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Chapter 4:
Numeration Systems
Mathematical Systems
A mathematical system is made up of
three components:
1.A set of elements;
2.One or more operations for combining the
elements;
3.One or more relations for comparing the
elements.
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)