The Chinese Numeration System and Place Value
Uy, Frederick L., Teaching Children Mathematics
I chi, ni, san... Otu, abouo ato... Eins, zwei, drei... Have you heard these words before? What might they mean? In case you are not familiar with these words, they mean "One, two, three..." in Japanese, Igbo, and German, respectively.
Two of the most fundamental topics in mathematics that children must learn are counting and the numeration system. Central to the Number and Operations Standard of NCTM's Principles and Standards is the development of number sense, which includes the ability to understand the base-ten number system (NCTM 2000). Reys (1998) explains that place value is the foundation of our numeration system. We use place value to read, represent, and operate on numbers and also to identify patterns, build number sense, and understand different numeration systems. This article presents the Chinese numeration system as an additional representation of numbers and place value for teachers to consider in their instruction. Sample learning activities using the Chinese numeration system for elementary students are provided.
The Chinese Numeration System Explained
Virtually every civilization and culture developed the concept of numbers and the formulation of a counting process (Eves 1990). Cultural differences led to the creation of numeration systems suited to the unique needs and purposes of those who used them (Cathcart et al. 2000). These needs ranged from tallying the number of animals kept to developing calendars. Many cultures created culturally specific numeration systems that included their own symbols, rules, and values. One of these is the Chinese numeration system.
The Chinese numeration system is a decimal (base-ten) system, unlike other systems such as the Babylonian (sexagesimal or base-sixty) or the Mayan (vigesimal or base-twenty). In the decimal system, counting is based on ten numerals, symbols are used to represent the numerals, and successive place values--such as 10; 100; 1,000; and 10,000-are powers of ten (Baumgart et al. 1989). Figure 1 shows the characters used in the Chinese numeration system.
What is unique about this numeration system? The written form indicates the number of units in each base, together with that base. For example, the number 72 is actually written as seven ten(s) and two[CHINESE CHARACTERS NOT REPRODUCIBLE IN ASCII]. In contrast, in the Hindu-Arabic system, the written numerals indicate digits that are positioned accordingly, and each digit is a representation of a multiple of some power of the chosen base (Eves 1990). Consider the number 564. The "5" has a value of 5([10.sup.2]), or 500; "6" has a value of 6([10.sup.1]), or 60; and "4" has a value of 4([10.sup.0]), or 4. In the Hindu-Arabic system, the base (10) is implied. Written in Chinese characters, 564 is [CHINESE CHARACTERS NOT REPRODUCIBLE IN ASCII] (5; 100; 6; 10; and 4), which literally means five hundred(s), six ten(s), and four.
In the Chinese numeration system, corresponding characters exist for 0-9 and for the multiples of 10, that is, 10; 100; 1,000; and so on. A number can easily be rewritten from Hindu-Arabic to Chinese by writing it in its expanded form, which gives the actual value of each digit. For primary students, it is suggested that a number be expanded as a sum of its parts (Cathcart et al. 2000). An example is 745 = 700 + 40 + 5, or 7 hundreds + 4 tens + 5. For intermediate students, the distributive property can be used: 745 = (7 x 100) + (4 x 10) + 5 or (7 x [10.sup.2]) + (4 X [10.sup.1]) + (5 x [10.sup.0]).
Consider the number 15 and its expanded form, 10 + 5. Using the table in figure 1, 10 is represented as [CHINESE CHARACTERS NOT REPRODUCIBLE IN ASCII] and 5 as [CHINESE CHARACTERS NOT REPRODUCIBLE IN ASCII]. Therefore, 15 written in Chinese will, appear as [CHINESE CHARACTERS NOT REPRODUCIBLE IN ASCII]. Another example is 79. In expanded form, 79 = 70 + 9. In Chinese, 70 is written as [CHINESE CHARACTERS NOT REPRODUCIBLE IN ASCII] and 9 as [CHINESE CHARACTERS NOT REPRODUCIBLE IN ASCII], and the combination of the two is [CHINESE CHARACTERS NOT REPRODUCIBLE IN ASCII]. …