left Euclidean division of ordinals
left Euclidean quotient of ordinals
left Euclidean remainder of ordinals
ordinals, nonnegative integers, or polynomials with positive integer coefficients
Div returns an expression sequence q, r such that a=b⋅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⋅q+r and r≺b, where ≺ 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.
a ≔ Ordinal⁡ω,1,3,2,2,5,0,4
b ≔ Ordinal⁡2,4,1,7,0,5
q,r ≔ Div⁡a,b
Any of the arguments can be an integer.
Error, (in Ordinals:-Div) division by zero
c ≔ Ordinal⁡2,4⁢x,1,y+10,0,z
Error, (in Ordinals:-Div) unable to divide
q,r ≔ Div⁡Eval⁡c,x=x+1,b
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.
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