[{"data":1,"prerenderedAt":2357},["ShallowReactive",2],{"post-complement":3,"home":2335},{"id":4,"title":5,"body":6,"date":2324,"description":2325,"extension":2326,"meta":2327,"navigation":2328,"path":2329,"seo":2330,"stem":2331,"tags":2332,"__hash__":2334},"blog\u002Fblog\u002Fcomplement.md","补数与补码",{"type":7,"value":8,"toc":2310},"minimark",[9,49,53,72,77,83,86,95,105,109,123,126,277,444,447,450,453,456,465,468,471,479,482,490,495,498,501,506,517,520,534,537,540,696,2130,2133],[10,11,12,13,17,18,21,22,25,26,29,32,33,36,37,40,41,44,45,48],"p",{},"始于《C++ Primer》中对超出无符号数范围的数的转换，转为无符号数后值为",[14,15,16],"code",{},"{num} mod {count}","，\n",[14,19,20],{},"{num}","即为原来的数，",[14,23,24],{},"{count}","为无符号数据类型可代表的数的总和，如：",[27,28],"br",{},[14,30,31],{},"unsigned char","可表示0~255共256个数, ",[14,34,35],{},"unsigned char a = -1","，实际上",[14,38,39],{},"a","的值为",[14,42,43],{},"255","，由",[14,46,47],{},"-1 mod 256","得出",[50,51,52],"h2",{"id":52},"模运算",[10,54,55,56,58,59,63,64,67,68,71],{},"商余定理：",[27,57],{},"\n给出",[60,61,62],"strong",{},"任何整数","A和一个",[60,65,66],{},"正整数","B，存在",[60,69,70],{},"唯一整数Q和R","，满足：",[10,73,74],{},[60,75,76],{},"A = B * Q + R，其中0 \u003C= R \u003C B",[10,78,79,80],{},"A除以B，Q是商，R是余数。以另一种形式书写就是",[60,81,82],{},"A mod B = R",[10,84,85],{},"求A mod B时，R = A - B * Q，求出商Q即可。而由于有多种取整方法，R的值随着取整方法的不同而变化",[87,88,89,92],"blockquote",{},[10,90,91],{},"ceil()，floor()，int()，round()分别为向上取整，向下取整，向零取整以及四舍五入",[10,93,94],{},"此处只讨论floor()和int()。并且正数取整时，这两个方法得到的结果相同，因此只讨论负数，即A \u003C 0时",[96,97,98,102],"ul",{},[99,100,101],"li",{},"向下取整，|Q| > |A\u002FB|，即BQ \u003C A，一个负数减去一个更小的负数得到R > 0",[99,103,104],{},"向零取整，|Q| \u003C |A\u002FB|，即BQ > A，R \u003C 0",[106,107,108],"h3",{"id":108},"运算性质",[96,110,111,114,117,120],{},[99,112,113],{},"(A + B) mod C = (A mod C + B mod C) mod C",[99,115,116],{},"(A - B) mod C = (A mod C - B mod C) mod C",[99,118,119],{},"(A * B) mod C = (A mod C * B mod C) mod C",[99,121,122],{},"A^B mod C     = ((A mod C)^B) mod C",[50,124,125],{"id":125},"同余",[10,127,128,129],{},"A与B模C同余，表示A mod C = B mod C，记作",[130,131,134,195],"span",{"className":132},[133],"katex",[130,135,138],{"className":136},[137],"katex-mathml",[139,140,142],"math",{"xmlns":141},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[143,144,145,190],"semantics",{},[146,147,148,152,156,159,162,165,169,181,184,187],"mrow",{},[149,150,151],"mi",{},"A",[153,154,155],"mo",{},"≡",[149,157,158],{},"B",[160,161],"mspace",{},[160,163],{"width":164},"0.4444em",[153,166,168],{"stretchy":167},"false","(",[146,170,171,175,178],{},[149,172,174],{"mathvariant":173},"normal","m",[149,176,177],{"mathvariant":173},"o",[149,179,180],{"mathvariant":173},"d",[160,182],{"width":183},"0.3333em",[149,185,186],{},"C",[153,188,189],{"stretchy":167},")",[191,192,194],"annotation",{"encoding":193},"application\u002Fx-tex","A\\equiv