition
${\ displaystyle \ cdot}$

${\ displaystyle a _ {(n)} b: = {\ begin {cases} a + b, & {\ text {if}} n = 1 \\ 0, & {\ text {if}} n = 2, b = 0 \\ 1, & {\ text {if}} n> 2, b = 0 \\\ left (a _ {(n)} (b-1) \ right) _ {(n-1)} a, & {\ text {other}} \ end {cases}}}$

However, this notation "collapses" for ; In contrast to hyper4, it no longer results in a power tower:
${\ displaystyle n = 4}$

${\ displaystyle a _ {(4)} b = a ^ {\ left (a ^ {(b-1)} \ right)}}$

How can and suddenly differ for? This is due to the associativity, a property that the operators and have (see also body ), but which the power operator lacks. (Generally is .)
${\ displaystyle a ^ {(n)} b}$${\ displaystyle a _ {(n)} b}$${\ displaystyle n> 3}$${\ displaystyle +}$${\ displaystyle \ cdot}$${\ displaystyle a ^ {b ^ {c}} = a ^ {(b ^ {c})} \ neq (a ^ {b}) ^ {c} = a ^ {b \ cdot c}}$

The other levels do not collapse in this way, which is why this family of operators, called "lower hyper-operators", is also of interest.