Ordinals
Base
convert ordinals between bases
Calling Sequence
Parameters
Returns
Description
Examples
Compatibility
Base(a, b, output=o)
a, b
-
ordinals, nonnegative integers, or polynomials with positive integer coefficients
o
(optional) literal keyword; either list (default) or inert
By default, a list of pairs , where each and is either an ordinal data structure, a nonnegative integer, or a polynomial with positive integer coefficients, and for all , where is the ordering of ordinals.
If output=inert is specified, then an inert sum of products of ordinal numbers using the inert operators &+, &. and &^, respectively, is returned.
The Base(a,b) calling sequence expresses the ordinal in terms of powers of the base instead of the standard base .
By default, the result is returned as a list of pairs such that
and for all . Use output=inert to return the above sum-of-products form instead; see the Returns section.
This representation is unique if . If or , a division by zero error is raised.
The exponents are not converted recursively; they are still represented in Cantor normal form (with respect to base ).
If , then all coefficients are either positive integers or polynomials with positive integer coefficients. In particular, if , then the and are just the exponents and coefficients of in the Cantor normal form. Otherwise, if , some of the coefficients will be proper ordinals .
The output representation is computed by calling the Log command repeatedly: if , then .
If one of a and b is a parametric ordinal and the logarithm cannot be taken, an error is raised.
Parametric examples.
Error, (in Ordinals:-Sub) unable to subtract 2+x from 2
When the base is constant.
If both and are integers, this is the usual base representation.
Example with nonconstant exponents.
The Ordinals[Base] command was introduced in Maple 2015.
For more information on Maple 2015 changes, see Updates in Maple 2015.
See Also
Ordinals[Log]
Ordinals[Ordinal]
Ordinals[Power]
value
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