A hereditary base representation expands not only the number in base , but also every exponent recursively in that same base. Changing the base at every depth defines the Goodstein base-change operation; replacing it by omega defines an ordinal rank of a Goodstein term.
Start at base two. At each nonzero step, change every base occurrence in the hereditary base representation to the next base and subtract one. Although the natural-number terms can grow, their ordinal ranks of a Goodstein term decrease strictly. Keep zero fixed once it is reached.
Every Goodstein sequence reaches zero. Before zero, the ordinal rank of a Goodstein term of each step is strictly smaller than the preceding one. An infinite sequence would contradict well-foundedness of the ordinals below epsilon zero.
Replacing every base occurrence by omega in a hereditary base representation gives a Cantor normal form below epsilon zero. Hereditary base change preserves this ordinal expression, and subtraction of one lowers it strictly. The ordinal is a termination measure even when the natural-number value increases.
The Goodstein function records the least step at which the Goodstein sequence from a given initial number reaches zero. With the initial step numbered zero, its values on are . Goodstein's theorem makes this effective termination-time function total.
For an integer base , a nonnegative integer has a finite positional representation with digits . Repeated Euclidean division gives the digits uniquely; appending zero high-place digits has no effect. The base need not be prime for uniqueness, but a prime base is useful in Lucas theorem and modular binomial coefficients.
The binary numeral system is positional notation of base two, using the digits zero and one.
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Positional notation is a system for representing numbers in which the position of each digit within a number determines its value based on a specific base or radix. This system allows for the efficient representation of large numbers using only a finite set of symbols (digits). ### Key Features of Positional Notation: 1. **Base (Radix)**: The base of the positional number system determines how many distinct digits are used and the value of each digit's position.