Chapter 4 Numeration Systems Numeration Systems A number

Chapter 4: Numeration Systems

Numeration Systems • A number system has a base. Our system is base 10, but other bases have been used (5, 20, 60) • Simple grouping system uses repetition of symbols, with each symbol denoting a power of the base (ex Egyptian) • Multiplicative grouping uses multipliers instead of repetition (ex Traditional Chinese)

Positional Systems In a positional system, each symbol (called a digit) conveys two things: 1) Face value: the inherent value of the symbol (so how many of a certain power of the base) 2) Place value: the power of the base which is associated with the position that the digit occupies in the numeral

Hindu-Arabic System • Our system, the Hindu-Arabic system, is a positional system with base 10. • Developed over many centuries, but traced to Hindus around 200 BC • Picked up by Arabs and transmitted to Spain • Finalized by Fibonacci in 13 th century • Widely accepted with invention of printing in 15 th century

Different Bases • Our number system is decimal, so the base is 10. The digits are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. • With a different base b, the digits are 0, 1, …, b-1. • Some special bases: 2 (binary), 8 (octal), 16 (hexadecimal)

What do we do with different number bases • Convert a number in a different base to decimal • Convert a decimal number to a different base • Add numbers with same base (be sure to carry if needed) • Subtract numbers with same base (be sure to regroup if needed)