172 CHAPTER 4 Systems of Numeration We are all familiar with the numerals we grew up with: 1, 2, 3, and so forth. Now think of another set of numerals. You may have thought of Roman numerals: I, II, III, and so forth. Some common modern-day uses of Roman numerals include clock and watch numbers, the copyright year on movie and television credits, outlines, and page numbers at the beginning of books. In this section, we will study several sets of numerals besides our familiar 1, 2, 3, .… Additive, Multiplicative, and Ciphered Systems of Numeration SECTION 4.1 LEARNING GOALS Upon completion of this section, you will be able to: 7 Write numbers using additive systems of numeration. 7 Write numbers using multiplicative systems of numeration. 7 Write numbers using ciphered systems of numeration. Why This Is Important Seeing numerals represented many different ways helps us to increase our number sense and gain a better understanding of how numbers work. Thinking about numerals in different ways can also help us to form connections or see patterns in data. Understanding patterns is important in many scientific areas such as predicting weather so we are prepared in an emergency situation. Artificial Intelligence (AI) can be used to identify patterns in very large sets of data to help solve problems in biology, in medicine and elsewhere. By studying other types of numerals, we can better understand the underlying concepts used in many applications of mathematics. Numbers and Numerals Just as the first attempts to write were made long after the development of speech, the first representation of numbers by symbols came long after people had learned to count. A tally system using physical objects, such as scratch marks in the soil or on a stone, notches on a stick, pebbles, or knots on a vine, was probably the earliest method of recording numbers. In ancient societies, such a tally system adequately served the limited need for recording livestock, agriculture, or whatever was counted. As some societies grew more complex, however, more efficient and accurate methods of calculating and keeping records were needed. Because tally systems are impractical and inefficient, societies developed symbols to replace them. The Egyptians, for example, used the symbol , and the Babylonians used the symbol to represent the number we symbolize by 10. A number is a quantity, and it answers the question “How many?” A numeral is a symbol such as , , or 10 used to represent the number. We think a number but write a numeral. The distinction between number and numeral will be made here only if it is helpful to the discussion. In language, relatively few letters of the alphabet are used to construct a large number of words. Similarly, in arithmetic, a small variety of numerals can be used to represent all numbers. In general, when representing a number, we use as few numerals as possible. One of the greatest accomplishments of humankind has been the development of systems of numeration, whereby all numbers are “created” from a few symbols. Without such systems, mathematics would not have developed to its present level. Definition: System of Numeration A system of numeration consists of a set of numerals and a scheme or rule for combining the numerals to represent numbers. EMJAY SMITH/Shutterstock
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