PNG  IHDRX cHRMz&u0`:pQ<bKGD pHYsodtIME MeqIDATxw]Wug^Qd˶ 6`!N:!@xI~)%7%@Bh&`lnjVF29gΨ4E$|>cɚ{gk= %,a KX%,a KX%,a KX%,a KX%,a KX%,a KX%, b` ǟzeאfp]<!SJmɤY޲ڿ,%c ~ع9VH.!Ͳz&QynֺTkRR.BLHi٪:l;@(!MԴ=žI,:o&N'Kù\vRmJ雵֫AWic H@" !: Cé||]k-Ha oݜ:y F())u]aG7*JV@J415p=sZH!=!DRʯvɱh~V\}v/GKY$n]"X"}t@ xS76^[bw4dsce)2dU0 CkMa-U5tvLƀ~mlMwfGE/-]7XAƟ`׮g ewxwC4\[~7@O-Q( a*XGƒ{ ՟}$_y3tĐƤatgvێi|K=uVyrŲlLӪuܿzwk$m87k( `múcE)"@rK( z4$D; 2kW=Xb$V[Ru819קR~qloѱDyįݎ*mxw]y5e4K@ЃI0A D@"BDk_)N\8͜9dz"fK0zɿvM /.:2O{ Nb=M=7>??Zuo32 DLD@D| &+֎C #B8ַ`bOb $D#ͮҪtx]%`ES`Ru[=¾!@Od37LJ0!OIR4m]GZRJu$‡c=%~s@6SKy?CeIh:[vR@Lh | (BhAMy=݃  G"'wzn޺~8ԽSh ~T*A:xR[ܹ?X[uKL_=fDȊ؂p0}7=D$Ekq!/t.*2ʼnDbŞ}DijYaȲ(""6HA;:LzxQ‘(SQQ}*PL*fc\s `/d'QXW, e`#kPGZuŞuO{{wm[&NBTiiI0bukcA9<4@SӊH*؎4U/'2U5.(9JuDfrޱtycU%j(:RUbArLֺN)udA':uGQN"-"Is.*+k@ `Ojs@yU/ H:l;@yyTn}_yw!VkRJ4P)~y#)r,D =ě"Q]ci'%HI4ZL0"MJy 8A{ aN<8D"1#IJi >XjX֔#@>-{vN!8tRݻ^)N_╗FJEk]CT՟ YP:_|H1@ CBk]yKYp|og?*dGvzنzӴzjֺNkC~AbZƷ`.H)=!QͷVTT(| u78y֮}|[8-Vjp%2JPk[}ԉaH8Wpqhwr:vWª<}l77_~{s۴V+RCģ%WRZ\AqHifɤL36: #F:p]Bq/z{0CU6ݳEv_^k7'>sq*+kH%a`0ԣisqにtү04gVgW΂iJiS'3w.w}l6MC2uԯ|>JF5`fV5m`Y**Db1FKNttu]4ccsQNnex/87+}xaUW9y>ͯ骵G{䩓Գ3+vU}~jJ.NFRD7<aJDB1#ҳgSb,+CS?/ VG J?|?,2#M9}B)MiE+G`-wo߫V`fio(}S^4e~V4bHOYb"b#E)dda:'?}׮4繏`{7Z"uny-?ǹ;0MKx{:_pÚmFמ:F " .LFQLG)Q8qN q¯¯3wOvxDb\. BKD9_NN &L:4D{mm o^tֽ:q!ƥ}K+<"m78N< ywsard5+вz~mnG)=}lYݧNj'QJS{S :UYS-952?&O-:W}(!6Mk4+>A>j+i|<<|;ر^߉=HE|V#F)Emm#}/"y GII웻Jі94+v뾧xu~5C95~ūH>c@덉pʃ1/4-A2G%7>m;–Y,cyyaln" ?ƻ!ʪ<{~h~i y.zZB̃/,雋SiC/JFMmBH&&FAbϓO^tubbb_hZ{_QZ-sύodFgO(6]TJA˯#`۶ɟ( %$&+V'~hiYy>922 Wp74Zkq+Ovn錄c>8~GqܲcWꂎz@"1A.}T)uiW4="jJ2W7mU/N0gcqܗOO}?9/wìXžΏ0 >֩(V^Rh32!Hj5`;O28؇2#ݕf3 ?sJd8NJ@7O0 b־?lldщ̡&|9C.8RTWwxWy46ah嘦mh٤&l zCy!PY?: CJyв]dm4ǜҐR޻RլhX{FƯanшQI@x' ao(kUUuxW_Ñ줮[w8 FRJ(8˼)_mQ _!RJhm=!cVmm ?sFOnll6Qk}alY}; "baӌ~M0w,Ggw2W:G/k2%R,_=u`WU R.9T"v,<\Ik޽/2110Ӿxc0gyC&Ny޽JҢrV6N ``یeA16"J³+Rj*;BϜkZPJaÍ<Jyw:NP8/D$ 011z֊Ⱳ3ι֘k1V_"h!JPIΣ'ɜ* aEAd:ݺ>y<}Lp&PlRfTb1]o .2EW\ͮ]38؋rTJsǏP@芎sF\> P^+dYJLbJ C-xϐn> ι$nj,;Ǖa FU *择|h ~izť3ᤓ`K'-f tL7JK+vf2)V'-sFuB4i+m+@My=O҈0"|Yxoj,3]:cо3 $#uŘ%Y"y죯LebqtҢVzq¼X)~>4L׶m~[1_k?kxֺQ`\ |ٛY4Ѯr!)N9{56(iNq}O()Em]=F&u?$HypWUeB\k]JɩSع9 Zqg4ZĊo oMcjZBU]B\TUd34ݝ~:7ڶSUsB0Z3srx 7`:5xcx !qZA!;%͚7&P H<WL!džOb5kF)xor^aujƍ7 Ǡ8/p^(L>ὴ-B,{ۇWzֺ^k]3\EE@7>lYBȝR.oHnXO/}sB|.i@ɥDB4tcm,@ӣgdtJ!lH$_vN166L__'Z)y&kH;:,Y7=J 9cG) V\hjiE;gya~%ks_nC~Er er)muuMg2;֫R)Md) ,¶ 2-wr#F7<-BBn~_(o=KO㭇[Xv eN_SMgSҐ BS헃D%g_N:/pe -wkG*9yYSZS.9cREL !k}<4_Xs#FmҶ:7R$i,fi!~' # !6/S6y@kZkZcX)%5V4P]VGYq%H1!;e1MV<!ϐHO021Dp= HMs~~a)ަu7G^];git!Frl]H/L$=AeUvZE4P\.,xi {-~p?2b#amXAHq)MWǾI_r`S Hz&|{ +ʖ_= (YS(_g0a03M`I&'9vl?MM+m~}*xT۲(fY*V4x@29s{DaY"toGNTO+xCAO~4Ϳ;p`Ѫ:>Ҵ7K 3}+0 387x\)a"/E>qpWB=1 ¨"MP(\xp߫́A3+J] n[ʼnӼaTbZUWb={~2ooKױӰp(CS\S筐R*JغV&&"FA}J>G֐p1ٸbk7 ŘH$JoN <8s^yk_[;gy-;߉DV{c B yce% aJhDȶ 2IdйIB/^n0tNtџdcKj4϶v~- CBcgqx9= PJ) dMsjpYB] GD4RDWX +h{y`,3ꊕ$`zj*N^TP4L:Iz9~6s) Ga:?y*J~?OrMwP\](21sZUD ?ܟQ5Q%ggW6QdO+\@ ̪X'GxN @'4=ˋ+*VwN ne_|(/BDfj5(Dq<*tNt1х!MV.C0 32b#?n0pzj#!38}޴o1KovCJ`8ŗ_"]] rDUy޲@ Ȗ-;xџ'^Y`zEd?0„ DAL18IS]VGq\4o !swV7ˣι%4FѮ~}6)OgS[~Q vcYbL!wG3 7띸*E Pql8=jT\꘿I(z<[6OrR8ºC~ډ]=rNl[g|v TMTղb-o}OrP^Q]<98S¤!k)G(Vkwyqyr޽Nv`N/e p/~NAOk \I:G6]4+K;j$R:Mi #*[AȚT,ʰ,;N{HZTGMoּy) ]%dHء9Պ䠬|<45,\=[bƟ8QXeB3- &dҩ^{>/86bXmZ]]yޚN[(WAHL$YAgDKp=5GHjU&99v簪C0vygln*P)9^͞}lMuiH!̍#DoRBn9l@ xA/_v=ȺT{7Yt2N"4!YN`ae >Q<XMydEB`VU}u]嫇.%e^ánE87Mu\t`cP=AD/G)sI"@MP;)]%fH9'FNsj1pVhY&9=0pfuJ&gޤx+k:!r˭wkl03׼Ku C &ѓYt{.O.zҏ z}/tf_wEp2gvX)GN#I ݭ߽v/ .& и(ZF{e"=V!{zW`, ]+LGz"(UJp|j( #V4, 8B 0 9OkRrlɱl94)'VH9=9W|>PS['G(*I1==C<5"Pg+x'K5EMd؞Af8lG ?D FtoB[je?{k3zQ vZ;%Ɠ,]E>KZ+T/ EJxOZ1i #T<@ I}q9/t'zi(EMqw`mYkU6;[t4DPeckeM;H}_g pMww}k6#H㶏+b8雡Sxp)&C $@'b,fPߑt$RbJ'vznuS ~8='72_`{q纶|Q)Xk}cPz9p7O:'|G~8wx(a 0QCko|0ASD>Ip=4Q, d|F8RcU"/KM opKle M3#i0c%<7׿p&pZq[TR"BpqauIp$ 8~Ĩ!8Սx\ւdT>>Z40ks7 z2IQ}ItԀ<-%S⍤};zIb$I 5K}Q͙D8UguWE$Jh )cu4N tZl+[]M4k8֦Zeq֮M7uIqG 1==tLtR,ƜSrHYt&QP윯Lg' I,3@P'}'R˪e/%-Auv·ñ\> vDJzlӾNv5:|K/Jb6KI9)Zh*ZAi`?S {aiVDԲuy5W7pWeQJk֤#5&V<̺@/GH?^τZL|IJNvI:'P=Ϛt"¨=cud S Q.Ki0 !cJy;LJR;G{BJy޺[^8fK6)=yʊ+(k|&xQ2`L?Ȓ2@Mf 0C`6-%pKpm')c$׻K5[J*U[/#hH!6acB JA _|uMvDyk y)6OPYjœ50VT K}cǻP[ $:]4MEA.y)|B)cf-A?(e|lɉ#P9V)[9t.EiQPDѠ3ϴ;E:+Օ t ȥ~|_N2,ZJLt4! %ա]u {+=p.GhNcŞQI?Nd'yeh n7zi1DB)1S | S#ًZs2|Ɛy$F SxeX{7Vl.Src3E℃Q>b6G ўYCmtկ~=K0f(=LrAS GN'ɹ9<\!a`)֕y[uՍ[09` 9 +57ts6}b4{oqd+J5fa/,97J#6yν99mRWxJyѡyu_TJc`~W>l^q#Ts#2"nD1%fS)FU w{ܯ R{ ˎ󅃏џDsZSQS;LV;7 Od1&1n$ N /.q3~eNɪ]E#oM~}v֯FڦwyZ=<<>Xo稯lfMFV6p02|*=tV!c~]fa5Y^Q_WN|Vs 0ҘދU97OI'N2'8N֭fgg-}V%y]U4 峧p*91#9U kCac_AFңĪy뚇Y_AiuYyTTYЗ-(!JFLt›17uTozc. S;7A&&<ԋ5y;Ro+:' *eYJkWR[@F %SHWP 72k4 qLd'J "zB6{AC0ƁA6U.'F3:Ȅ(9ΜL;D]m8ڥ9}dU "v!;*13Rg^fJyShyy5auA?ɩGHRjo^]׽S)Fm\toy 4WQS@mE#%5ʈfFYDX ~D5Ϡ9tE9So_aU4?Ѽm%&c{n>.KW1Tlb}:j uGi(JgcYj0qn+>) %\!4{LaJso d||u//P_y7iRJ߬nHOy) l+@$($VFIQ9%EeKʈU. ia&FY̒mZ=)+qqoQn >L!qCiDB;Y<%} OgBxB!ØuG)WG9y(Ą{_yesuZmZZey'Wg#C~1Cev@0D $a@˲(.._GimA:uyw֬%;@!JkQVM_Ow:P.s\)ot- ˹"`B,e CRtaEUP<0'}r3[>?G8xU~Nqu;Wm8\RIkբ^5@k+5(By'L&'gBJ3ݶ!/㮻w҅ yqPWUg<e"Qy*167΃sJ\oz]T*UQ<\FԎ`HaNmڜ6DysCask8wP8y9``GJ9lF\G g's Nn͵MLN֪u$| /|7=]O)6s !ĴAKh]q_ap $HH'\1jB^s\|- W1:=6lJBqjY^LsPk""`]w)󭃈,(HC ?䔨Y$Sʣ{4Z+0NvQkhol6C.婧/u]FwiVjZka&%6\F*Ny#8O,22+|Db~d ~Çwc N:FuuCe&oZ(l;@ee-+Wn`44AMK➝2BRՈt7g*1gph9N) *"TF*R(#'88pm=}X]u[i7bEc|\~EMn}P瘊J)K.0i1M6=7'_\kaZ(Th{K*GJyytw"IO-PWJk)..axӝ47"89Cc7ĐBiZx 7m!fy|ϿF9CbȩV 9V-՛^pV̌ɄS#Bv4-@]Vxt-Z, &ֺ*diؠ2^VXbs֔Ìl.jQ]Y[47gj=幽ex)A0ip׳ W2[ᎇhuE^~q흙L} #-b۸oFJ_QP3r6jr+"nfzRJTUqoaۍ /$d8Mx'ݓ= OՃ| )$2mcM*cЙj}f };n YG w0Ia!1Q.oYfr]DyISaP}"dIӗթO67jqR ҊƐƈaɤGG|h;t]䗖oSv|iZqX)oalv;۩meEJ\!8=$4QU4Xo&VEĊ YS^E#d,yX_> ۘ-e\ "Wa6uLĜZi`aD9.% w~mB(02G[6y.773a7 /=o7D)$Z 66 $bY^\CuP. (x'"J60׿Y:Oi;F{w佩b+\Yi`TDWa~|VH)8q/=9!g߆2Y)?ND)%?Ǐ`k/sn:;O299yB=a[Ng 3˲N}vLNy;*?x?