B\\pmod{C}",[130,196,200,225,243],{"className":197,"ariaHidden":199},[198],"katex-html","true",[130,201,204,209,214,218,222],{"className":202},[203],"base",[130,205],{"className":206,"style":208},[207],"strut","height:0.6833em;",[130,210,151],{"className":211},[212,213],"mord","mathnormal",[130,215],{"className":216,"style":217},[160],"margin-right:0.2778em;",[130,219,155],{"className":220},[221],"mrel",[130,223],{"className":224,"style":217},[160],[130,226,228,231,235,239],{"className":227},[203],[130,229],{"className":230,"style":208},[207],[130,232,158],{"className":233,"style":234},[212,213],"margin-right:0.0502em;",[130,236],{"className":237},[160,238],"allowbreak",[130,240],{"className":241,"style":242},[160],"margin-right:0.4444em;",[130,244,246,250,254,265,269,273],{"className":245},[203],[130,247],{"className":248,"style":249},[207],"height:1em;vertical-align:-0.25em;",[130,251,168],{"className":252},[253],"mopen",[130,255,257],{"className":256},[212],[130,258,260],{"className":259},[212],[130,261,264],{"className":262},[212,263],"mathrm","mod",[130,266],{"className":267,"style":268},[160],"margin-right:0.3333em;",[130,270,186],{"className":271,"style":272},[212,213],"margin-right:0.0715em;",[130,274,189],{"className":275},[276],"mclose",[96,278,279,441],{},[99,280,281,282,440],{},"C|(A-B)，\"|\"符号意味整除，C能整除(A-B)，即",[130,283,285,317],{"className":284},[133],[130,286,288],{"className":287},[137],[139,289,290],{"xmlns":141},[143,291,292,314],{},[146,293,294,308,311],{},[295,296,297,306],"mfrac",{},[146,298,299,301,304],{},[149,300,151],{},[153,302,303],{},"−",[149,305,158],{},[149,307,186],{},[153,309,310],{},"=",[149,312,313],{},"K",[191,315,316],{"encoding":193},"\\frac{A-B}{C}=K",[130,318,320,431],{"className":319,"ariaHidden":199},[198],[130,321,323,327,422,425,428],{"className":322},[203],[130,324],{"className":325,"style":326},[207],"height:1.2173em;vertical-align:-0.345em;",[130,328,330,334,419],{"className":329},[212],[130,331],{"className":332},[253,333],"nulldelimiter",[130,335,337],{"className":336},[295],[130,338,342,410],{"className":339},[340,341],"vlist-t","vlist-t2",[130,343,346,405],{"className":344},[345],"vlist-r",[130,347,351,372,383],{"className":348,"style":350},[349],"vlist","height:0.8723em;",[130,352,354,359],{"style":353},"top:-2.655em;",[130,355],{"className":356,"style":358},[357],"pstrut","height:3em;",[130,360,366],{"className":361},[362,363,364,365],"sizing","reset-size6","size3","mtight",[130,367,369],{"className":368},[212,365],[130,370,186],{"className":371,"style":272},[212,213,365],[130,373,375,378],{"style":374},"top:-3.23em;",[130,376],{"className":377,"style":358},[357],[130,379],{"className":380,"style":382},[381],"frac-line","border-bottom-width:0.04em;",[130,384,386,389],{"style":385},"top:-3.394em;",[130,387],{"className":388,"style":358},[357],[130,390,392],{"className":391},[362,363,364,365],[130,393,395,398,402],{"className":394},[212,365],[130,396,151],{"className":397},[212,213,365],[130,399,303],{"className":400},[401,365],"mbin",[130,403,158],{"className":404,"style":234},[212,213,365],[130,406,409],{"className":407},[408],"vlist-s","​",[130,411,413],{"className":412},[345],[130,414,417],{"className":415,"style":416},[349],"height:0.345em;",[130,418],{},[130,420],{"className":421},[276,333],[130,423],{"className":424,"style":217},[160],[130,426,310],{"className":427},[221],[130,429],{"className":430,"style":217},[160],[130,432,434,437],{"className":433},[203],[130,435],{"className":436,"style":208},[207],[130,438,313],{"className":439,"style":272},[212,213],"，与下面相同",[99,442,443],{},"A = B + K * C，A与B之间相差K个C。(K为整数)",[10,445,446],{},"同时，一个数X。