Among Liu's achievements are an emphasis on generalizations and proofs, incorporation of negative numbers into arithmetic, an early recognition of the notions of infinitesimals and limits, the Gaussian elimination method of solving simultaneous linear equations, calculations of solid volumes (including the use of Cavalieri's Principle), anticipation of Horner's Method, and a new method to calculate square roots. Like Archimedes, Liu discovered the formula for a circle's area; however he failed to calculate a sphere's volume, writing "Let us leave this problem to whoever can tell the truth."
Gaussian elimination, Cavalieri's Principle, Horner's Method都是我講科學計算方法課的內容。今天看來不是高大上的東西，但劉徽早在近1800年前就提出了這些算法，全都是代表當時最前沿的工作。更了不起的是，魏晉三國時他就已經有了負數，無限逼近，邏輯證明的理念。
