Community College of Rhode Island

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Math 1450: DEVELOPMENT OF THE NUMBER SYSTEM

3 Credit Course
Offered in Lecture Format
Prerequisite required (MATH 1430)

SYLLABUS

  • I. NUMERATION SYSTEMS
    • A. Historical development of, and computation with Egyptian, Babylonian, and Roman numerals
    • B. Base systems and computation in other bases
    • C. Modulo arithmetic (Gauss)
  • II. OUR COMPLEX NUMBER SYSTEMS
    • A. Complex number system Venn diagram
    • B. General definitions and history of the following sets
      • 1. Natural Numbers
        • a. Development of laws from set theoretical definitions of
          • i. Operations divisibility rules
          • ii. Factors and multiples
          • iii. Primes & twin primes
          • iv. Sieve of Eratosthenes
          • v. Composites
          • vi. Fundamental Theorem of Arithmetic
          • vii. Relative primes
          • viii. GCD
          • ix. LCM
          • x. Equivalent fractions
          • xi. Prime number generators (Euler)
          • xii. Background of factor tables
      • 2. Whole Numbers
        • a. Concept of zero as a number
        • b. Its symbol
        • c. Its role in multiplication and addition
      • 3. Integers a. Absolute value
        • b. Rules of operations
      • 4. Rationals
        • a. Denseness
        • b. Redefinition of division using reciprocals
        • c. Conversion from fraction to decimal and vice versa
      • 5. Irrationals
        • a. Powers and roots
        • b. Proof of irrationality of the square root of 2
        • c. Infinite decimals
        • d. Symbols for pi, e
      • 6. Imaginary
        • a. Even roots of negative numbers
        • b. Cyclic nature of power of i
        • c. Euler's symbols
      • 7. Complex
        • a. Complex plane
        • b. Graphing quadratic equation with complex solutions on cartesian coordinate plane
    • C. Topics for discussion (to each system as applicable)
      • 1. Order
      • 2. Finite vs. infinite
      • 3. Countable vs. uncountable
      • 4. Discrete vs. continuous
      • 5. Operations
        • a. Group properties and commutative group
        • b. Field properties
  • III. NUMBER THEORY
    • A. Pythagoreans
      • 1. Pentagram and golden ratio
      • 2. Figurate numbers
        • a. Tetratkys
        • b. Sum of arithmetic series
      • 3. Ratios between notes in music
    • B. Perfect, Abundant, Deficient, Friendly numbers as studied by the Pythagoreans, Fermat, Gauss, Euler.
    • C. Euclid's proof of an infinite number of primes
    • D. Basic number theory proofs
    • E. Casting away nines as an extension of Modulo 9 and as used to check addition and multiplication
      • 1. Proof of the divisibility rule of nine
    • F. Fermat's Theorems
      • 1. Little Theorem
      • 2. Last Theorem
    • G. Mersenne's Numbers
    • H. Goldbach's Conjecture
    • I. Krepreher's Constant
    • J. Wilson's Theorem
    • *K. Palindromes
    • *L. Syracuse Algorithm
    • *M. Stutterers

*Optional

NOTE: Howard Eve's An Introduction to the History of Math is an excellent resource text.


Last Updated: 3/18/13