8.2 KiB
Hyperoperation
WARNING: brain exploding article
{ This article contains unoriginal research with errors and TODOs. ~drummyfish }
Hyperoperations are mathematical operations that are generalizations of the basic arithmetic operations of addition, multiplication, exponentiation etc. When we realize that multiplication is just repeated addition and exponentiation is just repeated multiplication, it is possible to continue in the same spirit and keep inventing new operations by simply saying that a new operation means repeating the previously defined operation, so we define repeated exponentiation, which we call tetration, then we define repeated tetration, which we call pentation, etc.
There are infinitely many hyperoperations as we can go on and on in defining new operations, however we start with what seems to be the simplest operation we can think of: the successor operation (we may call it succ, +1, ++, next, increment, zeration or similarly). In the context of hyperoperations we call this operation hyper0. Successor is a unary operator, i.e. it takes just one number and returns the number immediately after it (suppose we're working with natural numbers). In this successor is a bit special because all the higher operations we are going to define will be binary (taking two numbers). After successor we define the next operation, addition (hyper1), or a + b, as repeatedly applying the successor operation b times on number a. After this we define multiplication (hyper2), or a * b, as a chain of b numbers as which we add together. Similarly we then define exponentiation (hyper3, or raising a to the power of b). Next we define tetration (hyper4, building so called power towers), pentation (hyper5), hexation (hyper6) and so on (heptation, octation, ...).
Indeed the numbers obtained by high order hyperoperations grow quickly as fuck.
An important note is this: there are multiple ways to define the hyperoperations, the most common one seems to be by supposing the right associative evaluation, which is what we're going to implicitly consider from now on. This means that once associativity starts to matter, we will be evaluating the expression chains FROM RIGHT, which may give different results than evaluating them from left (consider e.g. 2^(2^3) != (2^2)^3).
The following is a sum-up of the basic hyperoperations as they are commonly defined (note that many different symbols are used for these operations throughout literature):
| operation | symbol | meaning | commutative | associative |
|---|---|---|---|---|
| successor (hyper0) | succ(a) |
next after a | ||
| addition (hyper1) | a + b |
succ(succ(succ(...a...))), b succs |
yes | yes |
| multiplication (hyper2) | a * b |
0 + (a + a + a + ...), b as in brackets |
yes | yes |
| exponentiation (hyper3) | a ^ b |
1 * (a * a * a * ...), b as in brackets |
no | no |
| tetration (hyper4) | a ^^ b |
1 * (a ^ (a ^ (a ^ (...), b as in brackets |
no | no |
| pentation (hyper5) | a ^^^ b |
1 * (a^^ (a^^ (a^^ (...), b as in brackets |
no | no |
| hexation (hyper6) | a ^^^^ b |
1 * (a^^^(a^^^(a^^^(...), b as in brackets |
no | no |
| ... | no more | no more |
The following ASCII masterpiece shows the number 2 in the territory of these hyperoperations:
2 +1 +1 +1 +1 +1 +1 +1 ... successor
| __/ ________/ / 9
| / / ______________/
| / / /
2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 ... addition
| | __/ / 16
| | / ____________________/
| | / /
2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 ... multiplication
| | __/ 16 32 64 128 256
| | /
| | /
2 ^ (2 ^ (2 ^ (2 ^ (2 ^ (2 ^ (2 ^ (2 ... exponentiation
| | __/ 65536
| | / not sure about arrows here, numbers get too big, TODO
| | /
2 ^^(2 ^^(2 ^^(2 ^^(2 ^^(2 ^^(2 ^^(2 ... tetration
| | |
| | | not sure about arrows here either
| | |
2 ^^^(2 ^^^(2 ^^^(2 ^^^(2 ^^^(2 ^^^(2 ^^^(2 ... pentation
... a lot
Some things generally hold about hyperoperations, for example for any operation f = hyperN where N >= 3 and any number x it is true that f(1,x) = 1 (just as raising 1 to anything gives 1).
Hyperroot is the generalization of square root, i.e. for example for tetration the nth hyperroot of number a is such number x that tetration(x,n) = a.
Here's a C implementation of hyperoperations including a general hyperN operation (however note that even with 64 bit ints numbers overflow very quickly):
#include <stdio.h>
#include <inttypes.h>
#include <stdint.h>
// hyper0
uint64_t succ(uint64_t a)
{
return a + 1;
}
// hyper1
uint64_t add(uint64_t a, uint64_t b)
{
for (uint64_t i = 0; i < b; ++i)
a = succ(a);
return a;
// return a + b
}
// hyper2
uint64_t multiply(uint64_t a, uint64_t b)
{
uint64_t result = 0;
for (uint64_t i = 0; i < b; ++i)
result += a;
return result;
// return a * b
}
// hyper(n + 1) for n > 2
uint64_t nextOperation(uint64_t a, uint64_t b, uint64_t (*operation)(uint64_t,uint64_t))
{
if (b == 0)
return 1;
uint64_t result = a;
for (uint64_t i = 0; i < b - 1; ++i)
result = operation(a,result);
return result;
}
// hyper3
uint64_t exponentiate(uint64_t a, uint64_t b)
{
return nextOperation(a,b,multiply);
}
// hyper4
uint64_t tetrate(uint64_t a, uint64_t b)
{
return nextOperation(a,b,exponentiate);
}
// hyper5
uint64_t pentate(uint64_t a, uint64_t b)
{
return nextOperation(a,b,tetrate);
}
// hyper6
uint64_t hexate(uint64_t a, uint64_t b)
{
return nextOperation(a,b,pentate);
}
// hyper(n)
uint64_t hyperN(uint64_t a, uint64_t b, uint8_t n)
{
switch (n)
{
case 0: return succ(a); break;
case 1: return add(a,b); break;
case 2: return multiply(a,b); break;
case 3: return exponentiate(a,b); break;
default: break;
}
if (b == 0)
return 1;
uint64_t result = a;
for (uint64_t i = 0; i < b - 1; ++i)
result = hyperN(a,result,n - 1);
return result;
}
int main(void)
{
printf("\t0\t1\t2\t3\n");
for (uint64_t b = 0; b < 4; ++b)
{
printf("%" PRIu64 "\t",b);
for (uint64_t a = 0; a < 4; ++a)
printf("%" PRIu64 "\t",tetrate(a,b));
printf("\n");
}
return 0;
}
The code prints a table for tetration:
0 1 2 3
0 1 1 1 1
1 0 1 2 3
2 1 1 4 27
3 0 1 16 7625597484987
Lower Hyperoperations
These are the hyperoperations we get when we consider left associativity instead of right associativity. TODO: more about this
2 +1 +1 +1 +1 +1 +1 +1 ... successor
| __/ ________/ / 9
| / / ______________/
| / / /
2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 ... addition
| | __/ / 16
| | / ____________________/
| | / /
2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 ... multiplication
| | __/ 16 32 64 128 / 256
| | / ____________________/
| | / /
(2 ^ 2) ^ 2) ^ 2) ^ 2) ^ 2) ^ 2) ^ 2 ... exponentiation (left associative)
| | __/ 256 ~3*10^38
| | / ____________________________
| | / /
(2 ^^ 2) ^^ 2) ^^ 2) ^^ 2) ^^ 2) ^^ 2) ^^ 2 ... weird4
| |
| | TODO: arrows
| |
(2 ^^^ 2)^^^ 2)^^^ 2)^^^ 2)^^^ 2)^^^ 2)^^^ 2 ... weird5
...