探究珠算起源──以珠算作為數學工具之發展為例

Tracing the Origins of Zhusuan: A Historical Study of Its Development as a Mathematical Tool

「珠算（Zhusuan）」已於2013年12月4日「聯合國教科文組織」（UNESCO）列入《人類無形文化遺產代表作名錄》（簡稱非遺），說明曾經作為數學計算工具的「珠算」，在中國古代確實占有一席之地。然對於「珠算」的起源，珠算史家雖已有專文或專著探討相關論述，卻莫衷一是。同時相關文獻對於作為珠算運算工具之「算盤」（珠算器），所能彰顯的功能亦描述有限。本研究目的即在於透過既有的古籍文獻，分析「珠算」作為數學計算工具的起源歷程。意欲從珠算的以「珠」計「算」之有形樣貌，發展成數學運算技術的無形資產（非遺）脈絡，找出更為適合之廣義珠算起源。再根據文獻從「算盤」型式的演進，探討算盤是否還有其他仍未發現之有形或無形功能。本文進一步提出珠算作為運算工具的起源時期，從其發展脈絡分析應有更廣義的定義：算珠萌於周、文字顯於漢、珠算起於宋、大成集於明。從數學教育的角度觀察，珠算透過口訣背誦處理算術四則運算，相當於是運用眼看、耳聽、口說、手撥、心算等的多種感官輔助學習。因此，珠算可作為從具體到抽象算術思維的過渡橋樑。此外，算盤除了是珠算的運算工具外，也在進行除法運算時提供了便於存放餘數的功能，並能在運算過程中清楚表達大數與小數，又由於算盤因長期應用於古代中國百姓生活中，也成為現代一些哲理成語應用的代言。這些不僅反映古代中國對於珠算機械式使用的極致靈活度，也相對彰顯珠算與算盤在有形與無形資產中具有的價值。

The Zhusuan was inscribed on UNESCO’s Representative List of the Intangible Cultural Heritage of Humanity on December 4, 2013. This recognition highlights the historical significance of Zhusuan as a mathematical calculation tool in ancient China. However, abacus scholars of China held different opinions about the origin of Zhusuan, despite the existence of specialized publications and scholarly works. Additionally, limited attention has been paid to the full range of functions embodied by the abacus—the physical instrument of Zhusuan. The purpose of this paper is to analyze the origin process of Zhusuan as a mathematical calculation tool through existing ancient texts and historical literature. By tracing the transformation of Zhusuan—from its tangible form of calculating with beads (zhu) to its recognition as an intangible cultural asset characterized by mathematical techniques (ICH)—this research aims to establish a broader and more suitable interpretation of the origins of Zhusuan. In addition, based on historical literature, this study further explores the evolution of the Chinese abacus in form. It examines whether, beyond its known functions in arithmetic calculation and specific applications such as hexadecimal operation, the abacus may possess other tangible or intangible functions that have yet to be identified. This paper also proposes that the origin of Zhusuan as a calculation tool should have a broader definition according to its developmental contexts: the counting beads were first developed in the Zhou Dynasty, written records emerged in the Han Dynasty, Zhusuan itself started in the Song Dynasty, and eventually its algorithms fully matured in the Ming Dynaty. From the perspective of mathematics education, Zhusuan employs mnemonic recitation to perform arithmetic operations. It involves visual observation, auditory perception, verbal articulation, manual manipulation, and mental calculation, assisting learners to engage multiple senses in learning. Zhusuan serves as an effective transitional bridge from concrete manipulation to abstract arithmetic thinking. The abacus also provides the function of storing the remainder during divisions and the convenience of representing large and small numbers during calculations. These features reflect the remarkable adaptability of ancient users in manipulating mechanical tools. Through its long-standing presence in everyday life, the abacus has even influenced modern idioms and philosophical expressions, highlighting its enduring value in both tangible and intangible forms.