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Calling Sequence
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Base(a, b, output=o)
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Parameters
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a, b
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ordinals, nonnegative integers, or polynomials with positive integer coefficients
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o
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(optional) literal keyword; either list (default) or inert
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Returns
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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.
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If output=inert is specified, then an inert sum of products of ordinal numbers using the inert operators &+, &. and &^, respectively, is returned.
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Description
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The Base(a,b) calling sequence expresses the ordinal in terms of powers of the base instead of the standard base .
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By default, the result is returned as a list of pairs such that
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and for all . Use output=inert to return the above sum-of-products form instead; see the Returns section.
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This representation is unique if . If or , a division by zero error is raised.
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The exponents are not converted recursively; they are still represented in Cantor normal form (with respect to base ).
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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 .
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The output representation is computed by calling the Log command repeatedly: if , then .
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If one of a and b is a parametric ordinal and the logarithm cannot be taken, an error is raised.
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Examples
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Parametric examples.
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>
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When the base is constant.
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If both and are integers, this is the usual base representation.
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Example with nonconstant exponents.
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Compatibility
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The Ordinals[Base] command was introduced in Maple 2015.
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