By default, a list of pairs $\left[\left[e_1\,c_1\right]\,\left[e_2\,c_2\right]\,\mathrm{...}\right]$, where each $e_i$ and $c_i$ is either an ordinal data structure, a nonnegative integer, or a polynomial with positive integer coefficients, and $0\prec c_i\prec b$ for all $i$, where $\prec$ 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 $a$ in terms of powers of the base $b$ instead of the standard base $\mathbf{\omega }$.

By default, the result is returned as a list of pairs $\left[\left[e_1\,c_1\right]\,\left[e_2\,c_2\right]\,\mathrm{...}\right]$ such that

$e_1\succ e_2\succ \cdots$ and  $0\prec c_i\prec b$ for all $i$. Use output=inert to return the above sum-of-products form instead; see the Returns section.

This representation is unique if $b\succcurlyeq 2$. If $b=0$ or $b=1$, a division by zero error is raised.

The exponents $e_i$ are not converted recursively; they are still represented in Cantor normal form (with respect to base $\mathbf{\omega }$).

If $b\preccurlyeq \mathbf{\omega }$, then all coefficients $c_i$ are either positive integers or polynomials with positive integer coefficients. In particular, if $b\=\mathbf{\omega }$, then the $e_i$ and $c_i$ are just the exponents and coefficients of $a$ in the Cantor normal form. Otherwise, if $b\succ \mathbf{\omega }$, some of the coefficients will be proper ordinals $\succcurlyeq \mathbf{\omega }$.

The output representation is computed by calling the Log command repeatedly: if $l,q,r=\mathrm{Log}\left(a\,b\right)$, then $\mathrm{Base}\left(a\,b\right)=\left[\left[l\,q\right]\,\mathrm{op}\left(\mathrm{Base}\left(r\,b\right)\right)\right]$.

If one of a and b is a parametric ordinal and the logarithm cannot be taken, an error is raised.

$\mathrm{with}\left(\mathrm{Ordinals}\right)$

$\left[\mathrm{`+`}\&comma;\mathrm{`.`}\&comma;\mathrm{`<`}\&comma;\mathrm{<=}\&comma;\mathrm{Add}\&comma;\mathrm{Base}\&comma;\mathrm{Dec}\&comma;\mathrm{Decompose}\&comma;\mathrm{Div}\&comma;\mathrm{Eval}\&comma;\mathrm{Factor}\&comma;\mathrm{Gcd}\&comma;\mathrm{Lcm}\&comma;\mathrm{LessThan}\&comma;\mathrm{Log}\&comma;\mathrm{Max}\&comma;\mathrm{Min}\&comma;\mathrm{Mult}\&comma;\mathrm{Ordinal}\&comma;\mathrm{Power}\&comma;\mathrm{Split}\&comma;\mathrm{Sub}\&comma;\mathrm{`^`}\&comma;\mathrm{degree}\&comma;\mathrm{lcoeff}\&comma;\log \&comma;\mathrm{lterm}\&comma;\mathrm{\omega }\&comma;\mathrm{quo}\&comma;\mathrm{rem}\&comma;\mathrm{tcoeff}\&comma;\mathrm{tdegree}\&comma;\mathrm{tterm}\right]$

$a≔\mathrm{Ordinal}\left(\left[\left[5\&comma;1\right]\&comma;\left[4\&comma;4\right]\&comma;\left[2\&comma;2\right]\&comma;\left[1\&comma;1\right]\&comma;\left[0\&comma;3\right]\right]\right)$

$a≔{\mathbf{\omega }}^5\&plus;{\mathbf{\omega }}^4\cdot 4\&plus;{\mathbf{\omega }}^2\cdot 2\&plus;\mathbf{\omega }\&plus;3$

$b≔\mathrm{Ordinal}\left(\left[\left[2\&comma;1\right]\&comma;\left[0\&comma;3\right]\right]\right)$