~L&=xyӴ~}q{qE*IQ^^ͧvü{Huu=R|>JyUlZV, B~/YF!Y\u_ݼF{_C)LD]m {H 0ihhadd nUkf3oٺCvE\)QJi+֥@tDJkB$1!Đr0XQ|q?d2) Ӣ_}qv-< FŊ߫%roppVBwü~JidY4:}L6M7f٬F "?71<2#?Jyy4뷢<_a7_=Q E=S1И/9{+93֮E{ǂw{))?maÆm(uLE#lïZ  ~d];+]h j?!|$F}*"4(v'8s<ŏUkm7^7no1w2ؗ}TrͿEk>p'8OB7d7R(A 9.*Mi^ͳ; eeUwS+C)uO@ =Sy]` }l8^ZzRXj[^iUɺ$tj))<sbDJfg=Pk_{xaKo1:-uyG0M ԃ\0Lvuy'ȱc2Ji AdyVgVh!{]/&}}ċJ#%d !+87<;qN޼Nفl|1N:8ya  8}k¾+-$4FiZYÔXk*I&'@iI99)HSh4+2G:tGhS^繿 Kتm0 вDk}֚+QT4;sC}rՅE,8CX-e~>G&'9xpW,%Fh,Ry56Y–hW-(v_,? ; qrBk4-V7HQ;ˇ^Gv1JVV%,ik;D_W!))+BoS4QsTM;gt+ndS-~:11Sgv!0qRVh!"Ȋ(̦Yl.]PQWgٳE'`%W1{ndΗBk|Ž7ʒR~,lnoa&:ü$ 3<a[CBݮwt"o\ePJ=Hz"_c^Z.#ˆ*x z̝grY]tdkP*:97YľXyBkD4N.C_[;F9`8& !AMO c `@BA& Ost\-\NX+Xp < !bj3C&QL+*&kAQ=04}cC!9~820G'PC9xa!w&bo_1 Sw"ܱ V )Yl3+ס2KoXOx]"`^WOy :3GO0g;%Yv㐫(R/r (s } u B &FeYZh0y> =2<Ϟc/ -u= c&׭,.0"g"7 6T!vl#sc>{u/Oh Bᾈ)۴74]x7 gMӒ"d]U)}" v4co[ ɡs 5Gg=XR14?5A}D "b{0$L .\4y{_fe:kVS\\O]c^W52LSBDM! C3Dhr̦RtArx4&agaN3Cf<Ԉp4~ B'"1@.b_/xQ} _߃҉/gٓ2Qkqp0շpZ2fԫYz< 4L.Cyυι1t@鎫Fe sYfsF}^ V}N<_`p)alٶ "(XEAVZ<)2},:Ir*#m_YӼ R%a||EƼIJ,,+f"96r/}0jE/)s)cjW#w'Sʯ5<66lj$a~3Kʛy 2:cZ:Yh))+a߭K::N,Q F'qB]={.]h85C9cr=}*rk?vwV렵ٸW Rs%}rNAkDv|uFLBkWY YkX מ|)1!$#3%y?pF<@<Rr0}: }\J [5FRxY<9"SQdE(Q*Qʻ)q1E0B_O24[U'],lOb ]~WjHޏTQ5Syu wq)xnw8~)c 쫬gٲߠ H% k5dƝk> kEj,0% b"vi2Wس_CuK)K{n|>t{P1򨾜j>'kEkƗBg*H%'_aY6Bn!TL&ɌOb{c`'d^{t\i^[uɐ[}q0lM˕G:‚4kb祔c^:?bpg… +37stH:0}en6x˟%/<]BL&* 5&fK9Mq)/iyqtA%kUe[ڛKN]Ě^,"`/ s[EQQm?|XJ߅92m]G.E΃ח U*Cn.j_)Tѧj̿30ڇ!A0=͜ar I3$C^-9#|pk!)?7.x9 @OO;WƝZBFU keZ75F6Tc6"ZȚs2y/1 ʵ:u4xa`C>6Rb/Yм)^=+~uRd`/|_8xbB0?Ft||Z\##|K 0>>zxv8۴吅q 8ĥ)"6>~\8:qM}#͚'ĉ#p\׶ l#bA?)|g g9|8jP(cr,BwV (WliVxxᡁ@0Okn;ɥh$_ckCgriv}>=wGzβ KkBɛ[˪ !J)h&k2%07δt}!d<9;I&0wV/ v 0<H}L&8ob%Hi|޶o&h1L|u֦y~󛱢8fٲUsւ)0oiFx2}X[zVYr_;N(w]_4B@OanC?gĦx>мgx>ΛToZoOMp>40>V Oy V9iq!4 LN,ˢu{jsz]|"R޻&'ƚ{53ўFu(<٪9:΋]B;)B>1::8;~)Yt|0(pw2N%&X,URBK)3\zz&}ax4;ǟ(tLNg{N|Ǽ\G#C9g$^\}p?556]/RP.90 k,U8/u776s ʪ_01چ|\N 0VV*3H鴃J7iI!wG_^ypl}r*jɤSR 5QN@ iZ#1ٰy;_\3\BQQ x:WJv츟ٯ$"@6 S#qe딇(/P( Dy~TOϻ<4:-+F`0||;Xl-"uw$Цi󼕝mKʩorz"mϺ$F:~E'ҐvD\y?Rr8_He@ e~O,T.(ފR*cY^m|cVR[8 JҡSm!ΆԨb)RHG{?MpqrmN>߶Y)\p,d#xۆWY*,l6]v0h15M˙MS8+EdI='LBJIH7_9{Caз*Lq,dt >+~ّeʏ?xԕ4bBAŚjﵫ!'\Ը$WNvKO}ӽmSşذqsOy?\[,d@'73'j%kOe`1.g2"e =YIzS2|zŐƄa\U,dP;jhhhaxǶ?КZ՚.q SE+XrbOu%\GتX(H,N^~]JyEZQKceTQ]VGYqnah;y$cQahT&QPZ*iZ8UQQM.qo/T\7X"u?Mttl2Xq(IoW{R^ ux*SYJ! 4S.Jy~ BROS[V|žKNɛP(L6V^|cR7i7nZW1Fd@ Ara{詑|(T*dN]Ko?s=@ |_EvF]׍kR)eBJc" MUUbY6`~V޴dJKß&~'d3i WWWWWW
Current Directory: /usr/share/doc/python2-docs/html/library
Viewing File: /usr/share/doc/python2-docs/html/library/numbers.html
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd"> <html xmlns="http://www.w3.org/1999/xhtml"> <head> <meta http-equiv="X-UA-Compatible" content="IE=Edge" /> <meta http-equiv="Content-Type" content="text/html; charset=utf-8" /> <title>9.1. numbers — Numeric abstract base classes &#8212; Python 2.7.16 documentation</title> <link rel="stylesheet" href="../_static/classic.css" type="text/css" /> <link rel="stylesheet" href="../_static/pygments.css" type="text/css" /> <script type="text/javascript" id="documentation_options" data-url_root="../" src="../_static/documentation_options.js"></script> <script type="text/javascript" src="../_static/jquery.js"></script> <script type="text/javascript" src="../_static/underscore.js"></script> <script type="text/javascript" src="../_static/doctools.js"></script> <script type="text/javascript" src="../_static/sidebar.js"></script> <link rel="search" type="application/opensearchdescription+xml" title="Search within Python 2.7.16 documentation" href="../_static/opensearch.xml"/> <link rel="author" title="About these documents" href="../about.html" /> <link rel="index" title="Index" href="../genindex.html" /> <link rel="search" title="Search" href="../search.html" /> <link rel="copyright" title="Copyright" href="../copyright.html" /> <link rel="next" title="9.2. math — Mathematical functions" href="math.html" /> <link rel="prev" title="9. Numeric and Mathematical Modules" href="numeric.html" /> <link rel="shortcut icon" type="image/png" href="../_static/py.png" /> <link rel="canonical" href="https://docs.python.org/2/library/numbers.html" /> <script type="text/javascript" src="../_static/copybutton.js"></script> </head><body> <div class="related" role="navigation" aria-label="related navigation"> <h3>Navigation</h3> <ul> <li class="right" style="margin-right: 10px"> <a href="../genindex.html" title="General Index" accesskey="I">index</a></li> <li class="right" > <a href="../py-modindex.html" title="Python Module Index" >modules</a> |</li> <li class="right" > <a href="math.html" title="9.2. math — Mathematical functions" accesskey="N">next</a> |</li> <li class="right" > <a href="numeric.html" title="9. Numeric and Mathematical Modules" accesskey="P">previous</a> |</li> <li><img src="../_static/py.png" alt="" style="vertical-align: middle; margin-top: -1px"/></li> <li><a href="https://www.python.org/">Python</a> &#187;</li> <li> <a href="../index.html">Python 2.7.16 documentation</a> &#187; </li> <li class="nav-item nav-item-1"><a href="index.html" >The Python Standard Library</a> &#187;</li> <li class="nav-item nav-item-2"><a href="numeric.html" accesskey="U">9. Numeric and Mathematical Modules</a> &#187;</li> </ul> </div> <div class="document"> <div class="documentwrapper"> <div class="bodywrapper"> <div class="body" role="main"> <div class="section" id="module-numbers"> <span id="numbers-numeric-abstract-base-classes"></span><h1>9.1. <a class="reference internal" href="#module-numbers" title="numbers: Numeric abstract base classes (Complex, Real, Integral, etc.)."