The Mesopotamian system of sexagesimal counting numbers was based on the progressive series of units 1, 10, 1·60, 10·60, …. It may have been in use already before the invention of writing, with the mentioned units represented by various kinds of small clay tokens. After the invention of proto-cuneiform writing, c. 3300 BC, it continued to be used, with the successive units of the system represented by distinctive impressed cup- and disk-shaped number signs. Other kinds of “metrological” number systems in the proto-cuneiform script, with similar number signs, were used to denote area numbers, capacity numbers, etc. In a handful of known mathematical cuneiform texts from the latter half of the third millennium BC, the ancient systems of sexagesimal counting numbers and area numbers were still in use, alongside new kinds of systems of capacity numbers and weight numbers. Large area numbers, capacity numbers, and weight numbers were counted sexagesimally, while each metrological number system had its own kind of fractional units. In the system of counting numbers itself, fractions could be expressed as sixtieths, sixtieths of sixtieths, and so on, but also in terms of small units borrowed from the system of weight numbers. Among them were the “basic fractions” which we would understand as 1/3, 1/2, and 2/3. In a very early series of metro-mathematical division exercises and an equally early metro-mathematical table of squares (Early Dynastic III, c. 2400 BC), “quasi-integers” of the form “integer plus basic fraction” play a prominent role. Quasi-integers play an essential role also in a recently found atypical cuneiform table of reciprocals. The invention of sexagesimal numbers in place-value notation, in the Neo-Sumerian period c. 2000 BC, was based on a series of innovations. The range of the system of sexagesimal counting numbers was extended indefinitely both upward and downward, and the use of quasi-integers was abolished. Sexagesimal place-value numbers were used for all kinds of calculations in Old Babylonian mathematical cuneiform texts, c. 1700 BC, while traditional metrological numbers were retained in both questions and answers of the exercises. Examples of impressive computations of reciprocals of many-place regular sexagesimal place-value numbers, with no practical applications whatsoever, are known from the Old Babylonian period. In the Late Babylonian period (the latter half of the first millennium BC), such computations were still popular, performed by the same persons who constructed the many-place sexagesimal tables that make up the corpus of Late Babylonian mathematical astronomy.

中文翻译：

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美索不达米亚数学文本中三千年的六十进制数

美索不达米亚的六十进制计数系统基于单位 1, 10, 1 · 60, 10 · 60,...的递进系列。它可能在文字发明之前就已经被使用，其中提到的单位由各种小粘土代币表示。在原始楔形文字发明之后，c。公元前 3300 年，它继续使用，系统的连续单位以独特的压印杯和盘形数字符号表示。原始楔形文字中其他种类的“计量”数字系统，具有类似的数字符号，被用来表示面积数字、容量数字等。 在公元前三千年后半期的少数已知数学楔形文字文本中，古老的六十进制计数和面积计数系统仍在使用，以及新的容量数字和重量数字系统。大面积数、容量数和重量数以六十进制计算，而每个计量数系统都有自己的小数单位。在计数系统本身中，分数可以表示为六十度、六十度的六十度等，但也可以表示为从重量数字系统中借用的小单位。其中包括“基本分数”，我们将其理解为 1/3、1/2 和 2/3。在一系列非常早期的都市数学除法练习和同样早期的都市数学平方表（早期王朝 III，公元前 2400 年）中，“整数加基本分数”形式的“准整数”发挥了重要作用. 在最近发现的非典型楔形倒数表中，准整数也起着重要作用。在新苏美尔时期发明了用位值表示法的六十进制数 c。2000 BC，是基于一系列的创新。六十进制计数系统的范围向上和向下无限扩展，并取消了准整数的使用。在古巴比伦数学楔形文字文本中，6 进制位值数字被用于各种计算。公元前 1700 年，而传统的计量数字保留在练习的问题和答案中。从古巴比伦时期就知道了令人印象深刻的多位规则六进制位值数倒数计算的例子，但没有任何实际应用。在巴比伦晚期（公元前一千年后半期），这种计算仍然流行，