Arithmetic in Ancient China

October 3, 2009 at 12:00 am (Chinese Culture, Guest Writer, History, Science & Technology)

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.

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Epiphanies on Interstate-80

October 1, 2009 at 12:00 am (Current Affairs, Guest Writer, Travel)

An article by guest writer Maynard Chen. Refer to “About the Writer” at the end of the post.

February 2nd 2003, San Francisco, California

I started the engine, drove out of my loft in San Francisco and headed towards the Bay Bridge. Light mist enshrouded the bridge as I drove across. My thoughts were a mist of anticipation and apprehension. This was the first time I was driving across the North American continent.

The Tech Stock bubble had burst and Dot-com had morphed into Dot-bomb. Gone were the heady days of lucrative contracts in the technology job market that came with fancy take-out lunches from gourmet restaurants, and free massages to tempt us back into the office on weekends. Goodbye Tech Bubble.

Then 9-11 happened and in the aftermath, the Fed started pushing down interest rates to cushion the stock market crash. It was a time when one could refinance a housing loan without any closing costs to freeze interest rates for 30 years. Six months later one could do it again and ratchet down to lock in at an even lower rate. Housing prices began to soar. The banks made it so easy to take out home equity loans on the rising property values that people began to treat their homes as self replenishing ATM machines. Hello Housing Bubble.

As the owner of both a condo in Cambridge and a loft in San Francisco, I became worried when the housing market got so hot, that rentals started to drop even as housing prices were rising. This paradox came about because many would-be renters, able to qualify for easy financing, had decided to purchase rather than rent. I decided it was time to sell at least one of my properties. Since I had bought the Cambridge townhouse in 1981 it had made substantial capital gains. In order to lessen my tax burden, I had to move to Cambridge and live there for two years in order to qualify for the home owner’s deduction on Capital Gains Tax.

So it came to pass on that cold and wintry day, with my Toyota Camry packed to the gills with my computers and clothes and other worldly possessions, I was driving across the Bay Bridge embarking on a 3000 mile trans-continental journey from San Francisco to Boston. Even my front seat was completely filled with barang-barang, ruling out a tentative idea of taking a friend along to relieve the boredom of driving.

I considered two alternative ways to cross the USA. The southern route was longer but more sensible in winter because there was less likelihood of disruption by snow. I chose however the northern route because of the shorter distance and to avoid having to drive through regions that were reputed to be less friendly to people of color. A friend who was a member of AAA kindly obtained for me free maps for the entire trip. It was going to be essentially driving along Interstate 80 all the way. Security was also a major consideration since I could not hide any of the stuff in my car and I thought that it might be wiser to drive where climate would force most people to be indoors.

Besides the weather and security, my other concern for this trip was boredom, especially since I tend to suffer from highway hypnosis. I resolved to always pull over and take a nap whenever I felt drowsy. A friend made the brilliant suggestion of audio-books. So I assembled an eclectic stack of tapes borrowed from SF library and friends. Along the way I also planned to visit a few friends.

Berkeley was for me at the northern fringe of familiarity in the Bay Area, since I seldom visit Vallejo or the towns beyond. Towards evening I arrived in Sacramento where I spent the night in the apartment of an ex-colleague from Cambridge.

February 3rd 2003, On the road to Wendover, Utah

After Sacramento I began the ascent into the Sierras. This was where one was most likely to encounter snow. I began to listen to Red Azaleas by Anchee Min. It was a poignant semi-autobiographical story of her growing up in China during the Great Proletarian Cultural Revolution. As I drove past Emigrant Gap and Donner Summit(7239 ft) I thought of the early settlers who had passed along the route. The nearby pass is where the Donner party were stuck in the winter of 1846. Of the 81 emigrants, only 45 survived to reach California and some resorted to cannibalism. There couldn’t be anything more disparate and disconnected between the two chains of thought alternating in my mind: Anchee Min’s story of interpersonal struggles as a budding actress in China and the trials and tribulations of the early immigrants pitted against the weather while moving west to California.

