Maths
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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' or 'basis'. 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)
Particle Physics stuff Notes List ep1
This guide to imaginary numbers
Complex Numbers - This channel in a few weeks
HOW TO LEARN MATH: FOR STUDENTS