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: Face value: the inherent value of the symbol (so how many of a certain power of the base) 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 13th century Widely accepted with invention of printing in 15th 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)