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.

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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)

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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

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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.

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