><code class="xref py py-mod docutils literal notranslate"><span class="pre">numbers</span></code></a> — Numeric abstract base classes<a class="headerlink" href="#module-numbers" title="Permalink to this headline">¶</a></h1> <div class="versionadded"> <p><span class="versionmodified">New in version 2.6.</span></p> </div> <p>The <a class="reference internal" href="#module-numbers" title="numbers: Numeric abstract base classes (Complex, Real, Integral, etc.)."><code class="xref py py-mod docutils literal notranslate"><span class="pre">numbers</span></code></a> module (<span class="target" id="index-0"></span><a class="pep reference external" href="https://www.python.org/dev/peps/pep-3141"><strong>PEP 3141</strong></a>) defines a hierarchy of numeric <a class="reference internal" href="../glossary.html#term-abstract-base-class"><span class="xref std std-term">abstract base classes</span></a> which progressively define more operations. None of the types defined in this module can be instantiated.</p> <dl class="class"> <dt id="numbers.Number"> <em class="property">class </em><code class="descclassname">numbers.</code><code class="descname">Number</code><a class="headerlink" href="#numbers.Number" title="Permalink to this definition">¶</a></dt> <dd><p>The root of the numeric hierarchy. If you just want to check if an argument <em>x</em> is a number, without caring what kind, use <code class="docutils literal notranslate"><span class="pre">isinstance(x,</span> <span class="pre">Number)</span></code>.</p> </dd></dl> <div class="section" id="the-numeric-tower"> <h2>9.1.1. The numeric tower<a class="headerlink" href="#the-numeric-tower" title="Permalink to this headline">¶</a></h2> <dl class="class"> <dt id="numbers.Complex"> <em class="property">class </em><code class="descclassname">numbers.</code><code class="descname">Complex</code><a class="headerlink" href="#numbers.Complex" title="Permalink to this definition">¶</a></dt> <dd><p>Subclasses of this type describe complex numbers and include the operations that work on the built-in <a class="reference internal" href="functions.html#complex" title="complex"><code class="xref py py-class docutils literal notranslate"><span class="pre">complex</span></code></a> type. These are: conversions to <a class="reference internal" href="functions.html#complex" title="complex"><code class="xref py py-class docutils literal notranslate"><span class="pre">complex</span></code></a> and <a class="reference internal" href="functions.html#bool" title="bool"><code class="xref py py-class docutils literal notranslate"><span class="pre">bool</span></code></a>, <a class="reference internal" href="#numbers.Complex.real" title="numbers.Complex.real"><code class="xref py py-attr docutils literal notranslate"><span class="pre">real</span></code></a>, <a class="reference internal" href="#numbers.Complex.imag" title="numbers.Complex.imag"><code class="xref py py-attr docutils literal notranslate"><span class="pre">imag</span></code></a>, <code class="docutils literal notranslate"><span class="pre">+</span></code>, <code class="docutils literal notranslate"><span class="pre">-</span></code>, <code class="docutils literal notranslate"><span class="pre">*</span></code>, <code class="docutils literal notranslate"><span class="pre">/</span></code>, <a class="reference internal" href="functions.html#abs" title="abs"><code class="xref py py-func docutils literal notranslate"><span class="pre">abs()</span></code></a>, <a class="reference internal" href="#numbers.Complex.conjugate" title="numbers.Complex.conjugate"><code class="xref py py-meth docutils literal notranslate"><span class="pre">conjugate()</span></code></a>, <code class="docutils literal notranslate"><span class="pre">==</span></code>, and <code class="docutils literal notranslate"><span class="pre">!=</span></code>. All except <code class="docutils literal notranslate"><span class="pre">-</span></code> and <code class="docutils literal notranslate"><span class="pre">!=</span></code> are abstract.</p> <dl class="attribute"> <dt id="numbers.Complex.real"> <code class="descname">real</code><a class="headerlink" href="#numbers.Complex.real" title="Permalink to this definition">¶</a></dt> <dd><p>Abstract. Retrieves the real component of this number.</p> </dd></dl> <dl class="attribute"> <dt id="numbers.Complex.imag"> <code class="descname">imag</code><a class="headerlink" href="#numbers.Complex.imag" title="Permalink to this definition">¶</a></dt> <dd><p>Abstract. Retrieves the imaginary component of this number.</p> </dd></dl> <dl class="method"> <dt id="numbers.Complex.conjugate"> <code class="descname">conjugate</code><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="headerlink" href="#numbers.Complex.conjugate" title="Permalink to this definition">¶</a></dt> <dd><p>Abstract. Returns the complex conjugate. For example, <code class="docutils literal notranslate"><span class="pre">(1+3j).conjugate()</span> <span class="pre">==</span> <span class="pre">(1-3j)</span></code>.</p> </dd></dl> </dd></dl> <dl class="class"> <dt id="numbers.Real"> <em class="property">class </em><code class="descclassname">numbers.</code><code class="descname">Real</code><a class="headerlink" href="#numbers.Real" title="Permalink to this definition">¶</a></dt> <dd><p>To <a class="reference internal" href="#numbers.Complex" title="numbers.Complex"><code class="xref py py-class docutils literal notranslate"><span class="pre">Complex</span></code></a>, <a class="reference internal" href="#numbers.Real" title="numbers.Real"><code class="xref py py-class docutils literal notranslate"><span class="pre">Real</span></code></a> adds the operations that work on real numbers.