text: The modern base ten system uses ten symbols (0, 1, 2, 3, ...,9)\nwith place values that are powers of ten. These symbols are commonly called\nHindu-Arabic numerals. Although a base ten system seems natural to those\nwho have ten Þngers, in times past people have used many different kinds of\nnumerals and various bases in addition to base ten.\nOther numeration systems that have existed historically used base Þve,\nbase twenty, and base sixty. In fact almost all societies have used a number\nsystem based on 5, 10, or 20. This is understandable because of human\nanatomy. Counting on the Þve Þngers of one hand, all ten Þngers, or all\ntwenty digits is natural and useful. The quinary system (base 5) or vigesimal\n(base 20) or a combination of these two were used by early societies in the\nAmericas including almost all Eskimo tribes, some native North American\nsocieties, and almost all peoples indigenous to Central and South America.\nAlthough the quinary and vigesimal systems were also found in some societies\nin northern Siberia and Africa the decimal system used today dominated\nhistory. However, with advent of computers in this century bases two, eight\nand sixteen have come into prominence. Computers count best using only\ntwo digits. It is interesting to note that certain Polish engineers built a\ncomputer in the late 1950s and early 1960s that used a base of negative 2.\nIn order to understand numeration systems it is important to know the\ndistinction between place value systems and additive systems. In additive\nsystems (Roman numerals, for example) a numeration symbol has the same\nmeaning no matter where it is placed in writing the number. So, for example,\nthe Roman numeral V means Þve whether it is placed before or after the\nnumeral X. Thus XV and VX both represent Þfteen in an additive system,\nalthough as we discover VX would certainly be non-standard. The values\nten (X) and Þve (V) are added to obtain Þfteen.\nThe position of each numeral is important in positional (place value)\nnumber systems, but not in additive systems. For example, in order to\nrepresent Þfty or Þve hundred in the base ten place value system, the symbol\n5 is moved over to the left one or two places (50 or 500), putting a zero\nin the empty places. However, in order to represent Þfty or Þve hundred in\nan additive system multiple copies of the ten or hundred need to be used\n(XXXXX or CCCCC) or new symbols need to be invented (L and D in\nRoman numerals).\nPlace value (or positional) systems can continue to use the same set of\nsymbols since the value of the symbol changes with repositioning. The value\nof each symbol is multiplied by a power of the systems base. For example, in\nbase ten (decimal) different positions of the numeral Þve will represent Þve,\nÞfty, and Þve hundred. Hence, the symbol 5 in each of the following numbers\nrepresents different quantities because of its position.\n in 215 the Þve represents Þve (5 × 100 )\n in 354 the Þve represents Þfty (5 × 101 )\n in 567 the Þve represents Þve hundred (5 × 103 )\nThe number one hundred twenty four is represented as 124 in the base ten\nsystem, where the 1 signiÞes one hundred, the 2 signiÞes two tens and the\n4 signiÞes four ones. This number is written in expanded form as follows: