➖ 补码：让电脑做减法的聪明办法➖ Two's Complement: Subtraction by Adding
上一片里，字符住进了数字。可数字还有一半没安家：负数怎么办？更麻烦的是，💻 小电脑里根本没有减法器！Last leaf, characters moved into numbers. But half the numbers are still homeless: what about negatives? Worse — 💻 the little computer has no subtractor at all!
第 1 步：没有减法器，就把减法变成加法Step 1: No subtractor? Turn subtraction into addition
加法器是小电脑唯一的算术机器，那就让减法来迁就它。The adder is the little computer's only arithmetic machine — so subtraction must adapt to it.
5 - 3 = 5 + (-3)：把 5 和 -3 两张卡喂进加法器，结果一样是 2。所以问题只剩一个：负数 -3 该怎么存？5 − 3 = 5 + (−3): feed the cards 5 and −3 into the adder and out comes 2 all the same. So only one question remains: how do we store a negative like −3?
负数挤不进 0~255 的小格子？先看看格子转满一圈会发生什么。Negatives can't squeeze into cells of 0–255? First watch what happens when the cells spin one full turn.
第 2 步：8 位小转盘，转满就回到 0Step 2: The 8-bit dial spins full circle back to 0
8 个格子最多数到 255，再加 1 就转满一圈回到 0。这不是故障，是约定：多出来的进位直接丢掉，有限的格子循环用。Eight cells count up to 255 at most — plus one more spins full circle back to 0. That's not a bug, it's the deal: the extra carry is dropped, and the small set of cells loops forever.
回绕有什么用？想想：255 的上一个邻居是 0，它当 -1 用合不合适？What's the wrap good for? Think: 255 sits right before 0 — couldn't it play the role of −1?
第 3 步：负数住进补码Step 3: Negatives move into complements
从 0 往回倒：0 的上一个 255 当 -1，再上 254 当 -2。8 位里 -1 存成 11111111，-3 存成 253。5 - 3 就变成 5 + 253 = 258，回绕成 2。原码拿首位当符号，反码其余位取反，补码是反码再加 1。只有补码能让加法器直接算对。Count backwards from 0: the cell before 0, 255, plays −1; next 254 plays −2. In 8 bits, −1 is stored as 11111111 and −3 as 253. So 5 − 3 becomes 5 + 253 = 258, wrapping to 2. Sign-magnitude spends its first bit on the sign, ones' complement flips the rest, two's complement adds one more — and only two's complement lets the adder get it right directly.
🎮 你来当操作员（30 秒）🎮 Your turn: adder operator (30 seconds)
拖滑块调被加数 A，再押一把会不会回绕，带小电脑连过 4 关。Drag the slider to set addend A, bet on wrap or not, and walk the little computer through 4 rounds.
一句话记住它：补码 = 给负数办的加法器通行证，减法全变成加法。Remember it in one line: two's complement is the adder pass issued to negatives — every subtraction becomes an addition.
减法变加法：a - b = a + (-b)，加法器一台就够Subtraction becomes addition: a − b = a + (−b) — one adder is enough负数存补码：-1 存成 255（11111111），溢出回绕 255+1=0Negatives as complements: −1 stored as 255 (11111111); overflow wraps 255+1=0原码管符号、反码取反、补码再加 1，只有补码能直接加Sign-magnitude marks the sign, ones' complement flips, two's complement adds one — only it adds directly
内容参考 OpenStax 等公开教材，多来源核对 · AI 生成、人工审核 · 发现错误欢迎指正，帮这片叶子长得更好。Based on OpenStax and other open textbooks, cross-checked across sources · AI-generated, human-reviewed · Spotted a mistake? Tell us — help this leaf grow.
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