$b≔{\mathbf{\omega }}^2\&plus;3$

$\mathrm{Base}\left(a\&comma;b\right)$

$\left[\left[2\&comma;\mathbf{\omega }\&plus;3\right]\&comma;\left[1\&comma;{\mathbf{\omega }}^2\&plus;2\right]\&comma;\left[0\&comma;\mathbf{\omega }\&plus;3\right]\right]$

$l,q,r≔\mathrm{Log}\left(a\&comma;b\right)$

$l,q,r≔2,\mathbf{\omega }\&plus;3,{\mathbf{\omega }}^4\&plus;{\mathbf{\omega }}^2\cdot 2\&plus;\mathbf{\omega }\&plus;3$

$\mathrm{Base}\left(r\&comma;b\right)$

$\left[\left[1\&comma;{\mathbf{\omega }}^2\&plus;2\right]\&comma;\left[0\&comma;\mathbf{\omega }\&plus;3\right]\right]$

$\mathrm{Base}\left(a\&comma;b\&comma;\mathrm{output}=\mathrm{inert}\right)$

${\left({\mathbf{\omega }}^2\&plus;3\right)}^2\mathbf{\cdot }\left(\mathbf{\omega }\&plus;3\right)\mathbf{\&plus;}\left({\mathbf{\omega }}^2\&plus;3\right)\mathbf{\cdot }\left({\mathbf{\omega }}^2\&plus;2\right)\mathbf{\&plus;}\left(\mathbf{\omega }\&plus;3\right)$

$\mathrm{value}\left(\right)$

${\mathbf{\omega }}^5\&plus;{\mathbf{\omega }}^4\cdot 4\&plus;{\mathbf{\omega }}^2\cdot 2\&plus;\mathbf{\omega }\&plus;3$

$\mathrm{Base}\left(a\&comma;{\mathrm{\omega }}^2+2+x\right)$

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

$\mathrm{Base}\left(a\&comma;{\mathrm{\omega }}^2+3+x\right)$

$\left[\left[2\&comma;\mathbf{\omega }\&plus;3\right]\&comma;\left[1\&comma;{\mathbf{\omega }}^2\&plus;2\right]\&comma;\left[0\&comma;\mathbf{\omega }\&plus;3\right]\right]$

$\mathrm{Base}\left(a\&comma;{\mathrm{\omega }}^2+3+x\&comma;\mathrm{output}=\mathrm{inert}\right)$

${\left({\mathbf{\omega }}^2\&plus;\left(3+x\right)\right)}^2\mathbf{\cdot }\left(\mathbf{\omega }\&plus;3\right)\mathbf{\&plus;}\left({\mathbf{\omega }}^2\&plus;\left(3+x\right)\right)\mathbf{\cdot }\left({\mathbf{\omega }}^2\&plus;2\right)\mathbf{\&plus;}\left(\mathbf{\omega }\&plus;3\right)$

$\mathrm{value}\left(\right)$

${\mathbf{\omega }}^5\&plus;{\mathbf{\omega }}^4\cdot 4\&plus;{\mathbf{\omega }}^2\cdot 2\&plus;\mathbf{\omega }\&plus;3$

$\mathrm{Base}\left(a\&comma;{\mathrm{\omega }}^2+2\right)$

$\left[\left[2\&comma;\mathbf{\omega }\&plus;4\right]\&comma;\left[0\&comma;\mathbf{\omega }\&plus;3\right]\right]$

$\mathrm{Base}\left(a\&comma;{\mathrm{\omega }}^2+1\right)$

$\left[\left[2\&comma;\mathbf{\omega }\&plus;4\right]\&comma;\left[1\&comma;1\right]\&comma;\left[0\&comma;\mathbf{\omega }\&plus;3\right]\right]$

$\mathrm{Base}\left(a\&comma;{\mathrm{\omega }}^2\right)$

$\left[\left[2\&comma;\mathbf{\omega }\&plus;4\right]\&comma;\left[1\&comma;2\right]\&comma;\left[0\&comma;\mathbf{\omega }\&plus;3\right]\right]$