当A ≡ B(mod C)时，A与B相差K倍的C，有：",[10,448,449],{},"(X + A) ≡ (X + B) mod C",[50,451,452],{"id":452},"补数",[10,454,455],{},"假设现在时间为10点整，要想让时针指向2点，有两种方式：",[457,458,459,462],"ol",{},[99,460,461],{},"顺时针拨动4个小时",[99,463,464],{},"逆时针拨动8个小时",[10,466,467],{},"钟表时间以12为模，在模的范围内，两个相加等于模的数互为补数(complement)",[10,469,470],{},"在模的范围内做减法，即逆时针转动8个小时(-8 mod 12 = 4)与做加法，即顺时针转动4个小时(4 mod 12 = 4)得到的结果相同",[87,472,473,476],{},[10,474,475],{},"-8与4同余，有(X + (-8)) mod 12 = (X + 4) mod 12",[10,477,478],{},"因此可通过同余，令减法变成加法",[10,480,481],{},"在0~99中，以100为模，(30 + (- 40)) mod 100 = (30 + 60) mod 100，\"60\"可以替代\"-40\"进行加法运算，但是如果60用来表示-40，\n它也有原本的值，造成二义性",[10,483,484,485,489],{},"为解决该问题，将模范围内的数划分，0",[486,487,488],"del",{},"49区间的表示原来的值，而与之对应的补数来表示其负数，例如99的补数为1即\n99表示-1。当然，牺牲了一半的值表示负数，在以100为模的范围内，50","99的这些正数就无法表示了。要想取到50~99，可在\n更大的模中划分",[87,491,492],{},[10,493,494],{},"取模为C，一个半模以下的正整数A，补数为B。B用来表示-A，同时 B - (-A) = C - A + A = C，即B与(-A) mod C同余",[50,496,497],{"id":497},"二进制",[10,499,500],{},"二进制中划分范围后，半模以下的数最高位为0，半模以上的数最高位为1。这就是补码最高位为符号位的原因",[87,502,503],{},[10,504,505],{},"符号位的0, 1并不是为了表示正负，只是半模以上数的二进制最高位为1，该数为正数，用它表示其补数的相反数",[10,507,508,509,512,513,516],{},"假设有一个8位二进制数，它可表示2的8次方共256个数，即0",[486,510,511],{},"255其模为256。因此0","127表示正数，\n而128",[486,514,515],{},"255表示对应补数的相反数，如255的补数为1，即255表示-1。同时，128的补数为128即128用来表示-128，\n故8位二进制数的可表示范围为-128","127",[106,518,519],{"id":519},"补码",[10,521,522,523,526,527,530,531],{},"-1的原码为",[14,524,525],{},"1000 0001","故其补码为",[14,528,529],{},"1111 1111","对应255，正好是其绝对值的补数。也可得出-128的补码为\n其补数128的二进制形式",[14,532,533],{},"1000 0000",[106,535,536],{"id":536},"取反加一",[10,538,539],{},"n位二进制数，以2^n为模，取半模以下的数A，补数B = 2^n - A，B表示-A，即B的二进制为-A的补码",[87,541,542],{},[10,543,544,545,625,626,695],{},"A的最高位为0，令其",[130,546,548,569],{"className":547},[133],[130,549,551],{"className":550},[137],[139,552,553],{"xmlns":141},[143,554,555,566],{},[146,556,557],{},[558,559,560,563],"msub",{},[149,561,562],{},"k",[149,564,565],{},"i",[191,567,568],{"encoding":193},"k_i",[130,570,572],{"className":571,"ariaHidden":199},[198],[130,573,575,579],{"className":574},[203],[130,576],{"className":577,"style":578},[207],"height:0.8444em;vertical-align:-0.15em;",[130,580,582,586],{"className":581},[212],[130,583,562],{"className":584,"style":585},[212,213],"margin-right:0.0315em;",[130,587,590],{"className":588},[589],"msupsub",[130,591,593,616],{"className":592},[340,341],[130,594,596,613],{"className":595},[345],[130,597,600],{"className":598,"style":599},[349],"height:0.3117em;",[130,601,603,607],{"style":602},"top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;",[130,604],{"className":605,"style":606},[357],"height:2.7em;",[130,608,610],{"className":609},[362,363,364,365],[130,611,565],{"className":612},[212,213,365],[130,614,409],{"className":615},[408],[130,617,619],{"className":618},[345],[130,620,623],{"className":621,"style":622},[349],"height:0.15em;",[130,624],{},"代表其余n-1位，",[130,627,629,646],{"className":628},[133],[130,630,632],{"className":631},[137],[139,633,634],{"xmlns":141},[143,635,636,644],{},[146,637,638],{},[558,639,640,642],{},[149,641,562],{},[149,643,565],{},[191,645,568],{"encoding":193},[130,647,649],{"className":648,"ariaHidden":199},[198],[130,650,652,655],{"className":651},[203],[130,653],{"className":654,"style":578},[207],[130,656,658,661],{"className":657},[212],[130,659,562],{"className":660,"style":585},[212,213],[130,662,664],{"className":663},