Quotient

The integer quotient of $m$ by $n$ — for Gaussian integers, $m/n$ rounded to the nearest lattice point.

✓ checked against that system · ! checked, with disagreements (hover a chip)Wolfram LanguageQuotient!↗

Quotient(m, n)$\lfloor m/n \rfloor$ for integers; $m/n$ rounded half-even in each part for Gaussian integers

SageMathfloor((m) / (n))✓↗Oscarfld(ZZ(m), ZZ(n))Juliafloor(ZZRingElem, QQ(m) / QQ(n))Rustquotient(m, n)

Typed signatures

Type	Package	Applies
(number, number, number?) -> number	number-theory	always

Domain: Number theory

Details

• For integers, $\lfloor m/n \rfloor$, so $m = n\,\mathrm{Quotient}(m, n) + \mathrm{Mod}(m, n)$ with the remainder taking the sign of $n$.
• For Gaussian integers the quotient rounds instead, ties to even, which is what makes $\mathbb{Z}[i]$ Euclidean: the remainder then has smaller norm than $n$.
• compute-engine has no Quotient; this follows Wolfram's.

Plot

Examples

27 more kept as data, checked against the oracles

Basic #

live ↗

In$\mathrm{Quotient}(17, 5)$Out$3$

the floor, not truncation

live ↗

In$\mathrm{Quotient}(-7, 2)$Out$-4$

the remainder takes the sign of $n$

live ↗

In$\mathrm{Quotient}(17, -5)$Out$-4$

Scope #

$(7 + 5i)/(2 + i) = 3.8 + 0.6i$, rounded

live ↗

In$\mathrm{Quotient}(7+5i, 2+i)$Out$4+i$

Real arguments: $\lfloor 7.5/2\rfloor$

live ↗

In$\mathrm{Quotient}(7.5, 2)$Out$3$

Rational arguments: $\lfloor 21/2\rfloor$

live ↗

In$\mathrm{Quotient}(\frac{7}{2}, \frac{1}{3})$Out$10$

Exact past $2^{53}$: $\lfloor (25!-1)/24!\rfloor$, where a double would round up to 25

live ↗

In$\mathrm{Quotient}(25!-1, 24!)$Out$24$

Quotient(m, n, d) is $\lfloor (m-d)/n\rfloor$, so $m-n\,q$ lands in $[d, d+n)$

live ↗

In$\mathrm{Quotient}(17, 5, 3)$Out$2$

Listable: threads over a list

live ↗

In$\mathrm{Quotient}(\bigl\lbrack10, 20, 30\bigr\rbrack, 7)$Out$\bigl\lbrack1, 2, 4\bigr\rbrack$

Properties #

$m=n\,\mathrm{Quotient}(m,n)+\mathrm{Mod}(m,n)$. See Mod

live ↗

In$5\mathrm{Quotient}(17, 5)+17\bmod5$Out$17$

Possible issues #

$2.5 + 2.5i$ rounds to even in each part

live ↗

In$\mathrm{Quotient}(5+5i, 2)$Out$2+2i$

About

Examples	38
Checked against	Wolfram Language 35 agree2 differSageMath 11 agree26 errorSymPy 24 agree8 differ4 unevaluated1 errorOscar 5 agreeJulia 10 agree27 errorRust 5 agree6 error
Crosswalk	Wolfram LanguageSageMathOscarJuliaRust

Implementation

mappedoscarpackages/oracle/src/mappings.ts

mappedwolframpackages/oracle/src/mappings.ts

mappedwolframpackages/oracle/src/mappings.ts

mappedsympypackages/oracle/src/mappings.ts

SymPy has no Quotient: // on SymPy numbers is the floor of the exact quotient, so a Gaussian integer is floored part by part where ours rounds each part to the nearest.

mappedsagepackages/oracle/src/mappings.ts

mappedjuliapackages/oracle/src/mappings.ts

mappedrustpackages/oracle/src/mappings.ts

See also: Mod, GCD