$\mathrm{Base}\left(a\&comma;\mathrm{\omega }+4+x\right)$

$\left[\left[4\&comma;\mathbf{\omega }\&plus;3\right]\&comma;\left[3\&comma;\mathbf{\omega }\right]\&comma;\left[2\&comma;1\right]\&comma;\left[1\&comma;\mathbf{\omega }\right]\&comma;\left[0\&comma;\mathbf{\omega }\&plus;3\right]\right]$

$\mathrm{Base}\left(a\&comma;\mathrm{\omega }+3\right)$

$\left[\left[5\&comma;1\right]\&comma;\left[3\&comma;\mathbf{\omega }\right]\&comma;\left[2\&comma;1\right]\&comma;\left[1\&comma;\mathbf{\omega }\&plus;1\right]\right]$

$\mathrm{Base}\left(a\&comma;\mathrm{\omega }+2\right)$

$\left[\left[5\&comma;1\right]\&comma;\left[4\&comma;1\right]\&comma;\left[3\&comma;\mathbf{\omega }\right]\&comma;\left[2\&comma;1\right]\&comma;\left[1\&comma;\mathbf{\omega }\&plus;1\right]\&comma;\left[0\&comma;1\right]\right]$

$\mathrm{Base}\left(a\&comma;\mathrm{\omega }+1\right)$

$\left[\left[5\&comma;1\right]\&comma;\left[4\&comma;2\right]\&comma;\left[3\&comma;\mathbf{\omega }\right]\&comma;\left[2\&comma;2\right]\&comma;\left[0\&comma;2\right]\right]$

$\mathrm{Base}\left(a\&comma;\mathrm{\omega }\right)=\mathrm{op}\left(a\right)$

$\left[\left[5\&comma;1\right]\&comma;\left[4\&comma;4\right]\&comma;\left[2\&comma;2\right]\&comma;\left[1\&comma;1\right]\&comma;\left[0\&comma;3\right]\right]=\left[\left[5\&comma;1\right]\&comma;\left[4\&comma;4\right]\&comma;\left[2\&comma;2\right]\&comma;\left[1\&comma;1\right]\&comma;\left[0\&comma;3\right]\right]$

$\mathrm{Base}\left(a\&comma;5\right)$

$\left[\left[\mathbf{\omega }\cdot 5\&comma;1\right]\&comma;\left[\mathbf{\omega }\cdot 4\&comma;4\right]\&comma;\left[\mathbf{\omega }\cdot 2\&comma;2\right]\&comma;\left[\mathbf{\omega }\&comma;1\right]\&comma;\left[0\&comma;3\right]\right]$

$\mathrm{Base}\left(a\&comma;5\&comma;\mathrm{output}=\mathrm{inert}\right)$

${\left(5\right)}^{\mathbf{\omega }\cdot 5}\mathbf{\&plus;}{\left(5\right)}^{\mathbf{\omega }\cdot 4}\mathbf{\cdot }4\mathbf{\&plus;}{\left(5\right)}^{\mathbf{\omega }\cdot 2}\mathbf{\cdot }2\mathbf{\&plus;}{\left(5\right)}^{\mathbf{\omega }}\mathbf{\&plus;}3$

$\mathrm{Base}\left(a\&comma;4\right)$

$\left[\left[\mathbf{\omega }\cdot 5\&comma;1\right]\&comma;\left[\mathbf{\omega }\cdot 4\&plus;1\&comma;1\right]\&comma;\left[\mathbf{\omega }\cdot 2\&comma;2\right]\&comma;\left[\mathbf{\omega }\&comma;1\right]\&comma;\left[0\&comma;3\right]\right]$

