Before the 15th century, the mathematical treatises in China treating square root extraction did not describe explicitly how each digit of the root was estimated and failed to pinpoint how the digit is ascertained, not the method in the Mathematical Procedures in the Nine Chapters (九章算術 Jiuzhang suanshu), not in the Liu Hui’s or Li Chunfeng’s commentaries, nor can they be found in Yang Hui’s discussions. Moreover, in Yang Hui’s comments, the valuable concrete, transitional, numerical values that appear during the course of extraction openly contradict the place values bestowed on them by their positions on the counting board. In our studies, we discovered in Yang Hui’s/Jia Xian’s comments an additional way of keeping tract of the numerical values of the “divisors” throughout the course of the calculation even though that they are being moved to the right multiple times in the process. This additional way of keeping track of place values allows the root extraction process to faithfully reflect the steps in the geometric reasoning described in Liu Hui's commentary.

The Nine Chapters on the Art of Mathematics and its commentaries played crucial roles in ancient and medieval official mathematics curricula across East Asia. Notably, it was designated as one of the core textbooks in the official mathematics curricula and examinations of eight-century Tang China, Silla Korea, and Nara Japan. Each of these three governments adopted slightly different approaches in utilizing the Nine Chapters and other texts in teaching and assessments. By the 12th century, official mathematics examination systems in China and Japan had been discontinued for various reasons, leaving Korea's Koryŏ Dynasty (918–1392) as the sole preserver of an official mathematics examination system in East Asia. This paper delves into the content and possible structure of the mature mathematics examination system in 12th-century Koryŏ and discusses the implications for textbook usage within this framework.