Div returns an expression sequence q, r such that $a\=b\cdot q\+r$, where q and r are ordinals, nonnegative integers, or polynomials with positive integer coefficients, and r is as small as possible.

quo returns just q and rem returns just r.

The Div(a, b) calling sequence computes the unique ordinal numbers $q$ and $r$ such that $a\=b\cdot q\+r$ and $r\prec b$, where $\prec$ is the strict ordering of ordinals.

If $b=0$, a division by zero error is raised.

The ordinal $a$ is left divisible by $b$ if and only if $r=0$.

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

The quo and rem commands overload the corresponding top-level routines quo and rem, respectively. The top-level commands are still accessible via the :- qualifier, that is, :-quo and :-rem, respectively.

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

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

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

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

$b≔\mathrm{Ordinal}\left(\left[\left[2\&comma;4\right]\&comma;\left[1\&comma;7\right]\&comma;\left[0\&comma;5\right]\right]\right)$

$b≔{\mathbf{\omega }}^2\cdot 4\&plus;\mathbf{\omega }\cdot 7\&plus;5$

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

$q,r≔{\mathbf{\omega }}^{\mathbf{\omega }}\&plus;\mathbf{\omega }\cdot 2\&plus;1,{\mathbf{\omega }}^2\&plus;4$

$\mathrm{quo}\left(a\&comma;b\right)=q$

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

$\mathrm{rem}\left(a\&comma;b\right)=r$

${\mathbf{\omega }}^2\&plus;4={\mathbf{\omega }}^2\&plus;4$

$a=b·q+r$

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

$r<b$

$\mathrm{true}$

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

${\mathbf{\omega }}^{\mathbf{\omega }}\&plus;{\mathbf{\omega }}^3\cdot 2\&plus;{\mathbf{\omega }}^2\cdot 5\&plus;2\&comma;0$

$\mathrm{Div}\left(b\&comma;3\right)$

${\mathbf{\omega }}^2\cdot 4\&plus;\mathbf{\omega }\cdot 7\&plus;1\&comma;2$

$\mathrm{Div}\left(b\&comma;0\right)$

Error, (in Ordinals:-Div) division by zero

$c≔\mathrm{Ordinal}\left(\left[\left[2\&comma;4x\right]\&comma;\left[1\&comma;y+10\right]\&comma;\left[0\&comma;z\right]\right]\right)$

$c≔{\mathbf{\omega }}^2\cdot \left(4x\right)\&plus;\mathbf{\omega }\cdot \left(y+10\right)\&plus;z$

$\mathrm{Div}\left(c\&comma;b\right)$

Error, (in Ordinals:-Div) unable to divide

$q,r≔\mathrm{Div}\left(\mathrm{Eval}\left(c\&comma;\left[x=x+1\right]\right)\&comma;b\right)$

$q,r≔x+1,\mathbf{\omega }\cdot \left(y+3\right)\&plus;z$

The Ordinals[Div], Ordinals[quo] and Ordinals[rem] commands were introduced in Maple 2015.

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