numbers and counting: introduction 783 Martin Hall, Farmers, Kings, and Traders: The People of Southern Africa, 200–1860 (Chicago: University of Chicago Press, 1990) Erik Hildinger, Warriors of the Steppe: A Military History of Central Asia, 500 b.c to 1700 a.d (New York: Da Capo Press, 2001) Hugh Kennedy, Mongols, Huns and Vikings: Nomads at War (London: Cassell, 2002) Anatoly M Khazanov, Nomads and the Outside World, 2nd ed., trans Julia Crookenden (Madison: University of Wisconsin Press, 1994) Kathleen D Morrison and Laura L Junker, eds., Forager-Traders in South and Southeast Asia: Long-Term Histories (Cambridge, U.K.: Cambridge University Press, 2002) Robert Paine, Herds of the Tundra: A Portrait of Saami Reindeer Pastoralism (Washington, D.C.: Smithsonian Institution Press, 1994) Carla Rahn Phillips and William D Phillips, Jr., Spain’s Golden Fleece: Wool Production and the Wool Trade from the Middle Ages to the Nineteenth Century (Baltimore: Johns Hopkins University Press, 1997) Karl H Schlesier, Plains Indians, a.d 500–1500 (Norman: University of Oklahoma Press, 1994) Svat Soucek, A History of Inner Asia (New York: Cambridge University Press, 2000) Derrick J Stenning, Savannah Nomads (London: Oxford University Press, 1959) ▶ numbers and counting introduction Owing to our physiological makeup—we have 10 fingers and 10 toes—human beings naturally use the base 10 (decimal) numerical system The only other numerical system that was ever widely used before the 20th century was the base 60 (sexagesimal) system, which was developed by the ancient Sumerians and, in modified forms, is still used in calculating angles and in timekeeping To people who have gone through modern educational systems, simple mathematical operations might seem almost intuitive, and more complicated operations can be carried out quickly with the aid of writing But the situation in the Middle Ages was quite different One trick that people used to perform, which seemed miraculous to crowds of onlookers, was to calculate simple sums in their heads (such as 50 × 10) The ability of Pope Sylvester II (who imported Arabic numerals to Europe) to the same inspired the attribution of several books of magic to him At the beginning of the Middle Ages many different systems of writing numbers existed, varying from culture to culture In the Byzantine Empire and India, for instance, letters were assigned to stand for certain numbers, and large numbers often had to be written out in words (as if writing thousand instead of 1,000) Western Europe used the system of Roman numerals that is still employed in some archaizing contexts today (such as on clock faces) The forms of the Roman numerals might have arisen at least in part from finger counting But calculating was done on number tables or abacuses The use of the abacus was limited to specially trained professionals These devices used a base 10 system of numerals with place columns (for 1s, 10s, 100s, and so on) exactly as in modern numerical notation The sums in each column were marked with pebbles (calculi in Latin) or beads Calculations were done by adding or removing beads from the appropriate column and adding one to the next column to the right when a column reached 10 But before the Middle Ages there was no way of directly reproducing this system in writing The modern system of numbers now used throughout the world was developed in medieval India In the sixth century the Indian mathematician Aryabhata introduced a new system of numerals based on the Sanskrit alphabet (Before that time whole words were used to allegorically represent the numbers, such as fire for the number 3.) He differentiated consonants and vowels so that a vowel written after a consonant functioned as a power of 10 (and 100, 1,000, and so on) In the seventh century Brahmagupta became the first mathematician to use zero as a proper number rather than an empty placeholder (like a number-table column with no pebbles) With this innovation, it became possible to write down directly the decimal columns of the abacus, producing the modern form of numerical representation This discovery was spread throughout the Islamic world by the Persian philosopher al-Khwarizmi and then to western Europe by the reception of Arabic learning in 12th-century Spain The new mathematical concepts also were borrowed by China, in that case through anonymous merchants Because western Europeans received them from Islamic culture, we wrongly call the numbers of the 10-digit number system Arabic numerals, when they are actually derived from India Arabic speakers know them as Indian numerals They are based on some of the ancient Indian system of Brahmi numerals (which did not originally include the concept of decimal places) Many scholars today call them Indo-Arabic numerals to better reflect the historical reality The isolated peoples of the Americas developed their own systems of counting Most were base 10, derived from counting on the fingers But the system of counting in Central America was base 20, from counting on the hands and the feet In antiquity the Maya had developed a system of numerical notation fully equivalent to modern Arabic numerals and decimal places, even without the aid of a device like an abacus Since they used base 20, each of the numerals from to 19 was represented by an individual sign Rather than powers of 10 (10s, 100s, 1,000s, and so on), they used powers of 20 (20s, 400s, 8,000s, and so on) to express large numbers