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Ordinals

  

Base

  

convert ordinals between bases

 

Calling Sequence

Parameters

Returns

Description

Examples

Compatibility

Calling Sequence

Base(a, b, output=o)

Parameters

a, b

-

ordinals, nonnegative integers, or polynomials with positive integer coefficients

o

-

(optional) literal keyword; either list (default) or inert

Returns

• 

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.

Description

• 

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.

Examples

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

Parametric examples.

Error, (in Ordinals:-Sub) unable to subtract 2+x from 2

(9)

(10)

(11)

(12)

(13)

(14)

(15)

(16)

(17)

(18)

(19)

When the base is constant.

(20)

(21)

(22)

(23)

(24)

If both  and  are integers, this is the usual base  representation.

(25)

Example with nonconstant exponents.

(26)

(27)

(28)

(29)

(30)

Compatibility

• 

The Ordinals[Base] command was introduced in Maple 2015.

• 

For more information on Maple 2015 changes, see Updates in Maple 2015.

See Also

Ordinals

Ordinals[Log]

Ordinals[Ordinal]

Ordinals[Power]

value

 


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