Illustrating The Nine Chapters on the Mathematical Art - In the Classroom: Excess and Deficit

Chapter 7, "Excess and Deficit"

Chapter 7, "Excess and Deficit," provides a technique to solve what we today would describe as two linear equations in two unknowns via a method called “double false position” in the West. That this topic remains of interest to scholars can be seen in a presentation by Ji Zhigang of Shanghai Jiao Tong University, "The Rule of False Double Position: From Suàn Shù Sh­ū to Liber Abaci." Figure 13 shows a slide from Ji Zhigang's presentation that introduces a problem from the Nine Chapters that is solved by the method of double false position.

Figure 13. This slide from Ji Zhigang's presentation introduces a problem from the Nine Chapters that is to be solved by the method of double false position (photo by the author).

In the solution, the number of people is determined by computing \[\frac{3+4}{8-7}=7\] and the item price by \[\frac{8\cdot 4+7\cdot 3}{8-7}=53\] [Shen 360-361]. In my class discussion of Double False Position, we first show that the answer obtained by the procedure for this particular problem satisfies the story of the problem. We then derive the algebraic equations that describe the problem and use direct substitution to show that the answer will always be correct. Finally, in order to see how one might arrive at the formula given by Double False Position, we have the opportunity to discuss interpolation (now a largely lost skill) and also use analytic geometry to derive the formula. Seeing this variety of approaches makes this is a rich problem for future teachers.