diameter of a circle. Some Han mathematicians attempted to improve this number, such as Liu Xin, who is believed to have estimated pi to be 3.154. Later, Liu Hui attempted to improve the calculation by calculating pi to be 3.141024 (a low estimate of the number). Liu calculated this number by using polygons inside a hexagon as a lower limit compared to a circle. Zu Chongzhi later discovered the calculation of pi to be 3.1415926 < π < 3.1415927 by using polygons with 24,576 sides. This calculation would be discovered in Europe during the 16th century. There is no explicit method or record of how he calculated this estimate. Division and root extraction Basic arithmetic processes such as addition, subtraction, multiplication and division were present before the Han Dynasty. The Nine Chapters on the Mathematical Art take these basic operations for granted and simply instruct the reader to perform them. Han mathematicians calculated square and cube roots in a similar manner as division, and problems on division and root extraction both occur in Chapter Four of The Nine Chapters on the Mathematical Art. Calculating the square and cube roots of numbers is done through successive approximation, the same as division, and often uses similar terms such as dividend (shi) and divisor (fa) throughout the process. This process of successive approximation was then extended to solving quadratics of the second and third order, such as , using a method similar to Horner's method. The method was not extended to solve quadratics of the nth order during the Han Dynasty; however, this method was eventually used to solve these equations. Linear algebra The Book of Computations is the first known text to solve systems of equations with two unknowns. There are a total of three sets of problems within The Book of Computations involving solving systems of equations with the false position method, which again are put into practical terms. Chapter Seven of The Nine Chapters on the Mathematical Art also deals with solving a system of two equations with two unknowns with the false position method. To solve for the greater of the two unknowns, the false position method instructs the reader to cross-multiply the minor terms or zi (which are the values given for the excess and deficit) with the major terms mu. To solve for the lesser of the two unknowns, simply add the minor terms together. Chapter Eight