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Chapter 4: Numeration and Mathematical Systems

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

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Chapter 4: Numeration and Mathematical Systems

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  1. Chapter 4: Numeration and Mathematical Systems

  2. Mathematical Systems A mathematical system is made up of three components: • A set of elements; • One or more operations for combining the elements; • One or more relations for comparing the elements.

  3. 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)

  4. 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

  5. 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

  6. 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)

  7. 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)

  8. Symmetries of the Square • A mathematical system with elements {M, N, P, Q, R, S, T, V} and operation □ denoting composition. • Each element is an operation that is a symmetry of the square. • A □ B means do A first then B

  9. Elements of the Symmetries of the Square • M: rotate 90° • N: rotate 180° • P: rotate 270° • Q: rotate 360° • R: horizontal reflection • S: vertical reflection • T: diagonal reflection at top left corner • V: diag. reflection at top right corner

  10. Combining Table

  11. Properties of Mathematical Systems Given a system with elements a,b,c etc and operation * • Closure: a * b is defined for any a,b • Commutative: a * b = b * a for any a,b • Associative: a * (b * c) = (a * b) * c for any a,b,c • Identity: there is some element  such that for any a a *  = a =  * a • Inverse: for every a there is a b such that a * b =  (where  is the identity)

  12. Groups • A group is a mathematical system that satisfies the closure, associative, identity and inverse properties • Ex. Symmetries of the square

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