# Math 1450: DEVELOPMENT OF THE NUMBER SYSTEM

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**3 Credit Course**

Offered in Lecture Format

Prerequisite required (MATH 1430)

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

- a. Development of laws from set theoretical definitions of
- 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

- 1. Natural Numbers
- 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

- A. Pythagoreans

*Optional

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