实数之外另一重天地
Beyond real numbers lies another sphere
三类二维代数各守玄机
Three 2‑D algebras hold their own career
符号平方定下全部谜底
Each symbol’s square reveals the frontier
旋转、微分、双曲的轨迹
Rotation, derivative, hyperbola steer
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i 方负一，辗转轮回
i squared minus one, spins round and turns
ε方归零，微痕暗存
ε squared equals zero, tiny change adjourns
j方为一，双曲奔奔
j squared is one, hyperbolic burns
三条法则，各自星辰
Three simple rules, three separate turns
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复数旋起岁月涟漪
Complex numbers spin the ripples wide
Every turn unfolds another side
二元数藏微分秘语
Dual numbers hide the gradient inside
Infinitesimal, quietly abide
双曲数破边界藩篱
Split‑complex breaks the borderline tide
Three worlds in two dimensions side by side
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复数能除，圆上回旋
Complex divides, circles loop around
二元难除，增量轻扬
Dual can’t divide, small increments abound
双曲驰骋，渐近远方
Split‑complex runs toward asymptote ground
二维三境，不必相妨
Three realms in two, none need to bound
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一纸代数，万般万象
One algebra, ten thousand views
Three rules define our two‑dimensional truth
