4.2 Place-Value or Positional-Value Numeration Systems 181 Chances are you have seen one of the many games that require contestants to place digits in the correct order to guess the price of a car, appliance, vacation to Venice or elsewhere, or other fabulous prizes. Such games require the contestant to have an understanding of the concept of place-value that is used in our modern number system. In this section, we will study two other number systems that also rely on place-value. LEARNING GOALS Upon completion of this section, you will be able to: 7 Understand place-value numeration systems. 7 Use and understand Babylonian numerals. 7 Use and understand Mayan numerals Why This Is Important Understanding place-value helps you understand the difference between a paycheck for $500 and the $5000 price tag on a car you may like to purchase. By studying other place-value numeration systems, we will gain a better understanding of our own numeration system. Place-Value Numeration Systems Today the most common type of numeration system is the place-value system. The Hindu–Arabic numeration system, used in the United States and many other countries, is an example of a place-value system. In a place-value system , which is also called a positional-value system , the value of the symbol depends on its position in the representation of the number. For example, the 2 in 20 represents 2 tens, and the 2 in 200 represents 2 hundreds. A true positional-value system requires a base and a set of symbols, including a symbol for zero and one for each counting number less than the base. Although any number can be written in any base, the most common positionalvalue system is the base 10 system, which is called the decimal number system . The Hindus in India are credited with the invention of zero and the other symbols used in our system. The Arabs, who traded regularly with the Hindus, also adopted the system, thus the name Hindu–Arabic. Not until the middle of the fifteenth century, however, did the Hindu–Arabic numerals take the form we know today. Eighteenth-century mathematician Pierre Simon, Marquis de Laplace, speaking of the positional principle, said: “The idea is so simple that this very simplicity is the reason for our not being sufficiently aware of how much attention it deserves.” Did You Know? Babylonian Numerals The form Babylonian numerals took is directly related to their writing materials. Babylonians used a reed (later a stylus) to make their marks in wet clay. The end could be used to make a thin wedge, , which represents a unit, or a wider wedge, , which represents 10 units. The clay dried quickly, so the writings tended to be short but extremely durable. The Hindu–Arabic numerals and the positional-value system of numeration revolutionized mathematics by making addition, subtraction, multiplication, and division much easier to learn and very practical to use. Merchants and traders no longer had to depend on the counting board or abacus. The first group of mathematicians who computed with the Hindu–Arabic system, rather than with pebbles or beads on a wire, were known as the “algorists.” In the Hindu–Arabic system, the symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 are called digits . The base 10 system was developed from counting on fingers, and the word digit comes from the Latin word for fingers. The positional values in the Hindu–Arabic system are , (10) , (10) , (10) , (10) , 10, 1 5 4 3 2 … To evaluate a numeral in the Hindu–Arabic system, we multiply the digit on the right by 1. We multiply the second digit from the right by the base, 10. We multiply the third digit from the right by the base squared, 102 or 100. We multiply the fourth digit from the right by the base cubed, 103 or 1000, and so on. In general, we multiply the digit n places from the right by 10 . n 1− Therefore, we multiply the digit eight places from the right by 10 .7 Using the place-value rule, we can write a number in expanded form . Place-Value or Positional-Value Numeration Systems SECTION 4.2 Sailorr/Shutterstock
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