Hyper operator

The hyper operator is a continuation of the conventional mathematical operators of addition , multiplication and exponentiation . It is used to briefly display large numbers such as power towers .

${\displaystyle \operatorname {hyper} {\mathit {n}}(a,b)=\operatorname {hyper} (a,n,b)=a^{(n)}b=a\uparrow ^{n-2}b.}$

Derivation of the notation

Based on the observations

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

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

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

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:

${\displaystyle \operatorname {hyper4} (a,b)={}^{b}a}$.

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 . ${\displaystyle n>3}$

Knuth's arrow notation

→ Main article : Arrow notation

Another notation for the hyperoperator was developed by Donald Knuth , which is known as arrow notation. The definition is

${\displaystyle a\underbrace {\uparrow \dotsb \uparrow } _{k{\mbox{ mal}}}b:=\left\{{\begin{matrix}a^{b}&{\mbox{falls }}k=1\\\underbrace {a\underbrace {\uparrow \dotsb \uparrow } _{k-1{\mbox{ mal}}}a\underbrace {\uparrow \dotsb \uparrow } _{k-1{\mbox{ mal}}}\dotsb \underbrace {\uparrow \dotsb \uparrow } _{k-1{\mbox{ mal}}}a} _{b{\mbox{ mal}}}&{\mbox{sonst}}\end{matrix}}\right.}$

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 ${\displaystyle \cdot }$

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

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.

Examples

addition

${\displaystyle 3^{(1)}3=3+3=6.}$

multiplication

${\displaystyle 3^{(2)}3=3\cdot 3=3^{(1)}3^{(1)}3=3+3+3=9.}$

Exponentiation

${\displaystyle 3^{(3)}3=3^{3}=3^{(2)}3^{(2)}3=3\cdot 3\cdot 3=27.}$

Tetration

${\displaystyle {\begin{aligned}3^{(4)}3&=3^{3^{3}}\\&=3^{(3)}(3^{(4)}2)\\&=3^{(3)}(3^{(3)}(3^{(4)}1))\\&=3^{(3)}(3^{(3)}(3^{(3)}(3^{(4)}0)))\\&=3^{(3)}(3^{(3)}(3^{(3)}1))\\&=3^{3^{3^{1}}}\\&=3^{27}\\&=7.625.597.484.987.\end{aligned}}}$

It should be noted here that this applies, see also the power tower . ${\displaystyle 3^{3^{3}}=3^{(3^{3})}}$

Weblinks (English)

• What Lies Beyond Exponentiation?
• Lynz and the Clarkkkkson
• The Tetration Forum