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

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.