$\mathrm{Base}\left(a\&comma;3\right)$

$\left[\left[\mathbf{\omega }\cdot 5\&comma;1\right]\&comma;\left[\mathbf{\omega }\cdot 4\&plus;1\&comma;1\right]\&comma;\left[\mathbf{\omega }\cdot 4\&comma;1\right]\&comma;\left[\mathbf{\omega }\cdot 2\&comma;2\right]\&comma;\left[\mathbf{\omega }\&comma;1\right]\&comma;\left[1\&comma;1\right]\right]$

$\mathrm{Base}\left(a\&comma;2\right)$

$\left[\left[\mathbf{\omega }\cdot 5\&comma;1\right]\&comma;\left[\mathbf{\omega }\cdot 4\&plus;2\&comma;1\right]\&comma;\left[\mathbf{\omega }\cdot 2\&plus;1\&comma;1\right]\&comma;\left[\mathbf{\omega }\&comma;1\right]\&comma;\left[1\&comma;1\right]\&comma;\left[0\&comma;1\right]\right]$

$100=\mathrm{Base}\left(100\&comma;3\&comma;\mathrm{output}=\mathrm{inert}\right)$

$100={\left(3\right)}^5\mathbf{\&plus;}{\left(3\right)}^3\mathbf{\cdot }2\mathbf{\&plus;}3$

$b≔\mathrm{\omega }·2+3$

$b≔\mathbf{\omega }\cdot 2\&plus;3$

$b^b$

${\mathbf{\omega }}^{\mathbf{\omega }\cdot 2\&plus;3}\cdot 2\&plus;{\mathbf{\omega }}^{\mathbf{\omega }\cdot 2\&plus;2}\cdot 6\&plus;{\mathbf{\omega }}^{\mathbf{\omega }\cdot 2\&plus;1}\cdot 6\&plus;{\mathbf{\omega }}^{\mathbf{\omega }\cdot 2}\cdot 3$

$a≔\mathrm{Dec}\left(\right)+x$

$a≔{\mathbf{\omega }}^{\mathbf{\omega }\cdot 2\&plus;3}\cdot 2\&plus;{\mathbf{\omega }}^{\mathbf{\omega }\cdot 2\&plus;2}\cdot 6\&plus;{\mathbf{\omega }}^{\mathbf{\omega }\cdot 2\&plus;1}\cdot 6\&plus;{\mathbf{\omega }}^{\mathbf{\omega }\cdot 2}\cdot 2\&plus;{\mathbf{\omega }}^{\mathbf{\omega }\&plus;1}\&plus;x$

$r≔\mathrm{Base}\left(a\&comma;b\right)$

$r≔\left[\left[\mathbf{\omega }\cdot 2\&plus;2\&comma;\mathbf{\omega }\cdot 2\&plus;2\right]\&comma;\left[\mathbf{\omega }\cdot 2\&plus;1\&comma;\mathbf{\omega }\cdot 2\&plus;2\right]\&comma;\left[\mathbf{\omega }\cdot 2\&comma;\mathbf{\omega }\cdot 2\&plus;2\right]\&comma;\left[\mathbf{\omega }\&comma;\mathbf{\omega }\right]\&comma;\left[0\&comma;x\right]\right]$

$\mathrm{`+`}\left(\mathrm{seq}\left(b^{r\left[i,1\right]}·r\left[i,2\right]\&comma;i=1..\mathrm{nops}\left(r\right)\right)\right)$

${\mathbf{\omega }}^{\mathbf{\omega }\cdot 2\&plus;3}\cdot 2\&plus;{\mathbf{\omega }}^{\mathbf{\omega }\cdot 2\&plus;2}\cdot 6\&plus;{\mathbf{\omega }}^{\mathbf{\omega }\cdot 2\&plus;1}\cdot 6\&plus;{\mathbf{\omega }}^{\mathbf{\omega }\cdot 2}\cdot 2\&plus;{\mathbf{\omega }}^{\mathbf{\omega }\&plus;1}\&plus;x$

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

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