4.1 Outline

A.  Basic ideas
     1. number
     2. counting number
     3. numeral
     4. numeration system
B. Egyptian numeration system
     1. simple grouping system
     2. hieroglyphic symbols
     3. addition principle
     4. repetitive system
C. Roman numeration system
     1. Roman numerals
     2. subtractive principle
     3. multiplicative principle
D. Babylonian numeration system
     1. positional system
     2. cuneiform symbols
     3. properties of numeration systems
         a. definition
         b. simple grouping system
         c. positional system
         d. addition principle
         e. subtraction principle
         f. repetitive system
E. Other historical systems
     1. decimal
     2. Greek
     3. Mayan
     4. Chinese

4.1 Essential Ideas

A number is used to answer the question "How many?" and usually refers to numbers used to count objects:

{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...}

This set of numbers is called the set of counting numbers. A numeral is a symbol used to represent a number, and a numeration system consists of a set of basic symbols and some rules for making other symbols from them, the purpose being the identification of all numbers.