# Reflections on Chinese Numeration Systems

Author(s):
Frank J. Swetz (The Pennsylvania State University)

Figure 1. Ideogram for “to calculate,” as depicted by [Siu 2000].

The symbol above is an ideogram, a pictorial written character used to communicate a concept or idea. With a vivid imagination, and perhaps some prodding from an instructor, a viewer can discern two pairs of hands reaching inwards, involved in some activity with the rectangular object between them. This particular ideogram is from ancient China, and it denotes the act of numerical computing, of reckoning with a set of wooden rods often called counting rods, which are denoted by the central object in the figure.

Exposing students to another numeration system besides the standard Hindu-Arabic numerals that are most widely used today is generally considered beneficial for helping students understand the structures and operations of arithmetic. By default, perhaps, Roman numerals are a natural alternative used by schoolteachers and the instructors of preservice teachers to provide such a learning experience. To be sure, that numeration system remains culturally embedded in the West through movie credits, clock faces, stone inscriptions, and the like—signaling the long domination of Roman numerals in mathematical recordkeeping in Western Europe before they were displaced in the 15th century by the more “user-friendly” Hindu-Arabic numerals imported from the East [Swetz 1987]. However, the characteristics of Roman numerals pose pedagogical limitations in modern classrooms:

• The system does not use place value notation.
• Thus, in order to represent some numbers, additive or subtractive principles are required; for example, $4$ in the Roman notation is transcribed as IV and conceived conceptually as $5 - 1$, while $6$ is written as VI or $5 + 1$.
• In their representations, Roman numerals employ both tally symbols: I, II, . . . and letters of the alphabet: L, D, C, . . . which complicates the basic operations.
• Since Roman numerals do not readily lend themselves to computational processes, they were mainly used for recording solutions, as calculations were made on an abacus and the results written down.

This article presents the rod numerals of ancient China as an alternative numeration system that is much better suited for mathematics teaching and learning situations. The topic additionally permits students to encounter a culture they are likely to know little about. Computing rods can be traced back to at least the Warring States period of Chinese history (5th century BCE to 221 BCE), and they remained in use well into the 17th century [Volkov 2018]. The following sections summarize the history of this numeration system: how numerals were represented by counting rods, how computations were made, and how some mathematicians employed rod numerals in advanced mathematics. I end the article with suggestions that instructors of preservice elementary teachers can share with their students and ask a provocative question about how we think about numeration systems.

Frank J. Swetz (The Pennsylvania State University), "Reflections on Chinese Numeration Systems," Convergence (February 2022), DOI:10.4169/convergence20220220