Glossary

Number Systems

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

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.
Vector Space - Like a 2D plane. Can be built from 'unit vectors'. for example (1,0), (0,1). There was a youtube on the topic.