Chinese Shang Numerals Activity

Vertical Form of Shang Numerals:

__    __        ___      ____

ï          ïï        ïïï      ïïïï    ïïïïï       ï     ïï        ïïï      ïïïï

1          2           3           4           5          6      7           8           9

Horizontal Form of Shang Numerals:

—        —        —        —        —          |           |           |           |

—        —        —        —        —        —        —        —

—        —        —                    —        —        —

—        —                                —        —

—                                            —

1         2         3         4         5         6         7         8         9

Example 1: Write the number 3201 in Chinese Shang numerals.

Answer:                  (Two vertical forms must have a zero between them.)

Example 2: What number does this Chinese Shang number represent?

__                    ___

ïïï      ¾        ï         ¾        ïïï                              Answer:  31,618

I. Match each Shang number with the corresponding Indo-Arabic number:

_____1.                                                 A.        5040

­­­­_____2.                                                 B.         10,782

_____3.                                       C.        245,003

_____4.                                  D.        87

_____5.                                          E.         3693

II. Write each Indo-Arabic number as a Chinese Shang number:

1.     4301                                              1. ______________________________________

2.     10,284                                           2. ______________________________________

3.     769                                                3. ______________________________________

4.   242                                                4. ______________________________________

5.   30,025                                           5. ______________________________________

Instructor Notes

Objective: Students will experience an ancient method of writing numbers and will see number development from the perspective of another culture.  At the same time, this activity should reinforce students’ understanding of decimal place-value (positional) numeration systems and help them appreciate the importance of zero as a placeholder in such systems.

How to Use: After completing the activity, students may represent numbers of their own choosing using Shang numerals and/or may figure out how to add Shang numerals.

Background: Shang numerals date back to at least the fourth century BCE according to found artifacts, but historians believe they were used at least ten centuries before, during the Shang Dynasty (c. 16th-11th centuries BCE). The Shang numerals formed a base-10 place-value system of numeration. Numbers greater than 9 were represented by alternating the vertical and horizontal forms for the numerals 1 through 9. The vertical form was used for ones, hundreds, ten thousands, etc., and the horizontal form was used for tens, thousands, hundred thousands, etc. Counting rods were used for centuries to represent these numbers.

Other Ideas: Have students use toothpicks, popsicle sticks, or strips of paper to form the Chinese numerals. If you use strips of red and black paper for counting rods, then students can represent positive (red) and negative (black) numbers.

Solutions:

References: Activity from The Story of Negative Numbers, by J. Beery, G. Cochell, C. Dolezal, A. Sauk, and L. Shuey, in Historical Modules for the Teaching and Learning of Secondary Mathematics, Mathematical Association of America, Washington, D.C., 2003.

Cooke, Roger. The History of Mathematics. Wiley, New York, 1997, pages 223-225.

Katz, Victor J., A History of Mathematics: An Introduction, Addison-Wesley, Reading, Massachusetts, 1998, page 7.

Li Yan and Du Shiran, Chinese Mathematics, Clarendon Press, Oxford, 1987, pages 6-11.

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