Counters

Before the invention of the abacus, counters or tallies were used in China in calculating operations over long ages. Historical writings indicate they were already quite widespread during the Spring and Autumn and the Warring States periods (770-221 B.C.).The earliest counters were round sticks about 0.23 cm across and 12 cm long. Later on they became rectangular in cross-section and shortened to 7-8cm. The earliest counters thus far discovered were unearthed in 1971 in Qianyang County, Shaanxi Province. Dating from the Western Hart (206 B.C.-25 A.D.), they total 30 in number.Ancient counters, mostly made of bamboo and animal bone, are also found of iron, bronze, ivory and jade.To express numbers with the unnotched tallies, ancient Chinese followed the decimal system. That is to say, the same stick placed at the unit's place meant" 1 ", but placed at ten's place, meant "10". Arranged vertically, thus I, Iq, IH, ]ill, t1111, t, II, ill, and t117 (=1, 2, 3, 4, 5, 6, 7, 8 and 9), they were used to represent the figures at the unit's, hundred's, ten thousand's and million's places. Arranged horizontally, thus -, =, =, =-, -, J-, L, _-, and ~- _= they would represent the figures at the ten's, thousand's and hundred thousand's places. The two arrangements, alternating each with the other, could express any figure, no matter how long. In this way, - I~ = Ill1would mean "1984". When a digit happens to be zero, it would be left blank.Thus, == Ni ~]- would read "4308". The system proved easy to learn and master.The counters represented at first only positive numbers. But as more complicated calculations were called for, tallies began to be made in two different colours with red ones standing for positive numbers and black ones negative numbers.The counters, simple as they appear, were employed by the ancients not only to do the sums but also the extraction of square and cube roots, the solution of equations of higher degrees and the calculation pi.It is estimated that counter sticks were widely used in China for 1,500 years until gradually replaced by the abacus about the 15th century during the Ming Dynasty (1368-1644).
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