</p> <p>In short, those are: a conversion to <a class="reference internal" href="functions.html#float" title="float"><code class="xref py py-class docutils literal notranslate"><span class="pre">float</span></code></a>, <a class="reference internal" href="math.html#math.trunc" title="math.trunc"><code class="xref py py-func docutils literal notranslate"><span class="pre">math.trunc()</span></code></a>, <a class="reference internal" href="functions.html#round" title="round"><code class="xref py py-func docutils literal notranslate"><span class="pre">round()</span></code></a>, <a class="reference internal" href="math.html#math.floor" title="math.floor"><code class="xref py py-func docutils literal notranslate"><span class="pre">math.floor()</span></code></a>, <a class="reference internal" href="math.html#math.ceil" title="math.ceil"><code class="xref py py-func docutils literal notranslate"><span class="pre">math.ceil()</span></code></a>, <a class="reference internal" href="functions.html#divmod" title="divmod"><code class="xref py py-func docutils literal notranslate"><span class="pre">divmod()</span></code></a>, <code class="docutils literal notranslate"><span class="pre">//</span></code>, <code class="docutils literal notranslate"><span class="pre">%</span></code>, <code class="docutils literal notranslate"><span class="pre">&lt;</span></code>, <code class="docutils literal notranslate"><span class="pre">&lt;=</span></code>, <code class="docutils literal notranslate"><span class="pre">&gt;</span></code>, and <code class="docutils literal notranslate"><span class="pre">&gt;=</span></code>.</p> <p>Real also provides defaults for <a class="reference internal" href="functions.html#complex" title="complex"><code class="xref py py-func docutils literal notranslate"><span class="pre">complex()</span></code></a>, <a class="reference internal" href="#numbers.Complex.real" title="numbers.Complex.real"><code class="xref py py-attr docutils literal notranslate"><span class="pre">real</span></code></a>, <a class="reference internal" href="#numbers.Complex.imag" title="numbers.Complex.imag"><code class="xref py py-attr docutils literal notranslate"><span class="pre">imag</span></code></a>, and <a class="reference internal" href="#numbers.Complex.conjugate" title="numbers.Complex.conjugate"><code class="xref py py-meth docutils literal notranslate"><span class="pre">conjugate()</span></code></a>.</p> </dd></dl> <dl class="class"> <dt id="numbers.Rational"> <em class="property">class </em><code class="descclassname">numbers.</code><code class="descname">Rational</code><a class="headerlink" href="#numbers.Rational" title="Permalink to this definition">¶</a></dt> <dd><p>Subtypes <a class="reference internal" href="#numbers.Real" title="numbers.Real"><code class="xref py py-class docutils literal notranslate"><span class="pre">Real</span></code></a> and adds <a class="reference internal" href="#numbers.Rational.numerator" title="numbers.Rational.numerator"><code class="xref py py-attr docutils literal notranslate"><span class="pre">numerator</span></code></a> and <a class="reference internal" href="#numbers.Rational.denominator" title="numbers.Rational.denominator"><code class="xref py py-attr docutils literal notranslate"><span class="pre">denominator</span></code></a> properties, which should be in lowest terms. With these, it provides a default for <a class="reference internal" href="functions.html#float" title="float"><code class="xref py py-func docutils literal notranslate"><span class="pre">float()</span></code></a>.</p> <dl class="attribute"> <dt id="numbers.Rational.numerator"> <code class="descname">numerator</code><a class="headerlink" href="#numbers.Rational.numerator" title="Permalink to this definition">¶</a></dt> <dd><p>Abstract.</p> </dd></dl> <dl class="attribute"> <dt id="numbers.Rational.denominator"> <code class="descname">denominator</code><a class="headerlink" href="#numbers.Rational.denominator" title="Permalink to this definition">¶</a></dt> <dd><p>Abstract.</p> </dd></dl> </dd></dl> <dl class="class"> <dt id="numbers.Integral"> <em class="property">class </em><code class="descclassname">numbers.</code><code class="descname">Integral</code><a class="headerlink" href="#numbers.Integral" title="Permalink to this definition">¶</a></dt> <dd><p>Subtypes <a class="reference internal" href="#numbers.Rational" title="numbers.Rational"><code class="xref py py-class docutils literal notranslate"><span class="pre">Rational</span></code></a> and adds a conversion to <a class="reference internal" href="functions.html#int" title="int"><code class="xref py py-class docutils literal notranslate"><span class="pre">int</span></code></a>. Provides defaults for <a class="reference internal" href="functions.html#float" title="float"><code class="xref py py-func docutils literal notranslate"><span class="pre">float()</span></code></a>, <a class="reference internal" href="#numbers.Rational.numerator" title="numbers.Rational.numerator"><code class="xref py py-attr docutils literal notranslate"><span class="pre">numerator</span></code></a>, and <a class="reference internal" href="#numbers.Rational.denominator" title="numbers.Rational.denominator"><code class="xref py py-attr docutils literal notranslate"><span class="pre">denominator</span></code></a>. Adds abstract methods for <code class="docutils literal notranslate"><span class="pre">**</span></code> and bit-string operations: <code class="docutils literal notranslate"><span class="pre">&lt;&lt;</span></code>, <code class="docutils literal notranslate"><span class="pre">&gt;&gt;</span></code>, <code class="docutils literal notranslate"><span class="pre">&amp;</span></code>, <code class="docutils literal notranslate"><span class="pre">^</span></code>, <code class="docutils literal notranslate"><span class="pre">|</span></code>, <code class="docutils literal notranslate"><span class="pre">~</span></code>.