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hyperoperation.md
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{ This article contains unoriginal research with errors and TODOs, read at own risk. ~drummyfish }
Hyperoperations are [mathematical](math.md) operations that are generalizations/continuations 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.
Hyperoperations are [mathematical](math.md) operations that are generalizations/continuations of the basic arithmetic operations of addition, multiplication, exponentiation etc. Basically they're like the basic operations like plus but on steroids. 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](unary.md) operator, i.e. it takes just one number and returns the number immediately after it (suppose we're working with [natural numbers](natural_number.md)). 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 *a*s 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](power_tower.md)), pentation (*hyper5*), hexation (*hyper6*) and so on (heptation, octation, ...).