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.

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