An article by guest writer Oon Lay Yong. Refer to “About the Writer” at the end of the post.

Let us consider the basic arithmetic when children begin to learn. First, they are taught the names of the numerals and gradually they are shown how to use them to count. Next, they are taught how to write them and use them for addition, subtraction, multiplication and division. Arithmetic is considered a very important and necessary learning subject for all young children. As the children progress they will go on to learn algebra and geometry. When we know arithmetic well, algebra is a natural development of arithmetic. Geometry deals with space and the objects in space and its most well known early associations are with Euclid’s Elements. In this article we shall focus on arithmetic and explore its origins.

Our children begin by learning the names of the numerals, that is, one, two, three, four, … etc., and learning how to write them, namely, 1, 2, 3, 4, … etc. They then learn how to add, subtract, multiply and divide using them. All these operations are familiar to all of us. Let us list the essential properties of this numeral system: It has nine different signs, 1, 2, 3, 4, 5, 6, 7, 8, 9, and the “zero” sign 0. From these signs, any numeral however large can be written. For example, three thousand five hundred and ninety two is written as: 3592. This written numeral is said to exhibit a “place value” system. The place value where each numeral is positioned is of great importance. The numeral 2 is in the units place, 9 in the tens place, 5 in the hundreds place and 3 in the thousands place.

Let us look at how the numeral nine thousand and sixty five is written: 9065. 5 is in the units place, 6 is in the tens place and 9 in the thousands place. There is no digit in the hundreds place so the “zero digit” is written there. We can use this numeral system to perform numerous operations including the basic ones of addition, subtraction, multiplication and division. We can also use the system for fractions and their operations. There are also other properties such as expressing a numeral as large or as small as we wish and finding square root or cube root of a numeral.How did the concept of this numeral system originate? Do note here that we are emphasizing the concept of the numeral system and not the shape of the numerals.

The Chinese used this concept as early as the Warring States period (475 to 221 BC). It probably began with simple addition or subtraction by forming the numerals with the use of bones or sticks. With the passing of time, these “sticks” or rods (as I shall call them) became more refined, and the people who needed to do calculations would carry them in a holder or container. The first nine numerals were represented as follows:

As you can see, they had an ingenious way of representing numbers 6 to 9 by making a horizontal rod to denote the quantity 5 while a vertical rod denotes the quantity 1. With these nine numerals in place, they had another remarkable device to represent numerals greater than these. The digits of a numeral in units, tens, hundreds, thousands and so forth were placed side by side, with adjacent digits rotated, to tell each apart. The rotated digits would look like this:

In this case the vertical rod represents the quantity 5 and a horizontal rod the quantity 1. I quote here a written description of these numerals taken from Sun Zi suanjing 孙子算经 (The mathematical classic of Sun Zi) which was written around 400 AD. An English translation of the book can be found in “Fleeting Footsteps. Tracing the Conception of Arithmetic and Algebra” by Lam Lay Yong & Ang Tian Se.

“In the common method of computation with rods, one must first know the positions of the rod numerals. The units are vertical and the tens horizontal, the hundreds stand and the thousands prostrate; thousands and tens look alike and so do ten thousands and hundreds.”

By using this rotation of rods in alternate positions, they discovered that they could denote a numeral no matter how large it was. For example 75,169 and 706,528 would be as shown:

Note that in the notation of the numeral 706528, there is an empty space between 7 and 6. 7 is in the hundred thousands place; there is no digit in the ten thousands place which accounts for the blank space, and 6 is in the thousands place, followed by 5 in the hundreds place, 2 in the tens place and 8 in the units place. With this ingenious device, they had discovered a notation that could express any number no matter how large it was. What is of paramount importance is that each digit of the numeral has to occupy its correct position.There is no existing written account on how addition and subtraction were performed with the rod numerals. As these methods were very commonly and easily performed, they were probably considered too trivial to have them written. We can speculate how these are being performed. For example, in the addition of 16 and 7, this would probably be shown as follows: The numerals 16 and 7 would be placed on the board. The “board” could be any flat surface such as a table top. The numerals were probably displayed in this manner:

The first step is to add the 2 sets of vertical rods which give the quantity 3 represented by three vertical rods. The next step would be to add the two horizontal rods of the digits in the units place, knowing that each represents the quantity 5 so that their sum gives one tens, which is added to the existing horizontal rod on the left. The result 23 is shown below:

In the case of subtraction of 7 from 16, the two numerals are again displayed as above, the two fives are subtracted and thus removed leaving “2” subtracting from “11” above to give “9” as shown below:

Step-by-step descriptions on multiplication and division can be found in Sun Zi suanjing. With the invention of this marvelous numeral notation, the Chinese were able to know how to add, subtract, multiply and divide. Furthermore, the remainder in the division method led to the concept, formation and notation of a fraction. This in turn led to the addition, subtraction, multiplication and division of fractions, the methods of which were very similar to what is being taught to our school children today. All these operations were performed with counting rods.

Despite the vast time difference between the use of the rod numerals and our present numeral system, one cannot help but note that they share similar properties. One of the earliest and most well known book on mathematics in ancient China is Jiu zhang suanshu 九章算数 (Nine chapters on the mathematical art). Li Yan & Du Shiran in their book “Chinese Mathematics: A Concise History” stated that this book “constitutes a consummation and, at the same time, a work representative of the development of ancient Chinese mathematics from the Zhou and Qin to the Han dynasties (c. 11th century BC to 220 AD)”. (Li Yan & Du Shiran’s book has been translated into English by John N. Crossley & Anthony W. C. Lun).

Each of the nine chapters in the book has specific names. Chapter One is titled fang tian 方田 which involves the measurement of areas in square units. This chapter also shows the manipulations of fractions. The title of Chapter Two is su mi 粟米 which means “millet and rice”. It deals with problems on proportions especially on the exchange of cereals. Chapter Three is called cui fen 衰分 meaning “proportional distributions”. Chapter Four is called shao guang 少 广 (short width), Chapter Five shang gong 商功 (discussing work), Chapter Six jun shu 均输 (fair transportation), Chapter Seven ying bu zu 盈不足 (surplus and deficit), Chapter Eight fang cheng 方程 (rectangular tabulation, lit.square procedure), Chapter Nine gou gu 勾股 (right angled triangles, lit. the perpendicular sides of a right-angled triangle).

In the above I have emphasized that both our present arithmetic and the ancient Chinese arithmetic are built on numeral systems which have the same properties although the shape of the numerals are different. Besides the above two Chinese books, the ancient Chinese had also written numerous other mathematical texts. However, it is sufficient from the above two works to note that the Chinese were the initiators of the arithmetic that is still being taught to our school children today.

About the Writer

Oon Lay Yong is a retired professor of mathematics, formerly from the National University of Singapore.