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 .
![\ operatorname {hyper} {\ mathit {n}} (a, b) = \ operatorname {hyper} (a, n, b) = a ^ {{(n)}} b = a \ uparrow ^ {{n- 2 B.](https://wikimedia.org/api/rest_v1/media/math/render/svg/146aa27297845b3f198f35a05624f30fc01de83c)
Derivation of the notation
Based on the observations
![{\ displaystyle a + (b + 1) = 1 + \ left (a + b \ right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/642099b4c1fe9227b57e2d688eab17a9d2694d73)
![{\ displaystyle a \ cdot (b + 1) = a + \ left (a \ cdot b \ right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eecb812df4e7bbec4f9f0a2e4bc88098e6e849d5)
![{\ displaystyle a ^ {(b + 1)} = a \ cdot \ left (a ^ {b} \ right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c074e9fea9fb3b582ae0d2647f265aac5422733)
define a three-digit operator recursively (with )
![a, b, n \ geq 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/df61c6101bff2871b9b4861e1a148a04eb125698)
![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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c488f35da0c446885685b0bd35c0f201cd094c6)
and introduces the following terms:
![\ operatorname {hyper} {\ mathit {n}} (a, b) = \ operatorname {hyper} (a, n, b) = a ^ {{(n)}} b.](https://wikimedia.org/api/rest_v1/media/math/render/svg/67ac88a4b7d8b81adf6e88d444689678b80728f0)
(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)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9f97413c89fb0650588ccad7d2662602a1bc9946)
![b](https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3)
![\ cdot](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba2c023bad1bd39ed49080f729cbf26bc448c9ba)
![{\ displaystyle a ^ {(n)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9f97413c89fb0650588ccad7d2662602a1bc9946)
![{\ displaystyle \ operatorname {hyper} (a, n, b)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/45e439878dbdd346e5362b2b8f519801b7cc64c0)
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.
![b](https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3)
The family has been expanded for not for real numbers because there are several “obvious” ways to do this, but they are not associative .
![n> 3](https://wikimedia.org/api/rest_v1/media/math/render/svg/257030caae597fd034c2cbcff2cff9dfc4272d20)
Knuth's arrow notation
Another notation for the hyperoperator was developed by Donald Knuth , which is known as arrow notation. The definition is
![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 {otherwise}} \ end {matrix}} \ right.](https://wikimedia.org/api/rest_v1/media/math/render/svg/0bfd858bcc6f319834726829361fcbaf679c5966)
Another notation uses the symbol instead of the arrow . With the definition, the following applies
![\ uparrow](https://wikimedia.org/api/rest_v1/media/math/render/svg/ddb20b28c74cdaa09e1f101d426441da1996072f)
![{\ hat {{\ hbox {}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a073faff608f3fddf7633fc3f64c3cf91a958e40)
-
.
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
![\, a + b = (a + (b-1)) + 1](https://wikimedia.org/api/rest_v1/media/math/render/svg/f1fe7a3486dbf18243fbd5b83681e3f51520e3c9)
![a \ cdot b = (a \ cdot (b-1)) + a](https://wikimedia.org/api/rest_v1/media/math/render/svg/232936a7998431422dc33aab2fe8d7f1b9eed525)
-
,
because the links are + and commutative . This gives the definition
![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 {otherwise}} \ end {cases}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/610d6dd587b59e8267e2c4ac647858c4eb27e9a7)
However, this notation "collapses" for ; In contrast to hyper4, it no longer results in a power tower:
![n = 4](https://wikimedia.org/api/rest_v1/media/math/render/svg/d928ec15aeef83aade867992ee473933adb6139d)
![a _ {{(4)}} b = a ^ {{\ left (a ^ {{{(b-1)}} \ right)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bd8be625a2d0e401eb294bed9eb56cd423a016ab)
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 .)
![a ^ {{(n)}} b](https://wikimedia.org/api/rest_v1/media/math/render/svg/8d2f7a9035956b08c3e81f80768ab2094c3c43b5)
![a _ {{(n)}} b](https://wikimedia.org/api/rest_v1/media/math/render/svg/7bc9a53cac86349b61dd643f41930209394c6c2c)
![n> 3](https://wikimedia.org/api/rest_v1/media/math/render/svg/257030caae597fd034c2cbcff2cff9dfc4272d20)
![+](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe6ef363cd19902d1a7a71fb1c8b21e8ede52406)
![\ cdot](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba2c023bad1bd39ed49080f729cbf26bc448c9ba)
![a ^ {{b ^ {c}}} = a ^ {{(b ^ {c})}} \ neq (a ^ {b}) ^ {c} = a ^ {{b \ cdot c}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/73bb7c318f1b69c66584853f5193cceb396c1727)
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
multiplication
Exponentiation
Tetration
It should be noted here that this applies, see also the power tower .
![3 ^ {{3 ^ {{3}}}} = 3 ^ {{(3 ^ {{3}})}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/206d6db033c412c8a6189ca1e8c1318f2b30a9ae)
Weblinks (English)