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.

• 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)

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

• 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

• 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 )

• 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)