</p> </dd></dl> </div> <div class="section" id="notes-for-type-implementors"> <h2>9.1.2. Notes for type implementors<a class="headerlink" href="#notes-for-type-implementors" title="Permalink to this headline">¶</a></h2> <p>Implementors should be careful to make equal numbers equal and hash them to the same values. This may be subtle if there are two different extensions of the real numbers. For example, <a class="reference internal" href="fractions.html#fractions.Fraction" title="fractions.Fraction"><code class="xref py py-class docutils literal notranslate"><span class="pre">fractions.Fraction</span></code></a> implements <a class="reference internal" href="functions.html#hash" title="hash"><code class="xref py py-func docutils literal notranslate"><span class="pre">hash()</span></code></a> as follows:</p> <div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">__hash__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span> <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">denominator</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span> <span class="c1"># Get integers right.</span> <span class="k">return</span> <span class="nb">hash</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">numerator</span><span class="p">)</span> <span class="c1"># Expensive check, but definitely correct.</span> <span class="k">if</span> <span class="bp">self</span> <span class="o">==</span> <span class="nb">float</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span> <span class="k">return</span> <span class="nb">hash</span><span class="p">(</span><span class="nb">float</span><span class="p">(</span><span class="bp">self</span><span class="p">))</span> <span class="k">else</span><span class="p">:</span> <span class="c1"># Use tuple&#39;s hash to avoid a high collision rate on</span> <span class="c1"># simple fractions.</span> <span class="k">return</span> <span class="nb">hash</span><span class="p">((</span><span class="bp">self</span><span class="o">.</span><span class="n">numerator</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">denominator</span><span class="p">))</span> </pre></div> </div> <div class="section" id="adding-more-numeric-abcs"> <h3>9.1.2.1. Adding More Numeric ABCs<a class="headerlink" href="#adding-more-numeric-abcs" title="Permalink to this headline">¶</a></h3> <p>There are, of course, more possible ABCs for numbers, and this would be a poor hierarchy if it precluded the possibility of adding those. You can add <code class="docutils literal notranslate"><span class="pre">MyFoo</span></code> between <a class="reference internal" href="#numbers.Complex" title="numbers.Complex"><code class="xref py py-class docutils literal notranslate"><span class="pre">Complex</span></code></a> and <a class="reference internal" href="#numbers.Real" title="numbers.Real"><code class="xref py py-class docutils literal notranslate"><span class="pre">Real</span></code></a> with:</p> <div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="k">class</span> <span class="nc">MyFoo</span><span class="p">(</span><span class="n">Complex</span><span class="p">):</span> <span class="o">...</span> <span class="n">MyFoo</span><span class="o">.</span><span class="n">register</span><span class="p">(</span><span class="n">Real</span><span class="p">)</span> </pre></div> </div> </div> <div class="section" id="implementing-the-arithmetic-operations"> <h3>9.1.2.2. Implementing the arithmetic operations<a class="headerlink" href="#implementing-the-arithmetic-operations" title="Permalink to this headline">¶</a></h3> <p>We want to implement the arithmetic operations so that mixed-mode operations either call an implementation whose author knew about the types of both arguments, or convert both to the nearest built in type and do the operation there. For subtypes of <a class="reference internal" href="#numbers.Integral" title="numbers.Integral"><code class="xref py py-class docutils literal notranslate"><span class="pre">Integral</span></code></a>, this means that <a class="reference internal" href="../reference/datamodel.html#object.__add__" title="object.__add__"><code class="xref py py-meth docutils literal notranslate"><span class="pre">__add__()</span></code></a> and <a class="reference internal" href="../reference/datamodel.html#object.__radd__" title="object.__radd__"><code class="xref py py-meth docutils literal notranslate"><span class="pre">__radd__()</span></code></a> should be defined as:</p> <div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="k">class</span> <span class="nc">MyIntegral</span><span class="p">(</span><span class="n">Integral</span><span class="p">):</span> <span class="k">def</span> <span class="nf">__add__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span> <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">MyIntegral</span><span class="p">):</span> <span class="k">return</span> <span class="n">do_my_adding_stuff</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span> <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">OtherTypeIKnowAbout</span><span class="p">):</span> <span class="k">return</span> <span