Maths

Glossary

Number Systems

• ℕ - The set of all natural numbers. No zero. No negatives. Subtraction and division aren't always possible. ℕ = {1, 2, 3, ...}.
• ℤ - The set of all integers. Subtraction is always possible. ℤ = {..., -3, -2, -1, 0, 1, 2, 3, ...}
• ℚ - The set of all rational numbers (Quotient). Fractions. Division is always possible. ℚ is a field.
• ℝ - The set of all real numbers. Includes irrational and transcendental numbers. √(2), π, e, φ. ℝ is an extension of ℚ to a larger field.
• ℝ⁺ - The set of all positive reals.
• ℂ - The set of all complex numbers. i²=-1. Also a field.

• 𝔽ₚ - Set of integers modulo a prime p. Addition and multiplication also defined modulo p.
• ℚ(γ) - γ is the root of the polynomial x³-x-1. When γ³ appears it is replaced with γ+1. See: 2.2 in princeton companion of mathematics. AUTOMORPHISM

Algebraic Structures

• associative - a(bc) = (ab)c
• commutative - ab = ba

• homomorphism - Preserves the structure. f: X→Y. f(a)f(b)=f(c) and ab=c. A homomorphism of vector spaces is a linear map.
• isomorphism - A homomorphism with an inverse that is also a homomorphism. f: X→Y, g: Y→X
• automorphism - Galios Groups related. An isomorphism from a structure to itself.
• bijection - (isomorphisms are bijections?)

• Set - A collection of elements.
• Group - A set with a binary operation. The operation must be associative. If the operation is also commutative then it is an Abelian group. [TODO]: Get Axioms.
• Field - A set with 2 binary operations (+, ×). Both must be commutative and associative. And have an identity elements (+ is 0 and × is 1). Every element must have an inverse (x = -x, x = 1/x). Must follow the distributive law. Must be closed under addition, multiplication, taking of inverses (Results must in in the same set).
• Vector Space - Like a 2D or 3D plane. Can be built from 'unit vectors'. for example (1,0), (0,1). There was a Youtube on the topic. ℝ², ℝ³, ℝⁿ a vector space of n-dimensions over the field ℝ (ℝ is the scalar type).
• Scalar - A number used to multiply a vector. Infinite dimensional vector spaces exist such as when vectors are functions.
• Ring - Similar to a field but multiplication doesn't require an inverse. Multiplication might not be commutative.

• Eigenvector - A vector in a vector space that scales?

Videos

MOOCs

• Introduction to Higher Mathematics - Bill Shillito
• Coursera: Introduction to Mathematical Thinking - Stanford
• MIT OpenCourseware Maths
• MIT Linear Algebra

YouTube

• MIT Differential Equations
• Socratica - Abstract Algebra
• These Videos - Good explanations of advanced concepts.

Unsorted

HOW TO LEARN ADVANCED MATHEMATICS WITHOUT HEADING TO UNIVERSITY - PART 1 PART 3

4chan - Math Textbook Recommendations

Harvard Course of Abstract Algebra (apparently goes well with the Artin book)

List of books

Is this any good?

Order of abstract algebra

Particle Physics stuff Notes List ep1

Abstract Video Stuff

Vi Hart

Group Theory GT3

Mathologer

Higher Mathematics

mathisasport

This guide on algebra

This guide to imaginary numbers

Math intuition

These guides

Complex Numbers - This channel in a few weeks

These notes are recommended

HOW TO LEARN MATH: FOR STUDENTS

Projective Geometry

Perspective Geometry