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 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)