Linear Overview:
A Sequential Outline of Topics

Topic Headings** Description
  1. Bits, Bytes and Bases
    1. Positional Notation
    2. Roman Numerals
    3. From the Abacus to Computers
    4. Powers of 10
    5. Powers of 2
    6. Binary Numbers
    7. Hexadecimal Numbers

Learn some of the terminology surrounding number systems to the base N, with emphasis on those used in computing. Discuss positional notation systems with Roman Numerals for contrast. Research the history of computing devices, from the abacus until the present time. Preview exponential notation.

  1. Cardinality vs. Ordinality
    1. Numbers for Naming
    2. Addressing Schemes
    3. Relational Databases
    4. ASCII and Unicode
    5. Permutations
    6. Ordering a Set
    7. The Number Line

Develop cardinal sense by looking at symbols paired with unique binary identifiers as per ASCII and Unicode mappings, contrast with ordinal sense. Consider alphanumeric addressing schemes such as URIs for web-based resources. Disucuss the omnipresence of relational data bases in contemporary society. Review hexadecimal numbers. Look at greater-than, less-than and equals operators in the context of the number line as an organizing heuristic.

  1. Algorithms and Operations
    1. Basic Operations
    2. Modulo Arithmetic
    3. Groups, Rings, and Fields
    4. Basic Algorithms
    5. Sorting
    6. Huffman Compression
    7. Operations in Base 2
    8. Boolean Operations

Review basic operations and corresponding algorithms. Include exponentiation. Introduce computer programs as algorithms. Explore logic gates in hardware, and more complicated circuits such as the half-adder.

  1. Integers
    1. Modulo Arithmetic
    2. Primes and Composites
    3. Coprimes
    4. Groups and Galois Fields
    5. Factoring
    6. Fermat's Little Theorem
    7. Carmichael Numbers
    8. Cryptology and RSA
    9. Mathematical Induction

Contrast groups and fields using integer modulo multiplication and addition. Review raising to a power. Apply Fermat's Little Theorem to the task of determining primality. Review basic group theory. Discuss the RSA algorithm in the context of the field of cryptology, its history and applications. Preview extended Euclidean Algorithm in this context.

  1. Rational Numbers
    1. The Field of Rationals
    2. Terminology
    3. Stern-Brocot Tree
    4. GCD and LCD
    5. Euclid's Algorithm
    6. Reducing Fractions
    7. Fractions as Objects
    8. Converting to/from decimals

Build a fraction object to support basic operations, including division of fractions, with GCD and LCD as integral functions.

  1. Real Numbers
    1. Decimal Patterns
    2. The Cartesian Plane (R X R)
    3. Pythagorean Theorem
    4. Incommensurability
    5. Exponents and Logarithms
    6. Transcendental Numbers e and pi

Develop an understanding regarding the relatioship between decimals and reals. Preview trigonometry by introducing the Pythagorean Theorem and the distance formula. Introduce log and exp operations. Preview functions vs. inverse functions.

  1. Functions
    1. Operations on Sets of Objects
    2. Functions and Relations
    3. Domain and Range
    4. Recursive Functions
    5. Composition of Functions
    6. Inverse Functions
    7. Bijection, Surjection, Injection
    8. Linear Functions
    9. Polynomials
    10. Graphing Functions on the Cartesian Plane

Introduce the generic concept of a relation as a set of ordered (domain, range) pairs, functions as a subset of relation. Review permutations when discussing composition of functions. Preview polynomials. Discuss inverse functions, along with various common types of functions and their graphs on the XY plane

  1. Sequences and Sphere Packings
    1. Planar Figurate Numbers
    2. Spatial Figurate Numbers
    3. The Jitterbug Transformation
    4. SCP, BCC, CCP, HCP

Develop the relationship between number sequences and geometric figures, both on the plane and in space. Use the jitterbug transformation to bridge the cuboctahedron and icosahedron. Given these shapes may be developed from packed spheres, discuss common sphere packing arrangements and their applications in architecture, biology, chemistry and crystallography.

