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(1+r) 近似为乘 (1−r)：误差约 r²，三个刹车片与只用一次的规矩。",0,"0.00",21,"2026-09-24 00:37:46",false,true,"\u003Ch2>先看一道题，猜哪边快\u003C\u002Fh2>\n\u003Cpre>\u003Ccode>甲的算法：4260 ÷ 1.05 = ？\n\n乙的算法：4260 × 0.95 = ？\u003C\u002Fcode>\u003C\u002Fpre>\n\u003Cp>\u003Cstrong>乙明显快\u003C\u002Fstrong> —— 乘 0.95 就是\"先减 5%\"：4260 − 4260÷20 = 4260 − 213 = 4047，两步口算。\u003C\u002Fp>\n\u003Cp>\u003Cstrong>甲要老老实实做一次除法\u003C\u002Fstrong>，怎么截位都得多想几秒。\u003C\u002Fp>\n\u003Cp>\u003Cstrong>问题来了：这两个答案一样吗？\u003C\u002Fstrong>\u003C\u002Fp>\n\u003Cpre>\u003Ccode>精确值：4260 ÷ 1.05 = 4057.14\n乙的答案：4047\n差了约 0.25%\u003C\u002Fcode>\u003C\u002Fpre>\n\u003Cp>\u003Cstrong>不一样，但很接近。\u003C\u002Fstrong> 这 0.25% 的差距，就是本篇要讲清的全部内容：\u003Cstrong>这条近路什么时候能走、什么时候会摔\u003C\u002Fstrong>。\u003C\u002Fp>\n\u003Cp>\u003Cstrong>结论先给你\u003C\u002Fstrong>：\u003C\u002Fp>\n\u003Cblockquote>\u003Cp>\u003Cstrong>除以 (1+r) ≈ 乘以 (1−r)，代价是结果偏小约 r²。\u003C\u002Fstrong>\u003C\u002Fp>\u003Cp>\u003Cstrong>r 越小这条近路越平；r 超过 10%，路面就开始打滑。\u003C\u002Fstrong>\u003C\u002Fp>\u003C\u002Fblockquote>\n\u003Ch2>为什么能换：三行推导\u003C\u002Fh2>\n\u003Cp>不许只记结论，推导给你：\u003C\u002Fp>\n\u003Cpre>\u003Ccode>用二项式展开：\n1\u002F(1+r) = 1 − r + r² − r³ + …\n\nr 很小的时候，r² 往后的项都 tiny，砍掉：\n1\u002F(1+r) ≈ 1 − r\n\n两边同乘 A：\nA\u002F(1+r) ≈ A × (1−r)\u003C\u002Fcode>\u003C\u002Fpre>\n\u003Cp>\u003Cstrong>误差到底多大？\u003C\u002Fstrong> 砍掉的正是 r² 那一项：\u003C\u002Fp>\n\u003Cpre>\u003Ccode>真实差 = 1\u002F(1+r) − (1−r) = r²\u002F(1+r) ≈ r²\u003C\u002Fcode>\u003C\u002Fpre>\n\u003Cp>拿 r = 5% 验证：r² = 0.0025 = 0.25% —— \u003Cstrong>跟开头那道题的 0.25% 完全对上\u003C\u002Fstrong>。\u003C\u002Fp>\n\u003Cp>\u003Cstrong>所以误差表可以直接背\u003C\u002Fstrong>：\u003C\u002Fp>\n\u003Ctable>\u003Cthead>\u003Ctr>\u003Cth>r（增速）\u003C\u002Fth>\u003Cth>误差 ≈ r²\u003C\u002Fth>\u003Cth>意思是\u003C\u002Fth>\u003C\u002Ftr>\u003C\u002Fthead>\u003Ctbody>\u003Ctr>\u003Ctd>2%\u003C\u002Ftd>\u003Ctd>0.04%\u003C\u002Ftd>\u003Ctd>完全无感\u003C\u002Ftd>\u003C\u002Ftr>\u003Ctr>\u003Ctd>5%\u003C\u002Ftd>\u003Ctd>0.25%\u003C\u002Ftd>\u003Ctd>基本无感\u003C\u002Ftd>\u003C\u002Ftr>\u003Ctr>\u003Ctd>8%\u003C\u002Ftd>\u003Ctd>0.64%\u003C\u002Ftd>\u003Ctd>还行，但要知道自己在估\u003C\u002Ftd>\u003C\u002Ftr>\u003Ctr>\u003Ctd>10%\u003C\u002Ftd>\u003Ctd>1%\u003C\u002Ftd>\u003Ctd>边界，选项差距小于 1% 就别走这条近路\u003C\u002Ftd>\u003C\u002Ftr>\u003Ctr>\u003Ctd>20%\u003C\u002Ftd>\u003Ctd>4%\u003C\u002Ftd>\u003Ctd>\u003Cstrong>摔了\u003C\u002Fstrong> —— 误差比选项差距还大\u003C\u002Ftd>\u003C\u002Ftr>\u003C\u002Ftbody>\u003C\u002Ftable>\n\u003Cp>\u003Cstrong>注意方向\u003C\u002Fstrong>：1 − r \u003Cstrong>小于\u003C\u002Fstrong> 1\u002F(1+r)（因为砍掉的是正项 r²），所以：\u003C\u002Fp>\n\u003Cblockquote>\u003Cp>\u003Cstrong>用 A×(1−r) 代替 A\u002F(1+r)，结果一律偏小，偏小约 r²。\u003C\u002Fstrong>\u003C\u002Fp>\u003C\u002Fblockquote>\n\u003Cp>知道方向就有救 —— 算出来发现两个选项都像，\u003Cstrong>选偏大的那个\u003C\u002Fstrong>（第 9 篇《误差修正：什么时候必须停下来算准》讲过方向的作用，这里正好用上）。\u003C\u002Fp>\n\u003Cp>\u003Cstrong>方向还有一个反向用法\u003C\u002Fstrong>：如果题目问的是\"以下哪个\u003Cstrong>不可能\u003C\u002Fstrong>是基期\"，偏小约 r² 的方向感帮你判断估算值落在真值的哪一侧 —— \u003Cstrong>真值永远在估算值的上方一点点\u003C\u002Fstrong>（正增长情形）。估算 4412，真值就是 4421 这一带，绝不会是 4390。\u003Cstrong>方向感把\"一个近似数\"变成\"一个带符号的区间\"\u003C\u002Fstrong>，这是它比\"只算个大概\"值钱的地方。\u003C\u002Fp>\n\u003Ch2>刹车片一：r 的上限\u003C\u002Fh2>\n\u003Cp>\u003Cstrong>这条近路不是随时能上的。\u003C\u002Fstrong> 刹车标准很简单：\u003C\u002Fp>\n\u003Cpre>\u003Ccode>","none",{"id":233,"title":234},{"id":239,"title":240},1790222661209]