class="n">do_my_other_adding_stuff</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">)</span> <span class="k">else</span><span class="p">:</span> <span class="k">return</span> <span class="bp">NotImplemented</span> <span class="k">def</span> <span class="nf">__radd__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span> <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">MyIntegral</span><span class="p">):</span> <span class="k">return</span> <span class="n">do_my_adding_stuff</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span> <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">OtherTypeIKnowAbout</span><span class="p">):</span> <span class="k">return</span> <span class="n">do_my_other_adding_stuff</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span> <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Integral</span><span class="p">):</span> <span class="k">return</span> <span class="nb">int</span><span class="p">(</span><span class="n">other</span><span class="p">)</span> <span class="o">+</span> <span class="nb">int</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span> <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Real</span><span class="p">):</span> <span class="k">return</span> <span class="nb">float</span><span class="p">(</span><span class="n">other</span><span class="p">)</span> <span class="o">+</span> <span class="nb">float</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span> <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">other</span><span class="p">,</span> <span class="n">Complex</span><span class="p">):</span> <span class="k">return</span> <span class="nb">complex</span><span class="p">(</span><span class="n">other</span><span class="p">)</span> <span class="o">+</span> <span class="nb">complex</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span> <span class="k">else</span><span class="p">:</span> <span class="k">return</span> <span class="bp">NotImplemented</span> </pre></div> </div> <p>There are 5 different cases for a mixed-type operation on subclasses of <a class="reference internal" href="#numbers.Complex" title="numbers.Complex"><code class="xref py py-class docutils literal notranslate"><span class="pre">Complex</span></code></a>. I’ll refer to all of the above code that doesn’t refer to <code class="docutils literal notranslate"><span class="pre">MyIntegral</span></code> and <code class="docutils literal notranslate"><span class="pre">OtherTypeIKnowAbout</span></code> as “boilerplate”. <code class="docutils literal notranslate"><span class="pre">a</span></code> will be an instance of <code class="docutils literal notranslate"><span class="pre">A</span></code>, which is a subtype of <a class="reference internal" href="#numbers.Complex" title="numbers.Complex"><code class="xref py py-class docutils literal notranslate"><span class="pre">Complex</span></code></a> (<code class="docutils literal notranslate"><span class="pre">a</span> <span class="pre">:</span> <span class="pre">A</span> <span class="pre">&lt;:</span> <span class="pre">Complex</span></code>), and <code class="docutils literal notranslate"><span class="pre">b</span> <span class="pre">:</span> <span class="pre">B</span> <span class="pre">&lt;:</span> <span class="pre">Complex</span></code>. I’ll consider <code class="docutils literal notranslate"><span class="pre">a</span> <span class="pre">+</span> <span class="pre">b</span></code>:</p> <blockquote> <div><ol class="arabic simple"> <li>If <code class="docutils literal notranslate"><span class="pre">A</span></code> defines an <a class="reference internal" href="../reference/datamodel.html#object.__add__" title="object.__add__"><code class="xref py py-meth docutils literal notranslate"><span class="pre">__add__()</span></code></a> which accepts <code class="docutils literal notranslate"><span class="pre">b</span></code>, all is well.</li> <li>If <code class="docutils literal notranslate"><span class="pre">A</span></code> falls back to the boilerplate code, and it were to return a value from <a class="reference internal" href="../reference/datamodel.html#object.__add__" title="object.__add__"><code class="xref py py-meth docutils literal notranslate"><span class="pre">__add__()</span></code></a>, we’d miss the possibility that <code class="docutils literal notranslate"><span class="pre">B</span></code> defines a more intelligent <a class="reference internal" href="../reference/datamodel.html#object.__radd__" title="object.__radd__"><code class="xref py py-meth docutils literal notranslate"><span class="pre">__radd__()</span></code></a>, so the boilerplate should return <a class="reference internal" href="constants.html#NotImplemented" title="NotImplemented"><code class="xref py py-const docutils literal notranslate"><span class="pre">NotImplemented</span></code></a> from <a class="reference internal" href="../reference/datamodel.html#object.__add__" title="object.__add__"><code class="xref py py-meth docutils literal notranslate"><span class="pre">__add__()</span></code></a>. (Or <code class="docutils literal notranslate"><span class="pre">A</span></code> may not implement <a class="reference internal" href="../reference/datamodel.html#object.__add__" title="object.__add__"><code class="xref py py-meth docutils literal notranslate"><span class="pre">__add__()</span></code></a> at all.)</li> <li>Then <code class="docutils literal notranslate"><span class="pre">B</span></code>’s <a class="reference internal" href="../reference/datamodel.html#object.__radd__" title="object.