The Use of Negative Numbers

负数的使用

The introduction of negative numbers is a great contribution to mathe-matics by ancient Chinese mathematicians¹.In Nine Chapters on the Mathematical Art(Jiuzhang Suanshu)，negative numbers were used in the eighth chapter on solving systems of simultaneous equations. For instance, revenue num-bers are considered positive，while expense numbers are deemed negative;or sur-plus amounts are viewed as positive，while deficit² amounts are seen as negative.In a problem calculating grains，the increased grains are considered positive，and the lost grains，negative. At the time，calculation was done by the method of Suan Chou(counting rods)Red rods were used to denote positive coefficients，and black ones to denote negative ones. Or in another case，the normal position of Su-an Chou denoted positive，while an inclined position denoted negative.

负数的引进，是中国古代数学家对数学的一个巨大贡献。在《九章算术》的第八章“方程”中，就引人了负数解联立方程组。如负数出现在方程的系数和常数项中，把“卖(收人钱)”作为正，则“买(付出钱)”作为负，把“余钱”作为正，则“不足钱”作为负。在关于粮谷计算的问题中，是以益实(增加粮谷)为正，损失(减少粮谷)为负等。当时是用算筹来进行计算的，以红筹为正，黑筹为负；或将算筹首列作正、斜置作角。

Rules for the calculation of signed numbers were also given in Jiuzhang Suanshu.According to the book, the deduction⁴(or subtraction)of two numbers with the same sign(from another number) equals the deduction of the absolute values of the two numbers，while the deduction of two numbers with different signs equals the addition of the absolute values of the iwo numbers. Also一a positive number subtracted from zero gives a negative number,whereas a negative num-ber subtracted from zero gives a positive number. The addition of two numbers with different signs equals the deduction of their absolute values，while the addition of iwo numbers with the same sign equals the addition of their absolute values.Zero plus a positive number is still a positive number,and zero plus a negative number is still a negative number. Until the 17th century，it was the most complete depiction⁵ on the rules for adding and subtracting positive and negative一 numbers in the world.

在《九章算术》中，除了引进正负数的概念外，还完整地记载了正负数的运算法则:同号两数相减，等于其绝对值相减;异号两数相减，等于其绝对值相加;零减正数得负数，零减负数得正数。异号两数相加，等于其绝对值相减;同号两数相加，等于其绝对值相加;零加正数得正数，零加负数得负数。直到公元17世纪以前，这还是正负数加减运算最完整的叙述。

Negative numbers appeared very late in the West.Many noted³ mathemati-clans did not admit negative numbers，because they consider zero as“nothing”and could not understand that something could be even less than“nothing，”and so considered negative numbers“absurd.”It was only in the 17th century when Descartes invented the coordinate⁶ system，which gave a geometrical explanation and an actual meaning for negative numbers，that negative numbers began to be accepted gradually.

在西方，负数出现得很晚。许多著名数学家一直不承认负数。他们把零看作“没有”，他们不能理解比“没有”还要“少”的现象，因而认为负数是“荒谬的”。直到17世纪，笛卡儿创立了坐标系，负数获得了几何解释和实际意义，才逐渐得到了公认。

The introduction of negative numbers is an important contribution of Chinese mathematicians to world mathematics. With the introduction of negative numbers，the whole numbers and rational numbers became complete.

负数的引进，是中国古代数学家贡献给世界数学的一份宝贵财富。负数概念引进后，整数集和有理数集就完整地形成了。