Mathematical work

Liu Hui expressed mathematical results in the form of decimal fractions that utilized metrological units (i.e., related units of length with base 10 such as 1 chǐ = 10 cùn, 1 cùn = 10 fēn, 1 fēn = 10 lí, etc.); this led Liu Hui to express a diameter of 1.355 feet as 1 chǐ, 3 cùn, 5 fēn, 5 lí. Han Yen (fl. 780-804 CE) is thought to be the first mathematician that dropped the terms referring to the units of length and used a notation system akin to the modern decimal system and Yang Hui (c. 1238–1298 CE) is considered to have introduced a unified decimal system.

Liu provided a proof of a theorem identical to the Pythagorean theorem. Liu called the figure of the drawn diagram for the theorem the "diagram giving the relations between the hypotenuse and the sum and difference of the other two sides whereby one can find the unknown from the known."

In the field of plane areas and solid figures, Liu Hui was one of the greatest contributors to empirical solid geometry. For example, he found that a wedge with rectangular base and both sides sloping could be broken down into a pyramid and a tetrahedral wedge. He also found that a wedge with trapezoid base and both sides sloping could be made to give two tetrahedral wedges separated by a pyramid. He computed the volume of solid figures such as cone, cylinder, frustum of a cone, prism, pyramid, tetrahedron, and a wedge. However, he failed to compute the volume of a sphere and noted that he left it to a future mathematician to compute.

In his commentaries on The Nine Chapters on the Mathematical Art, he presented:

• An algorithm for the approximation of pi. While at the time, it was common practice to assume to equal 3, Liu utilized the method of inscribing a polygon within a circle to approximate to equal \frac{157}{50} on the basis of a 192-sided polygon. This method was similar to the one employed by Archimedes whereby one calculates the length of the perimeter of the inscribed polygon utilizing the properties of right-angled triangles formed by each half-segment. Liu subsequently utilized a 3072-sided polygon to approximate to equal 3.14159, which is a more accurate approximation than the one calculated by Archimedes or Ptolemy.

• Gaussian elimination.

• Cavalieri's principle to find the volume of a cylinder and the intersection of two perpendicular cylinders although this work was only finished by Zu Chongzhi and Zu Gengzhi. Liu's commentaries often include explanations why some methods work and why others do not. Although his commentary was a great contribution, some answers had slight errors which was later corrected by the Tang mathematician and Taoist believer Li Chunfeng.

• Through his work in the Nine Chapters, he could have been the first mathematician to discover and compute with negative numbers; definitely before Ancient Indian mathematician Brahmagupta started using negative numbers.

Surveying

Liu Hui also presented, in a separate appendix of 263 AD called Haidao Suanjing or The Sea Island Mathematical Manual, several problems related to surveying. This book contained many practical problems of geometry, including the measurement of the heights of Chinese pagoda towers. This smaller work outlined instructions on how to measure distances and heights with "tall surveyor's poles and horizontal bars fixed at right angles to them". With this, the following cases are considered in his work:

• The measurement of the height of an island opposed to its sea level and viewed from the sea

• The height of a tree on a hill

• The size of a city wall viewed at a long distance

• The depth of a ravine (using hence-forward cross-bars)

• The height of a tower on a plain seen from a hill

• The breadth of a river-mouth seen from a distance on land

• The width of a valley seen from a cliff

• The depth of a transparent pool

• The width of a river as seen from a hill

• The size of a city seen from a mountain.

Liu Hui's information about surveying was known to his contemporaries as well. The cartographer and state minister Pei Xiu (224–271) outlined the advancements of cartography, surveying, and mathematics up until his time. This included the first use of a rectangular grid and graduated scale for accurate measurement of distances on representative terrain maps. Liu Hui provided commentary on the Nine Chapter's problems involving building canal and river dykes, giving results for total amount of materials used, the amount of labor needed, the amount of time needed for construction, etc.

Although translated into English long beforehand, Liu's work was translated into French by Guo Shuchun, a professor from the Chinese Academy of Sciences, who began in 1985 and took twenty years to complete his translation.