[589],[130,665,667,687],{"className":666},[340,341],[130,668,670,684],{"className":669},[345],[130,671,673],{"className":672,"style":599},[349],[130,674,675,678],{"style":602},[130,676],{"className":677,"style":606},[357],[130,679,681],{"className":680},[362,363,364,365],[130,682,565],{"className":683},[212,213,365],[130,685,409],{"className":686},[408],[130,688,690],{"className":689},[345],[130,691,693],{"className":692,"style":622},[349],[130,694],{},"的值为0或者1",[10,697,698,1236,1582],{},[130,699,701,810],{"className":700},[133],[130,702,704],{"className":703},[137],[139,705,706],{"xmlns":141},[143,707,708,807],{},[146,709,710,712,714,722,725,733,736,743,745,751,753,756,758,760,762,775,777,789,791,793,795],{},[149,711,151],{},[153,713,310],{},[558,715,716,718],{},[149,717,562],{},[719,720,721],"mn",{},"0",[153,723,724],{},"∗",[726,727,728,731],"msup",{},[719,729,730],{},"2",[719,732,721],{},[153,734,735],{},"+",[558,737,738,740],{},[149,739,562],{},[719,741,742],{},"1",[153,744,724],{},[726,746,747,749],{},[719,748,730],{},[719,750,742],{},[153,752,735],{},[149,754,755],{"mathvariant":173},".",[149,757,755],{"mathvariant":173},[149,759,755],{"mathvariant":173},[153,761,735],{},[558,763,764,766],{},[149,765,562],{},[146,767,768,771,773],{},[149,769,770],{},"n",[153,772,303],{},[719,774,742],{},[153,776,724],{},[726,778,779,781],{},[719,780,730],{},[146,782,783,785,787],{},[149,784,770],{},[153,786,303],{},[719,788,730],{},[153,790,735],{},[719,792,721],{},[153,794,724],{},[726,796,797,799],{},[719,798,730],{},[146,800,801,803,805],{},[149,802,770],{},[153,804,303],{},[719,806,742],{},[191,808,809],{"encoding":193},"A=k_0*2^0+k_1*2^1+...+k_{n-1}*2^{n-2}+0*2^{n-1}",[130,811,813,831,888,935,990,1034,1054,1120,1173,1192],{"className":812,"ariaHidden":199},[198],[130,814,816,819,822,825,828],{"className":815},[203],[130,817],{"className":818,"style":208},[207],[130,820,151],{"className":821},[212,213],[130,823],{"className":824,"style":217},[160],[130,826,310],{"className":827},[221],[130,829],{"className":830,"style":217},[160],[130,832,834,837,878,882,885],{"className":833},[203],[130,835],{"className":836,"style":578},[207],[130,838,840,843],{"className":839},[212],[130,841,562],{"className":842,"style":585},[212,213],[130,844,846],{"className":845},[589],[130,847,849,870],{"className":848},[340,341],[130,850,852,867],{"className":851},[345],[130,853,856],{"className":854,"style":855},[349],"height:0.3011em;",[130,857,858,861],{"style":602},[130,859],{"className":860,"style":606},[357],[130,862,864],{"className":863},[362,363,364,365],[130,865,721],{"className":866},[212,365],[130,868,409],{"className":869},[408],[130,871,873],{"className":872},[345],[130,874,876],{"className":875,"style":622},[349],[130,877],{},[130,879],{"className":880,"style":881},[160],"margin-right:0.2222em;",[130,883,724],{"className":884},[401],[130,886],{"className":887,"style":881},[160],[130,889,891,895,926,929,932],{"className":890},[203],[130,892],{"className":893,"style":894},[207],"height:0.8974em;vertical-align:-0.0833em;",[130,896,898,901],{"className":897},[212],[130,899,730],{"className":900},[212],[130,902,904],{"className":903},[589],[130,905,907],{"className":906},[340],[130,908,910],{"class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