Place-Value or Positional-Value Numeration Systems 4.2 Place-Value or Positional-Value Numeration Systems

Place-Value System (or Positional Value System) The value of the symbol depends on its position in the representation of the number. It is the most common type of numeration system in the world today. The most common place-value system is the Hindu-Arabic numeration system. This is used in the United States.

Place-Value System A true positional-value system requires a base and a set of symbols, including a symbol for zero and one for each counting number less than the base. The most common place-value system is the base 10 system. It is called the decimal number system.

Hindu-Arabic System Digits: In the Hindu-Arabic system, the digits are 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Positions: In the Hindu-Arabic system, the positional values or place values are: … 105, 104, 103, 102, 10, 1.

Expanded Form To evaluate a number in this system, begin with the rightmost digit and multiply it by 1. Multiply the second digit from the right by base 10. Continue by taking the next digit to the left and multiplying by the next power of 10. In general, we multiply the digit n places from the right by 10n–1 in order to show expanded form.

Example: Expanded Form Write the Hindu-Arabic numeral in expanded form. a) 63 b) 3769 Solution: 63 = (6  101 ) + (3  1 ) or (6  10) + 3 3769 = (3  1000) + (7  100) + (6  10) + 9 or (3  103 ) + (7  102 ) + (6  101 ) + (9  1 )

Homework P. 184 #15 – 51 (x3)