Hindu-Arabic System and Its Relationship to Other Types of Systems

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Hindu-Arabic System and Its Relationship to Other Types of Systems

Section 1.1 Systems of Numeration and Additive Systems of Numeration We want to understand the system we use - Hindu-Arabic system and its relationship to other types of systems. A numeral is a symbol for a number. A system of numeration consists of a set of numerals and a rule for combining the numerals to represent numbers. For example, Numeral Type 2 Hindu-Arabi II Roman || Tally There have been four basic types of systems of numeration that different cultures developed in the course of human history. They are 1. Additive Systems 2. Alphabetic Systems 3. Multiplicative Systems 4. Position Value Systems In an additive system of numeration symbols are given for groups of ten and powers of ten. The value of a combination of symbols is the sum of the values of each symbol used. The order of the symbols does not matter in calculating the value of a group of symbols. The earliest systems that have been found in the archeological record are additive systems. Egyptian Numerals There was no symbol for zero. Example 1: Write 311 as an Egyptian numeral. Section 1.1 – Systems of Numeration 1 Example 2: Write the Egyptian numeral in Hindu-Arabic numerals. Roman Numerals The second additive system we will consider is the Roman system of numeration. This system was used in Rome and the Roman Empire and continued to be used in Europe until the 16th century. Rules for writing Roman numerals 1. A letter can only be repeated three times. 2. If one or more letters are placed after another letter of greater value, add that amount. 3. If a letter is placed before another letter of greater value, subtract that amount. Several rules apply for subtracting amounts from Roman numerals: a. Only subtract powers of ten (I, X, or C, but not V or L) For 95, do NOT write VC (100 – 5). DO write XCV (XC + V or 90 + 5) b. Only subtract one number from another. For 13, do NOT write IIXV (15 – 1 - 1). DO write XIII (X + I + I + I or 10 + 3) 4. A bar placed on top of a letter or string of letters increases the numeral's value by 1,000 times. XV = 15, XV = 15,000 (so it’s actually multiplicative too) Section 1.1 – Systems of Numeration 2 Let’s practice with subtracting Roman numerals: IV = IX = XL = XC = CD = CM = Example 3: Write the following as a Hindu-Arabic numeral. a. XVI b. DCXLVI c. XIX d. CXIIDXCIII Section 1.1 – Systems of Numeration 3 Example 4: Write the numeral as a Roman numeral. a. 94 b. 4285 c. 463 d. 2,849 We can find Roman numerals in the labeling of clocks and dating movies. Many inscriptions on monuments and buildings throughout Europe, and even some in the United States, use Roman numerals to represent calendar dates. Therefore, knowing how to interpret an ancient Roman numeral is important, even in our modern world. Section 1.1 – Systems of Numeration 4 .

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Roman Numerals
History of Numbers 1c. I can distinguish between an additive and positional system, and convert between Roman and Hindu-Arabic numbers. Roman Numerals The numeric system represented by Roman numerals originated in ancient Rome (753 BC–476 AD) and remained the usual way of writing numbers throughout Europe well into the Late Middle Ages. By the 11th century, the more efJicient Hindu–Arabic numerals had been introduced into Europe by way of Arab traders. Roman numerals, however, remained in commo use well into the 14th and 15th centuries, even in accounting and other business records (where the actual calculations would have been made using an abacus). Roman numerals are still used today, in certain contexts. See: Modern Uses of Roman Numerals Numbers in this system are represented by combinations of letters from the Latin alphabet. Roman numerals, as used today, are based on seven symbols: The numbers 1 to 10 are expressed in Roman numerals as: I, II, III, IV, V, VI, VII, VIII, IX, X. This an additive system. Numbers are formed by combining symbols and adding together their values. For example, III is three (three ones) and XIII is thirteen (a ten plus three ones). Because each symbol (I, V, X ...) has a Jixed value rather than representing multiples of ten, one hundred and so on (according to the numeral's position) there is no need for “place holding” zeros, as in numbers like 207 or 1066. Using Roman numerals, those numbers are written as CCVII (two hundreds, plus a ive and two ones) and MLXVI (a thousand plus a ifty plus a ten, a ive and a one).
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