Fortunately for me the weather held up, only a light dusting of snow. I turned on the windshield wipers to clear the snow, but something weird was happening. The more I sprayed fluid the dirtier it got! Upon closer examination I realized that the “dirt” was nothing more than the windshield cleaning fluid freezing and turning into slush. I had forgotten that my sunny California windshield wiper fluid did not contain anti-freeze. A new bottle of wiper fluid fixed the problem.

After driving over the Sierras, I crossed my first state border into Nevada – the casino state. The border was quite obvious because of the plethora of gigantic neons inviting you to try your luck and make(or more likely lose) your fortune. It was quite clear they were there to suck money out of California. Despite the glittering neons I did not stop except for gas, food and toilet breaks.

As I approached Wendover on the border with Utah, it was getting dark and I had resolved earlier to stop whenever it got dark so as to avoid getting into a situation where I might be stranded on the freeway at night. If I had engine problems, it would be bitterly cold, and if I was caught in a snowstorm I would not dare to leave the engine running anyway. In the Massachusetts blizzard of ’78, some people who were stranded on the freeway and left their engines on to keep the car warm, died of carbon monoxide poisoning because snow drifts blocked their exhaust vents.

On a lesser scale, at the border with Utah there were neon signs again attempting to suck money from the residents of Utah. I checked into a Motel 6 in Wendover. This is a great chain. The rooms are clean and comfortable, and the prices very reasonable. One advantage of the presence of a gaming industry for non-gamblers like me is the low price of accommodations and meals. I suspect they can afford to subsidize them so as to lure the crowds to the gaming tables.

February 4th 2003, On the road to Laramie, Wyoming

The next morning I passed by Salt Lake City and the Great Salt Lake. One really gets a sense of the size of the North American continent as the flat landscape affords a view that stretches from horizon to horizon.

Soon I crossed another state line. While driving through the bright snowy fields of Wyoming, I started to listen to “Interview with a Vampire”. There could not have been a more striking contrast between the sunny and open cowboy countryside I was passing through and the dark dank claustrophobic atmosphere of New Orleans as described in this novel. Lestat who was an immortal vampire was lonely, so he turned Louis into a vampire for companionship. Louis was a reluctant vampire who found it morally repulsive to kill humans for their blood, so he had to feed on the blood of animals in order to survive. Lestat however had no qualms about feeding upon the slaves in Louis’s plantation in Louisiana. Gradually Louis was persuaded by Lestat to feed on human blood.

I was very glad to pass the time with this tale of moral angst amongst vampires because otherwise the trip would have been numbingly boring. A three thousand mile journey is little more than endless cycles of the buildup and alleviation of biological and mechanical needs:

Begin Loop
If hungry, thirsty or in need of a toilet, then look for a rest stop
If the gas indicator is down to two thirds empty, then scan signs for cheap gas and fill up
Return to the beginning of this loop and repeat infinitely

Lestat fearing that Louis might leave him, turns a young girl into a vampire so as to give Louis a daughter – someone to care for. This leads to treachery as the vampire girl plots with Louis to murder Lestat and then flee to Europe. The novel was diverting without being the least bit scary and it made the day pass quickly. However as evening approached and the bright sunny landscape darkened into twilight, I found that the dark story became more powerful and affecting. At the same time, the gas tank was getting low and soon I would have to find the next Motel 6. As I was scanning for cheap gas, it gradually dawned upon me that gasoline to me was like blood to Louis. I could now understand at a visceral level the rising lust that vampires feel when the strength they derive from fresh blood is slowly drained and a parching thirst for fresh blood develops. To complete the metaphor, I suppose my spending the night in Motel 6 was the equivalent of resting in a coffin during the day for vampires. I laughed aloud at discovering this inadvertent metaphor, because fortunately, I did not have to kill people to fill my tank.

Driving alone day after day with no human interaction except at the most superficial level, is I suppose a form of sensory deprivation, and it is known that extended deprivation can lead to bizarre thoughts, even hallucinations.