__radd__"><code class="xref py py-meth docutils literal notranslate"><span class="pre">__radd__()</span></code></a> gets a chance. If it accepts <code class="docutils literal notranslate"><span class="pre">a</span></code>, all is well.</li> <li>If it falls back to the boilerplate, there are no more possible methods to try, so this is where the default implementation should live.</li> <li>If <code class="docutils literal notranslate"><span class="pre">B</span> <span class="pre">&lt;:</span> <span class="pre">A</span></code>, Python tries <code class="docutils literal notranslate"><span class="pre">B.__radd__</span></code> before <code class="docutils literal notranslate"><span class="pre">A.__add__</span></code>. This is ok, because it was implemented with knowledge of <code class="docutils literal notranslate"><span class="pre">A</span></code>, so it can handle those instances before delegating to <a class="reference internal" href="#numbers.Complex" title="numbers.Complex"><code class="xref py py-class docutils literal notranslate"><span class="pre">Complex</span></code></a>.</li> </ol> </div></blockquote> <p>If <code class="docutils literal notranslate"><span class="pre">A</span> <span class="pre">&lt;:</span> <span class="pre">Complex</span></code> and <code class="docutils literal notranslate"><span class="pre">B</span> <span class="pre">&lt;:</span> <span class="pre">Real</span></code> without sharing any other knowledge, then the appropriate shared operation is the one involving the built in <a class="reference internal" href="functions.html#complex" title="complex"><code class="xref py py-class docutils literal notranslate"><span class="pre">complex</span></code></a>, and both <a class="reference internal" href="../reference/datamodel.html#object.__radd__" title="object.__radd__"><code class="xref py py-meth docutils literal notranslate"><span class="pre">__radd__()</span></code></a> s land there, so <code class="docutils literal notranslate"><span class="pre">a+b</span> <span class="pre">==</span> <span class="pre">b+a</span></code>.</p> <p>Because most of the operations on any given type will be very similar, it can be useful to define a helper function which generates the forward and reverse instances of any given operator. For example, <a class="reference internal" href="fractions.html#fractions.Fraction" title="fractions.Fraction"><code class="xref py py-class docutils literal notranslate"><span class="pre">fractions.Fraction</span></code></a> uses:</p> <div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">_operator_fallbacks</span><span class="p">(</span><span class="n">monomorphic_operator</span><span class="p">,</span> <span class="n">fallback_operator</span><span class="p">):</span> <span class="k">def</span> <span class="nf">forward</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">):</span> <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="p">(</span><span class="nb">int</span><span class="p">,</span> <span class="n">long</span><span class="p">,</span> <span class="n">Fraction</span><span class="p">)):</span> <span class="k">return</span> <span class="n">monomorphic_operator</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span> <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="nb">float</span><span class="p">):</span> <span class="k">return</span> <span class="n">fallback_operator</span><span class="p">(</span><span class="nb">float</span><span class="p">(</span><span class="n">a</span><span class="p">),</span> <span class="n">b</span><span class="p">)</span> <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="nb">complex</span><span class="p">):</span> <span class="k">return</span> <span class="n">fallback_operator</span><span class="p">(</span><span class="nb">complex</span><span class="p">(</span><span class="n">a</span><span class="p">),</span> <span class="n">b</span><span class="p">)</span> <span class="k">else</span><span class="p">:</span> <span class="k">return</span> <span class="bp">NotImplemented</span> <span class="n">forward</span><span class="o">.</span><span class="vm">__name__</span> <span class="o">=</span> <span class="s1">&#39;__&#39;</span> <span class="o">+</span> <span class="n">fallback_operator</span><span class="o">.</span><span class="vm">__name__</span> <span class="o">+</span> <span class="s1">&#39;__&#39;</span> <span class="n">forward</span><span class="o">.</span><span class="vm">__doc__</span> <span class="o">=</span> <span class="n">monomorphic_operator</span><span class="o">.</span><span class="vm">__doc__</span> <span class="k">def</span> <span class="nf">reverse</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">a</span><span class="p">):</span> <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">Rational</span><span class="p">):</span> <span class="c1"># Includes ints.</span> <span class="k">return</span> <span class="n">monomorphic_operator</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span> <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">numbers</span><span class="o">.</span><span class="n">Real</span><span class="p">):</span> <span class="k">return</span> <span class="n">fallback_operator</span><span class="p">(</span><span class="nb">float</span><span class="p">(</span><span class="n">a</span><span class="p">),</span> <span class="nb">float</span><span class="p">(</span><span class="n">b</span><span class="p">))</span> <span class="k">elif</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">numbers</span><span class="o">.