Modulo

Many implementations use ''truncated division'', for which the quotient is defined by\n<math display=\"block\">q = \\operatorname{trunc}\\left(\\frac{a}{n}\\right)</math>\nwhere <math>\\operatorname{trunc}</math> is the [[Integral part|integral part function]] ([[Rounding#Rounding toward zero|rounding toward zero]]), i.e. the [[truncation]] to zero significant digits.\nThus according to equation ({{EquationNote|1}}), the remainder has the ''same sign as the dividend'' {{mvar|a}} so can take {{math|1=2{{!}}''n''{{!}} − 1}} values:\n<math display=\"block\">r = a - n \\operatorname{trunc}\\left(\\frac{a}{n}\\right)</math>\n</p>\n"},"2":{"wt":" [[File:Divmod floored.svg|class=skin-invert-image|thumb|upright=1.2|Quotient and remainder using floored division]]\n<p>[[Donald Knuth]]<ref>{{cite book|first=Donald. E. |last=Knuth |title=The Art of Computer Programming |url=https://archive.org/details/artofcomputerpro0003knut |url-access=registration |publisher=Addison-Wesley |year=1972}}</ref> promotes ''floored division'', for which the quotient is defined by\n<math display=\"block\">q = \\left\\lfloor\\frac{a}{n}\\right\\rfloor</math>\nwhere <math>\\lfloor\\,\\rfloor</math> is the [[floor function]] ([[Rounding#Rounding down|rounding down]]).\nThus according to equation ({{EquationNote|1}}), the remainder has the ''same sign as the divisor'' {{mvar|n}}:\n<math display=\"block\">r = a - n \\left\\lfloor\\frac{a}{n}\\right\\rfloor</math>\n</p>\n"},"3":{"wt":"[[File:Divmod Euclidean.svg|class=skin-invert-image|thumb|upright=1.2|Quotient and remainder using Euclidean division]]\n<p>Raymond T. Boute<ref>{{cite journal |last = Boute |first = Raymond T. |title = The Euclidean definition of the functions div and mod |journal = ACM Transactions on Programming Languages and Systems |volume = 14 |issue = 2 |pages = 127–144 |publisher = ACM Press (New York, NY, USA) |date = April 1992 |url = http://portal.acm.org/citation.cfm?id=128862&coll=portal&dl=ACM |doi = 10.1145/128861.128862| hdl = 1854/LU-314490 |s2cid = 8321674 |hdl-access = free}}</ref> promotes ''[[Euclidean division]]'', for which the <strong>non-negative remainder</strong> <math>r \\in \\{0, 1, 2...\\}</math> is defined by\n<math display=\"block\">r := a - nq \\ \\mathrm{such\\ that} \\ {\\color{red}{0 \\leq r}} < |n|.</math> (Emphasis added.)\nUnder this definition, we can say the following about the quotient <math>q</math>:\n<math display=\"block\">\\begin{align}\nq &= \\frac{a - r}{n} \\in \\mathbb{Z} \\\\\n&= \\text{sgn}(n) \\cdot \\frac{a-r}{|n|} \\\\\n&= \\text{sgn}(n) \\cdot \\left( \\frac{a}{|n|} - \\frac{r}{|n|} \\right) \\\\\n&= \\text{sgn}(n) \\cdot \\left\\lfloor \\frac{a}{\\left|n\\right|} \\right\\rfloor\n\\end{align}\n</math>\nwhere {{math|sgn}} is the [[sign function]], <math>\\lfloor\\,\\rfloor</math> is the [[floor function]] ([[Rounding#Rounding down|rounding down]]), and <math>\\frac{a}{|n|} \\in \\mathbb{Q}</math>, <math>\\frac{r}{|n|} \\in \\mathbb{Q}</math> are [[rational numbers]].\n</p>\n<p>Equivalently, one may instead define the quotient <math>q \\in \\mathbb{Z}</math> as follows:\n<math display=\"block\">q := \\sgn(n) \\left\\lfloor\\frac{a}{\\left|n\\right|}\\right\\rfloor =\n\\begin{cases}\n \\left\\lfloor\\frac{a}{n}\\right\\rfloor & \\text{if } n > 0 \\\\\n \\left\\lceil\\frac{a}{n}\\right\\rceil & \\text{if } n < 0 \\\\\n\\end{cases}</math>\nwhere <math>\\lceil\\,\\rceil</math> is the [[ceiling function]] ([[Rounding#Rounding up|rounding up]]).\nThus according to equation ({{EquationNote|1}}), the remainder <math>r</math> is <strong>non-negative</strong>:\n<math display=\"block\">r = a - nq = a - |n| \\left\\lfloor\\frac{a}{\\left|n\\right|}\\right\\rfloor</math>\n</p>\n"},"4":{"wt":" [[File:Divmod rounding.svg|class=skin-invert-image|thumb|upright=1.2|Quotient and remainder using rounded division]]\n<p>Common Lisp and [[IEEE 754-1985|IEEE 754]] use ''rounded division'', for which the quotient is defined by\n<math display=\"block\">q = \\operatorname{round}\\left(\\frac{a}{n}\\right)</math>\nwhere {{math|round}} is the [[Rounding|round function]] ([[Rounding#Rounding half to even|rounding half to even]]).\nThus according to equation ({{EquationNote|1}}), the remainder falls between <math>-\\frac{n}{2}</math> and <math>\\frac{n}{2}</math>, and its sign depends on which side of zero it falls to be within these boundaries:\n<math display=\"block\">r = a - n \\operatorname{round}\\left(\\frac{a}{n}\\right)</math>\n</p>\n"},"5":{"wt":" [[File:Divmod ceiling.svg|class=skin-invert-image|thumb|upright=1.2|Quotient and remainder using ceiling division]]\n<p>Common Lisp also uses ''ceiling division'', for which the quotient is defined by\n<math display=\"block\">q = \\left\\lceil\\frac{a}{n}\\right\\rceil</math>\nwhere ⌈⌉ is the [[ceiling function]] ([[Rounding#Rounding up|rounding up]]).\nThus according to equation ({{EquationNote|1}}), the remainder has the ''opposite sign of that of the divisor'':\n<math display=\"block\">r = a - n \\left\\lceil\\frac{a}{n}\\right\\rceil</math>\n</p>\n"}},"i":0}}]}'>

•   Quotient (r) as functions of dividend (trunc is the integral part function (rounding toward zero), i.e. the truncation to zero significant digits. Thus according to equation (1), the remainder has the same sign as the dividend r=a−ntrunc⁡(an)

• Quotient and remainder using floored division

  Donald Knuth promotes floored division, for which the quotient is defined by n: r:=a−nq such that 0≤r<|n|. (Emphasis added.) Under this definition, we can say the following about the quotient sgn is the sign function, r|n|∈ℚ are rational numbers.

  Equivalently, one may instead define the quotient ⌈⌉ is the ceiling function (rounding up). Thus according to equation (1), the remainder q=round⁡(an) where n2, and its sign depends on which side of zero it falls to be within these boundaries: r=a−n⌈an⌉