Editing Tetration (section)

== Introduction ==
The first four [[hyperoperation]]s are shown here, with tetration being considered the fourth in the series. The [[unary operation]] [[Successor function|succession]], defined as <math>a' = a + 1</math>, is considered to be the zeroth operation.

#[[Addition]] <math display="block">a + n = a + \underbrace{1 + 1 + \cdots + 1}_n</math> {{mvar|n}} copies of 1 added to {{mvar|a}} combined by succession.
#[[Multiplication]] <math display="block">a \times n = \underbrace{a + a + \cdots + a}_n</math> {{mvar|n}} copies of {{mvar|a}} combined by addition.
#[[Exponentiation]] <math display="block">a^n = \underbrace{a \times a \times \cdots \times a}_n</math> {{mvar|n}} copies of {{mvar|a}} combined by multiplication.
#Tetration <math display="block">{^{n}a} = \underbrace{a^{a^{\cdot^{\cdot^{a}}}}}_n</math> {{mvar|n}} copies of {{mvar|a}} combined by exponentiation, right-to-left.

Importantly, nested exponents are interpreted from the top down: {{tmath|a^{b^c} }} means {{tmath|a^{\left(b^c \right)} }} and not {{tmath|\left(a^b \right)^c.}}

Succession, <math>a_{n+1} = a_n + 1</math>, is the most basic operation; while addition (<math>a + n</math>) is a primary operation, for addition of natural numbers it can be thought of as a chained succession of <math>n</math> successors of <math>a</math>; multiplication (<math>a \times n</math>) is also a primary operation, though for natural numbers it can analogously be thought of as a chained addition involving <math>n</math> numbers of <math>a</math>. Exponentiation can be thought of as a chained multiplication involving <math>n</math> numbers of <math>a</math> and tetration (<math>^{n}a</math>) as a chained power involving <math>n</math> numbers <math>a</math>. Each of the operations above are defined by iterating the previous one;<ref name="uwu">Neyrinck, Mark. [http://skysrv.pha.jhu.edu/~neyrinck/extessay.pdf An Investigation of Arithmetic Operations.] Retrieved 9 January 2019.</ref> however, unlike the operations before it, tetration is not an [[elementary function]].

The parameter <math>a</math> is referred to as the '''base''', while the parameter <math>n</math> may be referred to as the '''height'''. In the original definition of tetration, the height parameter must be a natural number; for instance, it would be illogical to say "three raised to itself negative five times" or "four raised to itself one half of a time." However, just as addition, multiplication, and exponentiation can be defined in ways that allow for extensions to real and complex numbers, several attempts have been made to generalize tetration to negative numbers, real numbers, and complex numbers. One such way for doing so is using a recursive definition for tetration; for any positive [[real number|real]] <math>a > 0</math> and non-negative [[integer]] <math>n \ge 0</math>, we can define <math>\,\! {^{n}a}</math> recursively as:<ref name="uwu" />
: <math>{^{n}a} := \begin{cases} 1 &\text{if }n=0 \\ a^{\left(^{(n-1)}a\right)} &\text{if }n>0 \end{cases}</math>
The recursive definition is equivalent to repeated exponentiation for [[Natural number|natural]] heights; however, this definition allows for extensions to the other heights such as <math>^{0}a</math>, <math>^{-1}a</math>, and <math>^{i}a</math> as well – many of these extensions are areas of active research.