Between Rock Springs and Rawlins, I passed over the Continental Divide of the Americas. I stopped to take a photograph of the sign that marked the line. Behind me the rivers drained into the Pacific and before me the rivers flowed into the Atlantic and the Gulf of Mexico.

Finally I arrived in Laramie where I would spend the night in Motel 6 again. I used to watch a TV series called Laramie when I was in primary school. Never would I have dreamt then that decades later I would spend a night in this town.

February 5th 2003, On the road to Des Moines, Iowa

The next morning I turned on the car radio and there was Colin Powell giving a speech at the UN on Iraq, so I switched off my audiobooks. It was an impressive and convincing speech. He had intricately detailed information about Iraq’s manufacture of weapons of mass destruction. There were satellite photos of mobile biological weapons factories and taped conversations of people actively engaged in concealing WMD from the inspectors. The case for war was getting stronger.

Upon further reflection however, there was something that didn’t quite add up. He was trying to argue that the inspections were a failure because Saddam was successfully hiding his weapons of mass destruction from the inspectors, and he could prove that because he had detailed evidence of their existence and locations. However if the US did indeed have such prodigious and detailed intelligence about these weapons, why didn’t they pass it on to the inspectors so that they could catch Sadddam red-handed? The weapons could then be confiscated and destroyed and the inspections would be a success. Of course then there would be no rationale for an invasion to remove them. He was like the salesmen selling both invincible spears that could penetrate anything, and impenetrable shields.

Just a week before this trip, I had read an essay by Peter Lee in the Smirking Chimp website*, in which he wonderfully captured the essence of the Iraq situation in a single sentence.

“Today the new orthodoxy of war and pre-emption and empire is slouching towards Baghdad waiting to be born.”

In the late afternoon while heading towards Des Moines, I was again anxiously scouting for gas to slake the recurring thirst of my Toyota Camry. I suddenly realized that even though I personally didn’t have to kill anyone for gas, the country was preparing for war, to slake the collective thirst of millions of automobiles like mine.

February 6th 2003, On the road to Ann Arbor, Michigan

While I was hurtling along Interstate 80 towards Ann Arbor, Michigan. I heard an unfamiliar noise. I glanced up and saw V shaped lines slowly traversing the grey and overcast skies. It was a flock of honking geese on their annual migration to warmer climes. I felt a certain affinity for them. Fearing the impending chill of an economic winter, I too was migrating. We were each coasting on individual trajectories towards private destinies…. just as this great nation that I was traversing was slouching towards Baghdad.

Epilogue

I finally reached Boston safely after two detours to visit old schoolmates in Ann Arbor and Toronto. Buoyed by ever more creative financing, the housing market continued to rise during the two years I lived in my Cambridge townhouse, allowing me to sell at a good price and avoid paying a lot of capital gains tax. Then the bubble finally burst. My timing was sheer luck because I thought the housing market was going to crash much sooner, but then I did not anticipate the creativity of the mortgage industry inventing balloon loans, no doc mortgages and liar loans.

Peter Lee’s prophecy came to pass. The unborn beast reached Baghdad when US troops moved into Iraq on March 20th 2003 to hunt for WMD. Two years later the CIA admitted in a final report that no WMD were found in Iraq. Powell told Walters that he felt “terrible” about the claims made in his UN speech. When asked whether it would tarnish his reputation he replied: “Of course it will. It’s a blot. I’m the one who presented it on behalf of the United States to the world, and [it] will always be a part of my record. It was painful. It’s painful now.”

* The essay by Peter Lee is entitled “A few good men … and Condi too” by Peter Lee. Posted on January 23rd 2003 in The Smirking Chimp website. It is no longer available for download, but I saved a copy.

About the Writer

Maynard Chen was a software consultant working in Silicon Valley from 1997 to 2003. He has now relocated to Singapore. © Maynard Chen, All rights reserved 2009.

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