numbers and counting: Asia and the Pacific utes in an hour and partially in the number of degrees for an equilateral triangle and in the 360 degrees in a circle The Babylonians were able to use their system for reciprocal numbers, square roots, cubes, and cube roots of numbers There is evidence that mathematics was taught in schools, and some tablets survive with mathematical problems and with the working out of pure mathematical theories The Assyrians followed the same system as the Babylonians; indeed, they seem to have added nothing to the Babylonian advances in mathematics They used the numbers for purely applied purposes, such as for building temples and palaces, for computing taxation owed to their rulers, and for military reasons As a result, when the Achaemenid Persian Empire (538–331 b.c.e.) was established, it was from Babylonian science and numbering that it drew its inspiration Although the Persians did embark on a number of premathematical inquiries, their primary focus was applied mathematics There were many other numeral systems that operated in the Near East at the time, including that used by the Jews The Hebraic numeral system uses the additive principle, in which the numeric values of the individual letters are added together to form the whole Different numerals denote each number from through 10 and then 20, 30, 40, and so on until 100; different numbers are used for 200, 300, 400, 500, and so forth There are 27 Hebrew numerals in all The Hittites also had their own numbering system, as did the Phoenicians By the time Alexander the Great conquered the Achaemenid Persian Empire in the 330s b.c.e., Greek numerals tended to dominate notation in the ancient world In many ways it had come about as Greek learning permeated much of the Near East, even before Alexander’s conquest This remained the case throughout the rest of the ancient world, so much so that Greek mathematicians dominated the field during the Roman Empire ASIA AND THE PACIFIC BY FRANK J SWETZ The development of Asian counting systems and mathematics was greatly influenced by two pervasive civilizations: the Hindu empires of the Indian subcontinent and the dynastic empires of imperial China Through conquest, trade, and religious evangelization these civilizations spread their customs, beliefs, and rituals over much of the Asian region In particular, Korea, Annam (northern Vietnam), and Japan became the intellectual heirs of China Contacts resulted in the adoption of Chinese mathematical practices From the Shang Dynasty (ca 1500–ca 1045 b.c.e.) onward the Chinese had been using a decimal system of counting and recording numbers The earliest evidence of such numbers has been found on Shang oracle shells and bones used for fortune-telling More formal systems of numeration appear in bronze inscriptions of the following Zhou Dynasty (ca 1045–ca 256 b.c.e.) Finally, during the Han Dynasty 801 (ca 202–ca 220 b.c.e.), this decimal numeration system was standardized By the early centuries of the Common Era, Chinese use of number symbols had evolved into four distinct systems: a form for common use, a more elaborate form for legal and administrative documents, a commercial form that allowed for quick recording, and a scientific, countingrod, or rod-numeral, form that lent itself to calculations on a computing board It was the latter set, the rod numerals, that was most influential in terms of mathematical facility and cultural transmission and eventual adoption in the territories influenced by China Rod numerals actually represented configurations of a set of wooden computing rods employed on a counting board They were basically tally symbols representing numbers; thus would be represented by three vertical rods, recorded as a numeral by three vertical strokes In designating digits, this process continued up to recording by using five vertical strokes; for numbers above a horizontal stroke would be added, serving as a cap over the vertical strokes This horizontal stroke represented a count of The vertical strokes added below it would then represent units to be added to the So a horizontal stroke covering one vertical stroke would represent the number and a horizontal stroke with four vertical strokes would stand for the number These symbols were used to designate collections of units—100s, 10,000s, and so on (alternating powers of 10) For the remaining alternating powers of 10—10s, 1,000s, 100,000s, and so on—the orientation of the strokes was changed; vertical strokes were replaced by horizontal strokes and horizontal by vertical Thus, alternating 10’s positions in a decimal-based numeral would be symbolized by contrasting sets of strokes In this system numerals would be written from left to right, with the highest power of 10 occupying the left-most position in the numeral An empty space in the numeral indicated an empty position on a counting board (a “zero”) A detailed description of Chinese counting rods and their computational procedures is given in Sun zi suan jing (The Mathematical Classic of Master Sun), written about the year 400 c.e This rod counting and numeration system was most efficient and was used throughout Asia for many years until it was replaced by the abacus Common Chinese numerals were (and still are) written in a vertical column with the highest place value at the top These columns of characters are unique in that they serve as numerals, counting symbols, and number words or phrases There are separate characters for each grouping of 10s—10, 100s, 1,000s—and, in a numeral, each 10’s grouping would be preceded by a counting character; thus a reader would note two characters (words), such as three hundreds or five tens In such a numeration system there is no need for a “zero” symbol as a placeholder, the name of the particular 10’s place is just missing from the designation The positional value scheme of the Chinese numerals was very efficient and flexible, allowing for the recording of large numbers, up to 107; it could even, conceptually, express