</span><span class="n">Complex</span><span class="p">):</span> <span class="k">return</span> <span class="n">fallback_operator</span><span class="p">(</span><span class="nb">complex</span><span class="p">(</span><span class="n">a</span><span class="p">),</span> <span class="nb">complex</span><span class="p">(</span><span class="n">b</span><span class="p">))</span> <span class="k">else</span><span class="p">:</span> <span class="k">return</span> <span class="bp">NotImplemented</span> <span class="n">reverse</span><span class="o">.</span><span class="vm">__name__</span> <span class="o">=</span> <span class="s1">&#39;__r&#39;</span> <span class="o">+</span> <span class="n">fallback_operator</span><span class="o">.</span><span class="vm">__name__</span> <span class="o">+</span> <span class="s1">&#39;__&#39;</span> <span class="n">reverse</span><span class="o">.</span><span class="vm">__doc__</span> <span class="o">=</span> <span class="n">monomorphic_operator</span><span class="o">.</span><span class="vm">__doc__</span> <span class="k">return</span> <span class="n">forward</span><span class="p">,</span> <span class="n">reverse</span> <span class="k">def</span> <span class="nf">_add</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">):</span> <span class="sd">&quot;&quot;&quot;a + b&quot;&quot;&quot;</span> <span class="k">return</span> <span class="n">Fraction</span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">numerator</span> <span class="o">*</span> <span class="n">b</span><span class="o">.</span><span class="n">denominator</span> <span class="o">+</span> <span class="n">b</span><span class="o">.</span><span class="n">numerator</span> <span class="o">*</span> <span class="n">a</span><span class="o">.</span><span class="n">denominator</span><span class="p">,</span> <span class="n">a</span><span class="o">.</span><span class="n">denominator</span> <span class="o">*</span> <span class="n">b</span><span class="o">.</span><span class="n">denominator</span><span class="p">)</span> <span class="fm">__add__</span><span class="p">,</span> <span class="fm">__radd__</span> <span class="o">=</span> <span class="n">_operator_fallbacks</span><span class="p">(</span><span class="n">_add</span><span class="p">,</span> <span class="n">operator</span><span class="o">.</span><span class="n">add</span><span class="p">)</span> <span class="c1"># ...</span> </pre></div> </div> </div> </div> </div> </div> </div> </div> <div class="sphinxsidebar" role="navigation" aria-label="main navigation"> <div class="sphinxsidebarwrapper"> <h3><a href="../contents.html">Table Of Contents</a></h3> <ul> <li><a class="reference internal" href="#">9.1. <code class="docutils literal notranslate"><span class="pre">numbers</span></code> — Numeric abstract base classes</a><ul> <li><a class="reference internal" href="#the-numeric-tower">9.1.1. The numeric tower</a></li> <li><a class="reference internal" href="#notes-for-type-implementors">9.1.2. Notes for type implementors</a><ul> <li><a class="reference internal" href="#adding-more-numeric-abcs">9.1.2.1. Adding More Numeric ABCs</a></li> <li><a class="reference internal" href="#implementing-the-arithmetic-operations">9.1.2.2. Implementing the arithmetic operations</a></li> </ul> </li> </ul> </li> </ul> <h4>Previous topic</h4> <p class="topless"><a href="numeric.html" title="previous chapter">9. Numeric and Mathematical Modules</a></p> <h4>Next topic</h4> <p class="topless"><a href="math.html" title="next chapter">9.2. <code class="docutils literal notranslate"><span class="pre">math</span></code> — Mathematical functions</a></p> <div role="note" aria-label="source link"> <h3>This Page</h3> <ul class="this-page-menu"> <li><a href="../_sources/library/numbers.rst.txt" rel="nofollow">Show Source</a></li> </ul> </div> <div id="searchbox" style="display: none" role="search"> <h3>Quick search</h3> <div class="searchformwrapper"> <form class="search" action="../search.html" method="get"> <input type="text" name="q" /> <input type="submit" value="Go" /> <input type="hidden" name="check_keywords" value="yes" /> <input type="hidden" name="area" value="default" /> </form> </div> </div> <script type="text/javascript">$('#searchbox').show(0);</script> </div> </div> <div class="clearer"></div> </div> <div class="related" role="navigation" aria-label="related navigation"> <h3>Navigation</h3> <ul> <li class="right" style="margin-right: 10px"> <a href="../genindex.html" title="General Index" >index</a></li> <li class="right" > <a href="../py-modindex.html" title="Python Module Index" >modules</a> |</li> <li class="right" > <a href="math.html" title="9.2. math — Mathematical functions" >next</a> |</li> <li class="right" > <a href="numeric.html" title="9. Numeric and Mathematical Modules" >previous</a> |</li> <li><img src="../_static/py.png" alt="" style="vertical-align: middle; margin-top: -1px"/></li> <li><a href="https://www.python.org/">Python</a> &#187;</li> <li> <a href="../index.html">Python 2.7.16 documentation</a> &#187; </li> <li class="nav-item nav-item-1"><a href="index.html" >The Python Standard Library</a> &#187;</li> <li class="nav-item nav-item-2"><a href="numeric.html" >9. Numeric and Mathematical Modules</a> &#187;</li> </ul> </div> <div class="footer"> &copy; <a href="../copyright.html">Copyright</a> 1990-2019, Python Software Foundation. <br /> The Python Software Foundation is a non-profit corporation. <a href="https://www.python.org/psf/donations/">Please donate.</a> <br /> Last updated on Mar 27, 2019. <a href="../bugs.html">Found a bug</a>? <br /> Created using <a href="http://sphinx.pocoo.org/">Sphinx</a> 1.7.6. </div> </body> </html>