Sunzi Suanjing: From Counting Objects to an Algorithm for Remainders

There is no reliable direct record of the author’s life or the exact year in which Sunzi Suanjing was completed. Researchers estimate its date from institutions, weights and measures, terminology, and other textual clues, generally discussing it in the context of the third to fifth centuries CE. This entry uses an approximate range of 200–500 solely to indicate a broad period of formation. It cannot establish a particular publication year, and the name Sunzi cannot be equated with Sun Wu, the author associated with The Art of War.

The surviving text consists of three juan. The first explains the foundations of calculation, while the middle and final juan develop methods through problems and procedural instructions. Prices, grain, numbers of people, distances, and areas feature among its subjects, showing the connections between mathematics and everyday administration. Its later inclusion in the mathematical canon and use in teaching belong to another stage of transmission, and do not prove that all surviving contents were originally organized according to later curricular institutions.

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The book stresses the need to recognize place value before calculating and explains an arrangement in which rods alternate between vertical and horizontal orientations. Units, tens, hundreds, and subsequent positions use alternating directions to prevent neighboring rod numerals from becoming confused. The same rod configuration represents ten times as much in the tens position as in the units position. Empty positions also have meaning: the absence of rods does not permit the numbers on either side to be pushed together arbitrarily.

The multiplication and division procedures also specify where the numbers being operated on and the results should be placed, and when used portions should be moved or removed. Algorithms therefore combine numerical relationships with physical operations. Writing a result as a line of Arabic numerals is convenient today, but leaves out the process by which the original readers calculated through spatial arrangement. Reading ancient mathematical books requires attention to both verbal instructions and changes in rod positions.

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The book first lists units of length, capacity, weight, and other quantities, then practices conversions through specific problems. In buying and selling, distribution, or calculations of size, numbers must be understood together with their units. Mixing zhang with chi or jin with liang makes an answer meaningless even if the multiplication or division is performed correctly. Ancient unit systems could also vary by period and region; researchers cannot assume they were identical to present-day Chinese customary measures.

The sequence of problem, answer, and procedure offers learners two levels of checking: whether the prescribed steps yield the given numerical answer, and whether the resulting quantity fits what the problem asks. An answer is not merely an isolated number; it must explain how much length, volume, or money it represents. These foundational exercises establish a computational language for the more abstract counting problems that follow.

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The famous problem in the final juan gives the results of grouping the same collection of objects: counting in threes leaves two, in fives leaves three, and in sevens leaves two, and the total number is sought. The book’s method takes seventy, twenty-one, and fifteen respectively, multiplies each by the corresponding remainder, adds the results, and subtracts an appropriate multiple of 105.

Working through the given numbers, seventy times two is 140, twenty-one times three is sixty-three, and fifteen times two is thirty, giving a sum of 233. Subtracting two lots of 105 yields twenty-three. Dividing twenty-three by three, five, and seven indeed leaves remainders of two, three, and two. This is a reconstruction of the original problem’s calculation, not an attribution of authorship or a historical event based on the later story of Han Xin counting his troops.

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In modern terms, seventy is divisible by five and seven but leaves a remainder of one when divided by three; twenty-one is divisible by three and seven but leaves one when divided by five; and fifteen is divisible by three and five but leaves one when divided by seven. Each term therefore supplies one remainder condition without affecting the other two. Subtracting 105 after combining the terms preserves every condition because it is a common multiple of three, five, and seven.

Twenty-three is only the smallest positive answer; adding any whole number of multiples of 105 still satisfies all three conditions. This explanation helps modern readers understand the procedure, but does not mean that the ancient book proved, in modern notation, a complete theorem covering all moduli. The original text provides a calculation for particular divisors. Algorithms and theory for more general cases were developed further in later mathematical history.

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The concise procedures and rhythmic phrasing of Sunzi Suanjing lent themselves to oral teaching and memorization. Memorizing them did not itself confer understanding: readers still needed to grasp place value, units, and the purpose of each step to calculate correctly with changed numbers. Later remainder problems acquired new names, mnemonic formulas, and more general algorithms. These developments show the problem’s capacity for extension, but should not all be projected back into the early text.

The editing and annotation of mathematical classics by Li Chunfeng and others during the Tang dynasty brought Sunzi Suanjing into a new setting of teaching and transmission. Song printed editions, manuscripts, and textual criticism from the Qing onward also shaped the wording available today. Its importance is twofold: it preserves detailed clues to calculation with counting rods and demonstrates how apparently scattered counting information can be combined into a definite result. Uncertainty about its date does not prevent precise analysis of these particular methods.

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