  1. Series
    1. Sigma Notation
    2. Pi Notation
    3. Factorials
    4. Infinite Series
    5. Continued Fractions

Introduce sigma and pi notation, factorials. Review recursion. Look at continued fractions, especially for roots, e and pi. Review Stern-Brocot tree.

  1. Pascal's Triangle
    1. Permutations vs. Arrangements
    2. Finding Sequences
    3. Fibonacci Sequence
    4. Polynomials as Objects in a Ring
    5. Binomial Theorem
    6. Gaussian Distribution
    7. Basic Statistics Concepts
    8. Bernoulli Numbers
    9. Pascal's Tetrahedron and Trinomials

Discuss the difference between permutations and arrangements and derive the relevant formulae. Generate Pascal's Triangle and locate sequences. Revisit polynomial objects in a ring algebra and dervive the Binomial Theorem. Relate Pascal's Triangle to the Gaussian distribution and discuss probability and statistical concepts. Review factorial, sigma and pi notations.

  1. Complex Numbers
    1. Roots of Negative Numbers
    2. Complex Plane
    3. Fractals
    4. Roots of Polynomials
    5. Galois Theory

Discuss the complex plane, convergence, fractals. Review recursion. Preview Galois Theory regarding roots of polynomials, building on group theory concepts.

  1. Basic Trig
    1. Parts of a Triangle
    2. Types of Triangles
    3. Properties of a Tetrahedron
    4. Basic Trig Functions
    5. Graphing Trig Functions
    6. Some Trig Identities
    7. Polygons and Symmetry
    8. Complex Numbers and Trig
    9. Fourier Transformation

Discuss the parts and types of triangles and strategies for computing their measures, given partial information. Define the trig functions and their inverse functions. Preview polyhedra when discussing polygons. Look at complex numbers in conjunction with the unit circle.

  1. Polyhedra
    1. Fuller's Concentric Hierarchy
    2. Euler's Law
    3. Descartes' Deficit
    4. Computing Angles and Distances
    5. Coordinate Geometry
    6. Rotational Symmetry
    7. Symmetry Groups
    8. Data Structures and Objects

Introduce Fuller's concentric hierarchy while reviewing sphere packing concepts and the octet truss. Use Euler's Law and Descartes' deficit to determine various measures. Discuss various coordinate systems and the concept of symmetry groups. Begin building polyhedra as objects.

  1. Simulations
    1. Feedback Networks
    2. Dynamical Systems
    3. Markov Chains
    4. State Machines
    5. Cellular Automata

Explore simulations of complex systems, including those exhibiting non-linear and/or chaotic features. Deepen understanding of modeling techniques and the relationship between models and actual phenomena.

  1. Vector Algebra
    1. Historical Overview
    2. Coordinate Systems
    3. Vectors as Objects
    4. Vector Operations
    5. Matrix Operations
    6. Rotation Matrices
    7. Polyhedra as Objects

Build vector objects after looking at the historical challenges addressed by vector mathematics, in physics especially. Review polyhedra in light of these new tools. Consider polyhedra as objects with inherited methods for scaling, rotation and translation.

  1. Some Calculus Concepts
    1. The Study of Change
    2. Newton's Method
    3. Derivative
    4. Integral
    5. Area under a Curve
    6. Length of a Curve
    7. LaGuerre's Method
    8. Vector Calculus

Introduce the calculus using the motion picture analogy. Use Newton's Method to find nth roots of a real number, developing the geometric relationship between the derivative and slope at a point. Introduce integration as the inverse of differentiation. Introduce hyperbolic trig functions when discussing the catenary (a common curve). Review roots of polynomials in the complex plane in the context of LaGuerre's Method. Preview some vector calculus concepts.

** underlined topic headings link to essays

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