Tetration

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Review of Exponentiation

Exponentiation is the process of raising a base to a power, commonly denoted x^y. There are many ways of saying this, the most common form is: "x to the y" which comes from the longer way of saying it when y is an integer n: "x to the n-th power."

In the beginning, exponentiation was defined as iterated multiplication. Iteration is the process of repeating a function several times, so this would be a function x*x*x*...*x where there are n x's. This forms the basis of exponentiation for integers, but what about real numbers? To define exponentiation for real numbers, mathematicians had to find certain rules that were satisfied by integer exponentiation, and use them to generalize it to a continuously iterated multiplication. The rules that were found centuries ago are now known as the laws of exponentiation:

(xm)(xn) = x(m + n)	additive property of powers
(xm)n = x(mn)	multiplicative property of powers

These laws allow exponentiation to be extended to rational numbers fairly easily, and from there to real numbers by choosing rational numbers that are closer and closer to a real number you want to use. For example, if you wanted to find x^(3/5), then you could use the multiplicative property, and notice that: (x^(3/5))⁵ = x³. So x^(3/5) is the value y that satisfies: y⁵ = x³, because (x^(3/5))⁵ = x^(3/5)*5 = x³.

Extending the definition of exponentiation to real numbers allows inverse functions of exponentiation, namely roots and logarithms. Eventually, after the investigation of logarithms, the same rules are found to apply to them as well. These rules have become known as the laws of logarithms:

logx(mn) = logx(m) + logx(n)	additive property of logarithms
logx(mn) = n logx(m)	multiplicative property of logarithms

These properties form the basis of exponential and logarithmic arithmetic and algebraic manipulations. There are sometimes two additional laws that are based on each of the two above, but they are more of a short-cut, because they can be derived from the properties above:

(xm)/(xn) = x(m - n)	subtractive property of powers
logx(m/n) = logx(m) - logx(n)	subtractive property of logarithms

Under Construction.

Under Construction.

Introduction to Tetration

Tetration is the analytic iteration of an exponential function evaluated at one.

         What is the analytic iteration of an exponential function you may ask? Good question. Lets break that down into the words that compose it. For when its constituent words are understood, it is easy to understand the whole phrase:

         Analytic is a fancy way of saying infinitely smooth, which is also a special case of the term continuous. Continuous things are usually contrasted with Discrete things, and both are analogous to the roles in mathematics that the real numbers and the integers play respectively. The integers are discrete because you can't go between them without "jumping". With real numbers on the other hand, you can go between any number by going through other real numbers, which makes the real numbers continuous.

         Iteration is the process of applying a function to itself. In order for this to happen, the output of a function must be the same kind of thing as its input. When you iterate a function n times, you are giving its output to its input n times. In mathematical notation:

f n(z) = f · f · f · ... · f · z   (with n  f s)

         where "·" represents composition, or equivalently:

f 0(z) = z
f 1(z) = f (z)
f n(z) = f ( f n−1(z))
f n(z) = f n−1( f (z))

         By nature, iteration is discrete, because you can only apply a function once, not half a time. The only way you can go between iterations is by "jumping" between them. This makes iteration discrete. The definition of tetration uses the word continuous to mean a different kind of iteration that isn't discrete. Continuous iteration would go between iterations without "jumping" so you can find the iterates between the usual ones. So it is understood that continuous iteration is a generalization of discrete iteration.

         Exponentiation is the process of taking a base and raising it to a power. Usually the following names are used for different ways of doing this, depending on which variable you are iterating with respect to:

pown(z) = zn	a power function
expb(z) = bz	an exponential function

         Both of which are forms of exponentiation, but taking one to a power isn't very interesting, because you always get one in the end. Iterated exponentiation then, would be the iteration of an exponential function a certain number of times. Applying all of this to the definition of tetration we understand iterated exponential to be the function:

expn(z) = e^e^e^...^z   (with n e's)	using the base of the natural logarithm e, or
expnb(z) = b^b^b^...^z   (with n b's)	using any base b

         This function is only a discrete iteration, though, and we want continuous iteration, so we must somehow generalize this definition to get analytic iterated exponentials. Once we have, we can apply the definition, and the definition of tetration says to evaluate the "analytic iteration of an exponential function" at 1, so evaluating this function at z = 1 gives tetration:

yx = expyx(1)

where f ⁿ(z) = f(f ^{n - 1}(z)) and exp_x(a) = x^a and exp^y_x(a) = (exp_x)^y(a)

About Association

         Care must be taken when performing iteration. The difference between power functions and exponential functions manifest in their vastly different iterates. If you don't know which one you're using, for example, if you wanted to do "3^3^3", make sure you put your parentheses in the correct place, and evaluation will become obvious:

((3)^3)^3 = 27^3 = 19683	the "bottom-up" method
3^(3^(3)) = 3^27 = 7625597484987	the "top-down" method

         The first way of doing it, or the "bottom-up" method, which is not an iterated exponential function, but an iterated power function, has a continuous iterate which is simply expressible in terms of exponentiation. There will be more about this in the hyper-operator page. When we talk about tetration, we are only talking about iterated exponentials, or the "top-down" method, because this makes a new operation. Of these two operations, iterated powers can be expressed by using the notation we used above, and iterated exponentials can be expressed using a new notation used on this website as:

powcb(a) = a^b^c	iterated powers
expcb(a) = cba	iterated exponentials

The reason why this notation was chosen for iterated exponentials is due to three primary motivations:

The yx notation has been around since 1901	(see history)
Assigning a = 1 in cba gives cb1 = cb	standard tetration notation
Assigning c = 1 in cba gives 1ba = ba	standard exponentiation notation

         As you can see, assigning one to either of the superscripts, makes them disappear, and "collapse" to earlier notations. It is this feature that makes this notation very natural, and easy to understand. The only difficulty that could come about is in being careful with parentheses, because depending on where you put the parentheses, you could get very different expressions, because in general:

cba ≠ c(ba)
cba ≠ (cb)a

So take care when using this notation, and when necessary, use parentheses.

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