NationMaster - Encyclopedia: Hyper operator

(See Knuth's up-arrow notation and Conway chained arrow notation.) In mathematics, Knuths up-arrow notation is a notation for very large integers introduced by Donald Knuth in 1976. ... Jump to: navigation, search Conway chained arrow notation, created by mathematician John Conway, is a means of expressing certain extremely large numbers. ...

For n = 4 we have hyper4 or tetration, super-exponentiation or power towers in terms of an extension of standard operators: Tetration (also exponential map, hyperpower, power tower, super-exponentiation, and hyper4) is iterated exponentiation, the first hyper operator after exponentiation. ... In mathematics, an operator is some kind of function; if it comes with a specified type of operand as function domain, it is no more than another way of talking of functions of a given type. ...

$operatorname{hyper4} (a, b) = operatorname{hyper}(a, 4, b) = a ^ {(4)} b = {a uparrowuparrow b = atop { }} !!!!!!!{{underbrace{a^{a^{cdot^{cdot^{a}}}}}} atop {bmbox{ copies of }a}}=ato bto 2$

See also Tables of values. In mathematics, Knuths up-arrow notation is a notation for very large integers introduced by Donald Knuth in 1976. ...

1 Derivation of the notation
2 Evaluation from left to right
3 See also
4 External links

Derivation of the notation

It can be seen as an answer to the question "what's next in this sequence: summation (+), multiplication (×), exponentiation (^),…?" Noting that This is a page about mathematics. ... Addition is one of the basic operations of arithmetic. ... In its simplest form, multiplication is the sum of a list of identical numbers. ... In mathematics, exponentiation is a process generalized from repeated (or iterated) multiplication, in much the same way that multiplication is a process generalized from repeated addition. ...

$a + b = 1 + (a + (b - 1))$
$a times b = a + (a times (b - 1))$
$a ^ b = a times (a ^ {(b - 1)})$

recursively define an infix triadic operator (making n=0 correspond to the successor function): In mathematics, an operator is some kind of function; if it comes with a specified type of operand as function domain, it is no more than another way of talking of functions of a given type. ... A successor function is the label in the literature for what is actually an operation. ...

$a ^ {(n)} b= left{ begin{matrix} b+1, & mbox{if }n=0 a, & mbox{if }n=1,b=0 0, & mbox{if }n=2,b=0 1, & mbox{if }n>2,b=0 a ^ {(n-1)} ( a ^ {n} (b - 1)) & mbox{otherwise} end{matrix} right.$

then define $operatorname{hypermathit{n}} (a, b) = a ^ {(n)} b$ and $operatorname{hyper}(a, n, b) = a ^ {(n)} b$

This gives:

$operatorname{hyper1} (a, b) = operatorname{hyper}(a, 1, b) = a ^ {(1)} b = a+b$

$operatorname{hyper2} (a, b) = operatorname{hyper}(a, 2, b) = a ^ {(2)} b = ab$

$operatorname{hyper3} (a, b) = operatorname{hyper}(a, 3, b) = a ^ {(3)} b = a^b$

$operatorname{hyper4} (a, b) = operatorname{hyper}(a, 4, b) = a ^ {(4)} b = {a uparrowuparrow b = atop { }} !!!!!!!{{underbrace{a^{a^{cdot^{cdot^{a}}}}}} atop {bmbox{ copies of }a}}$

as further explained in the separate article tetration. Tetration (also exponential map, hyperpower, power tower, super-exponentiation, and hyper4) is iterated exponentiation, the first hyper operator after exponentiation. ...

Known aliases for hyper4 include superpower, superdegree, and powerlog; other notation, $operatorname{hyper4}(a,b)={}^{b}a$ .

The family has not been extended from natural numbers to real numbers in general for n>3, due to nonassociativity in the "obvious" ways of doing it. In mathematics, associativity is a property that a binary operation can have. ...

Evaluation from left to right

An alternative for these operators is obtained by evaluation from left to right. Since

$a + b = (a + (b - 1)) + 1$
$atimes b = (atimes (b-1))+a$
$a^b = (a^{(b-1)})times a$

define (with subscripts instead of superscripts) $a (n + 1) b = (a (n + 1) (b - 1)) (n) a$ with $a (1) b = a + b$ , $a (2) 0 = 0$ , and $a (n) 0 = 1$ for $n > 2$

But this suffers a kind of collapse, failing to form the "power tower" traditionally expected of hyper4: $a_{(4)}b = a^{(a^{(b-1)})}$

How can $a (n) b$ be so different from $a (n) b$ for n>3? This is because of a symmetry called associativity that's defined into + and × (see field) but which ^ lacks. It is more apt to say the two (n)s were decreed to be the same for n<4. (On the other hand, one can object that the field operations were defined to mimic what had been "observed in nature" and ask why "nature" suddenly objects to that symmetry…) Square with symmetry group D4 Symmetry is a characteristic of geometrical shapes, equations, and other objects; we say that such an object is symmetric with respect to a given operation if this operation, when applied to the object, does not appear to change it. ... In mathematics, associativity is a property that a binary operation can have. ... Jump to: navigation, search In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...

The other degrees do not collapse in this way, and so this family has some interest of its own as lower (perhaps lesser or inferior) hyper operators.

For example:

moser = (..(2^^2)^^..2)^^2 (258 numbers 2)

In mathematics, Mosers polygon notation is a means of expressing certain extremely large numbers. ...

External links

The Dictionary of Large Numbers (Dictionary's author Robert Munafo claims the (n) notation as his own—it's all right to use it, but it's not a standard.)
Lynz and the Clarkkkkson
On extending hyper4 to nonintegers

Category: Arithmetic

Results from FactBites:

Knuth's up-arrow notation - Wikipedia, the free encyclopedia (702 words)
	is useful (and also for descriptions with a variable number of arrows), or equivalently, hyper operators.
	It is generally suggested that Knuth's arrow should be used for relatively smaller magnitude numbers, and the chained arrow or hyper operators for larger ones.
	This is significant since the arrow operators are not commutative.

Hyper operator - Wikipedia, the free encyclopedia (316 words)
	The hyper operators forming the hypern family are as follows:
	The family has not been extended from natural numbers to real numbers in general for n>3, due to nonassociativity in the "obvious" ways of doing it.
	An alternative for these operators is obtained by evaluation from left to right.

More results at FactBites »

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Contents

Derivation of the notation GA_googleFillSlot("encyclopedia_square");

Evaluation from left to right

See also

External links

Derivation of the notation