Knuth's Up-arrow Notation - Numeration Systems Based On The Hyperoperation Sequence

Numeration Systems Based On The Hyperoperation Sequence

R. L. Goodstein, with a system of notation different from Knuth arrows, used the sequence of hyperoperators here denoted by to create systems of numeration for the nonnegative integers. Letting superscripts denote the respective hyperoperators, the so-called complete hereditary representation of integer n, at level k and base b, can be expressed as follows using only the first k hyperoperators and using as digits only 0, 1, ..., b-1:

• For 0 ≤ n ≤ b-1, n is represented simply by the corresponding digit.
• For n > b-1, the representation of n is found recursively, first representing n in the form

where x_k, ..., x₁ are the largest integers satisfying (in turn)

...

.

Any x_i exceeding b-1 is then re-expressed in the same manner, and so on, repeating this procedure until the resulting form contains only the digits 0, 1, ..., b-1.

The remainder of this section will use, rather than superscripts, to denote the hyperoperators.

Unnecessary parentheses can be avoided by giving higher-level operators higher precedence in the order of evaluation; thus,

level-1 representations have the form, with X also of this form;

level-2 representations have the form, with X,Y also of this form;

level-3 representations have the form, with X,Y,Z also of this form;

level-4 representations have the form, with X,Y,Z,T also of this form;

and so on.

The representations can be abbreviated by omitting any instances of etc.; for example, the level-3 base-2 representation of the number 6 is, which abbreviates to .

Examples: The unique base-2 representations of the number 266, at levels 1, 2, 3, 4, and 5 are as follows:

.