(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.0' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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The usage of list programming \ is limited in this notebook to the very simple Range, Length and Select \ commands that perform exactly what their name suggests. No hidden programming \ is necessary." }], "Text", FontWeight->"Bold"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ "Addition in ", Cell[BoxData[ \(\(\(\ \ \)\(Z\_n\)\)\)]] }], "Subsection"], Cell[CellGroupData[{ Cell["Integer division and rest modulo n", "Subsubsection"], Cell[CellGroupData[{ Cell[BoxData[ \(Mod[129, 20]\)], "Input"], Cell[BoxData[ \(9\)], "Output"] }, Closed]], Cell[BoxData[ \(Z[n_] := Range[n] - 1\)], "Input", InitializationCell->True], Cell[CellGroupData[{ Cell[BoxData[ \(Z[20]\)], "Input"], Cell[BoxData[ \({0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19}\)], "Output"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Addition", "Subsubsection"], Cell[BoxData[ \(\(\(add[k_, m_, n_]\)\(:=\)\(Mod[k + \ m, n]\)\(\ \)\)\)], "Input", FontColor->RGBColor[1, 0, 0]], Cell["\<\ The result of this operation is not influenced by adding multiples \ of n since these divide out in the integer division by n :\ \>", "Text",\ FontWeight->"Bold"], Cell[CellGroupData[{ Cell[BoxData[ \(add[4 + \ n, 5\ - \ 2\ n, n]\)], "Input"], Cell[BoxData[ \(Mod[9 - n, n]\)], "Output"] }, Closed]], Cell["The full set of all sums is given by the addition table :", "Text", FontWeight->"Bold"], Cell[BoxData[ \(addTable[n_] := Table[add[\(Z[n]\)[\([i]\)], \(Z[n]\)[\([j]\)], n\ \ ]\n\t\t\t, {i, Length[Z[n]]}, {j, Length[Z[n]]}\n\t\t\t]\)], "Input", FontColor->RGBColor[1, 0, 1]], Cell["which can be written in short form using the Outer command", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(add2Table[n_] := Outer[add[#1, #2, n] &, Z[n], Z[n]]\)], "Input"], Cell[BoxData[ \(General::"spell1" \(\(:\)\(\ \)\) "Possible spelling error: new symbol name \"\!\(add2Table\)\" is \ similar to existing symbol \"\!\(addTable\)\"."\)], "Message"] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(add2Table[17] // TableForm\)], "Input"], Cell[BoxData[ TagBox[GridBox[{ {"0", "1", "2", "3", "4", "5", "6", "7", "8", "9", "10", "11", "12", "13", "14", "15", "16"}, {"1", 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] [.01131 .85053 0 4.5 ] [.01131 .86376 -18 -4.5 ] [.01131 .86376 0 4.5 ] [.01131 .87698 -18 -4.5 ] [.01131 .87698 0 4.5 ] [.01131 .89021 -18 -4.5 ] [.01131 .89021 0 4.5 ] [.01131 .90344 -18 -4.5 ] [.01131 .90344 0 4.5 ] [.01131 .91667 -18 -4.5 ] [.01131 .91667 0 4.5 ] [.01131 .92989 -18 -4.5 ] [.01131 .92989 0 4.5 ] [.01131 .94312 -18 -4.5 ] [.01131 .94312 0 4.5 ] [.01131 .95635 -18 -4.5 ] [.01131 .95635 0 4.5 ] [.01131 .96958 -18 -4.5 ] [.01131 .96958 0 4.5 ] [ 0 0 0 0 ] [ 1 1 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .03042 .02381 m .03042 .03006 L s [(1)] .03042 .01131 0 1 Mshowa .04365 .02381 m .04365 .03006 L s [(3)] .04365 .01131 0 1 Mshowa .05688 .02381 m .05688 .03006 L s [(5)] .05688 .01131 0 1 Mshowa .07011 .02381 m .07011 .03006 L s [(7)] .07011 .01131 0 1 Mshowa .08333 .02381 m .08333 .03006 L s [(9)] .08333 .01131 0 1 Mshowa .09656 .02381 m .09656 .03006 L s [(11)] .09656 .01131 0 1 Mshowa .10979 .02381 m .10979 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