家世

郭書春主張，劉徽有可能是梁敬王劉定國之孫菑鄉侯劉逢喜的後裔。吳文俊《中國數學史大系》亦同意此說法。

籍貫

關於劉徽的資料甚少，正史和其他史籍中都沒有此人的傳記資料。其生平資料主要可見於其本人的著作和零散的史料，依靠學者推測。籍貫方面，《宋史·禮志》中記述：「於是中書舍人張邦昌定算學：封.......魏劉徽淄鄉男」。學者據此記載有不同的看法。

山東鄒平縣說。《中國大百科全書》（第二版）、郭書春、沈康主此說。郭書春《劉徽籍貫考》中，據《漢書》、《元豐九域志》、《金史》、《山東通志》等史籍，主張淄鄉在今天山東鄒平縣境。沈康以山東鄒平縣的地里位置，和劉徽《海島算經》題文的地里描述，同樣主此說。

山東淄博淄川說。吳文俊《中國數學史大系》 （此節由白尚恕執筆）則持此說，《中國數學史大系》認為今山東淄博市淄川區有淄城鎮，有可能就是宋金時的淄鄉鎮。

此外，學者有其他的看法。嚴敦傑《劉徽簡傳》中認為：「祀典各封爵並不十九均據各人里貫為之，其中見於史傳的而有里貫的一般相符，如沒有里貫的一是推測，二是按郡望。劉徽的籍貫既沒有直接材料，則封為淄鄉男不出這兩種可能性，所以還是定不可考為好。」，主張淄鄉不一定是他的籍貫。李迪認為「這話（《宋史》中的記載）不完全可靠，但也不應完全否定」，認為可初步斷定劉徽是山東人。

生平

關於劉徽的生平，史書中沒有記載。從《九章算術注》中提及的書名來看，他精通四書五經和諸子。在其《九章算術注·序》提及「立表於洛陽之城，令高八尺，南北各盡平地，同日度正中之景」，可見他曾親身到洛陽，並可能洛陽參加過歷法討論和進行過天文測量。

此外，《隋書·經籍志》中記述：「《魯史欹器圖》，儀同劉徽注。」。《中國數學史大系》據此認為他有可能做過儀同官，並在任上完成《魯史欹器圖》一卷。清代姚振宗《隋書經籍志考證》中指，凡是隋代人所著的書，《經籍志》都只書官名再加上書名，不加上朝代；而不是隋人的才註明朝代，因此此劉徽應是劉暉之誤。嚴敦傑認為，此處的「劉徽」應該是「劉暉」，是隋人而非魏人。

數學成就

劉徽為《九章算術》作注，其中他提出用割圓術計算圓周率的方法，以內接正六邊形開始，逐次倍加邊數的方法，逐步逼近圓周率。《九章算術》僅以π=3，劉徽則計算出正192邊形的面積，先得到圓周率的近似值為\pi=\frac{157}{50}=3.14，和晉武庫王莽銅律嘉量比較，覺得「此術微小」，于是再用圓周率捷法計算出正3072邊形的面積，求得\pi=\frac{3927}{1250}=3.1416。作此書注時，他還依據其「割補術」為證勾股定理，另闢蹊徑作青朱出入圖。圖雖失傳，但據其「出入相補、以盈補虛」原理，後人參照書中類似方法還原了此圖。

劉徽的注釋兼用圖形和模型作說明，以圖形相互拼湊方法解決各種面積計算問題，相當於一般平面幾何學中所用的平移與疊合的方法；並用直截面積的方法來計算立體體積。他指出《九章算術》計算球體體積方法錯誤，並引入了「牟合方蓋」（垂直相交的兩個圓柱體的共同部分的體積）這一著名的幾何模型，認為只有「牟合方蓋」與球體積之比才正好等於正方形與其內切圓的面積之比，也就是：

::球體積 :\, 牟合方蓋體積 = \pi:4 \,

但劉徽沒有給出牟合方蓋的體積公式，所以也就得不出球體的體積公式。

劉徽並在《九章算術注》提出重差術，應用中國傳統的出入相補原理，以多達4次的觀測，測量山高水深等數值。在唐代，有關重差術的注文被抽出單行，因其第一題是測量海島高度和距離的問題，故題為《海島算經》，成為《算經十書》之一。劉徽創造的四次重差觀測術，被吳文俊稱為「使中國測量學達到登峰造極的地步」，美國數學家弗蘭克·斯委特茲讚譽這使得「中國在數學測量學的成就，超越西方約一千年」。