4.4 Looking Back at Early Numeration Systems

The Egyptian Numeration System The Egyptians used the oldest numeration system called hieroglyphic notation.

The Roman Numeration System Roman Numeral I V X L C D M Hindu-Arabic Numeral 1 5 10 50 100 500 1000 The Roman numerals were used until the eighteenth century and are still commonly used today for outlining, on clocks, and in numbering some pages in books.

Example 1: Using the Egyptian Numeration System Write the following numeral as a Hindu-Arabic numeral: Solution: Using the table, find the value of each of the Egyptian numerals. Then add them. 1,000,000 + 10,000 + 10,000 + 10 + 10 + 10 + 1 + 1 + 1 + 1 = 1,020,034

Example 2: Using the Egyptian Numeration System Write 1752 as an Egyptian numeral. Solution: First break down the Hindu-Arabic numeral into quantities that match the Egyptian numerals: 1752 = 1000 + 700 + 50 + 2 = 1000 + 100 + 100 + 100 + 100 + 100 + 100 + 100 + 10 + 10 + 10 + 10 + 10 + 1 + 1 Now use the table to find the Egyptian symbol that matches each quantity. Thus, 1752 can be expressed as

The Roman Numeration System Roman Numeral I V X L C D M Hindu-Arabic Numeral 1 5 10 50 100 500 1000 Used until the eighteenth century Still commonly used today for outlining, on clocks, and in numbering some pages in books Super Bowl numbering

The Roman Numeration System If the symbols decrease in value from left to right, then add their values to obtain the value of the Roman numeral as a whole. If the symbols increase in value from left to right, then subtract the value of the symbol on the left from the symbol on the right to obtain the value of the Roman numeral as a whole.

Example 3: Using Roman Numerals Write CLXVII as a Hindu-Arabic numeral. Solution: Because the numerals decrease in value from left to right, we add their values to find the value of the Roman numeral as a whole. CLXVII = 100 + 50 + 10 + 5 + 1 + 1 = 167

Example 4: Using Roman Numerals Write MCMXCVI as a Hindu-Arabic numeral. Solution:

Your Turn Write the following Roman numerals in Hindu-Arabic numerals. MMCXVI = 1000 + 1000 + 100 + 10 + 5 + 1 = 2116 MCMDXII = 1000 + 900 + 500 + 10 + 2 = 2412 MDCCIXIII = 1000 + 500 + 100 + 100 + 9 + 3 = 1712

The Traditional Chinese Numeration System

Using the Traditional Chinese Numeration System Write 3264 as a Chinese numeral.

The Traditional Japanese Numeration 一 二 三 四 五 六 七 八 九 十 百 千 万 Hindu-Arabic 1 2 3 4 5 6 7 8 9 10 100 1,000 10,000 (104) Traditional Japanese 億 (oku) 兆 (chou) 京 (kei) Hindu-Arabic 100,000,000 (108) 1,000,000,000,000 (1012) 10 (1016) E.g., World population: 7.2 109 = 72億 Honda Civic Hybrid: ¥2,500,000 = ¥250万 ≈ $25,000

Japanese Currency ¥1000 ¥5000 ¥10,000 ¥1 ¥10 ¥50

Unit Prefix 1,000 103 kilo- 1,000,000 106 mega- 1,000,000,000 109 giga- 1,000,000,000,000 1012 tera- 1,000,000,000,000,000 1015 peta- 1,000,000,000,000,000,000 1018 exa-

Unit Prefix 0.1 10-1 deci- 0.01 10-2 centi- 0.001 10-3 milli- 0.000 001 10-6 micro- 0.000 000 001 10-9 nano- 0.000 000 000 001 10-12 pico-

The Ionic Greek Numeration System The ancient Greeks used letters from their alphabet for numerals. The symbols are written right next to one another.

Using the Ionic Greek Numeration System Write ψλδ as a Hindu-Arabic numeral. Solution: Retrieving what each Greek numeral represents, ψ =700, λ=30, δ=4, next we add the digits left to right according to their positions. ψλδ = 700 + 30 + 4 = 734 Thus, ψλδ represents 734 in Hindu-Arabic numerals.