Editing Maths (section)

=Number Systems & Groups=
* 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, ...}. ℤ is an infinite '''ring'''.
* ℤₘ - The set of integers modulo m. A '''finite ring'''.
* ℚ - The set of all '''rational numbers''' (Quotient). Fractions. Division is always possible. ℚ is a '''field''' and an infinite ring.
* 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'''. Also an infinite '''ring'''.
* ℝ⁺ - The set of all '''positive reals'''.
* ℂ - The set of all '''complex numbers'''. 𝒾²=−1. Is a '''field''' and an infinite '''ring'''. Is not ordered because there is no way to tell which complex number is > another.
* ℍ - The set of all '''Quaternions'''. Are '''noncommutative'''. 𝒾𝒿=𝓀, 𝒿𝒾=−𝓀. Not a field.
* 𝕆 - '''Octonions'''. An 8 dimension number system. '''Noncommutative''' and '''nonassociative'''. Not a field.
* 𝔽ₚ - Set of '''integers modulo a prime p'''. Addition and multiplication also defined modulo p. A '''field'''.

* 𝕂 - Field of real or complex numbers.
* 𝔽 - Finite field.
* 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
* ℙ - Primes.
* ℙⁿ - n-dimensional '''[https://en.wikipedia.org/wiki/Projective_space projective space]'''. ℝⁿ/scaling. If you can rescale one vector to another, they are the same.
* 𝔸ⁿ - '''[https://en.wikipedia.org/wiki/Affine_space Affine Space]'''. A n-dimensional real vector space without origin. ℝⁿ/translations. 2 Objects points and vectors. [https://www.youtube.com/watch?v=uAFzRRiLSWA ~3:00min in]
* ℤ/pℤ - The set of integers modulo some prime p. Or just ℤ/p for short.
* p-adic - Changes the measurement metric to one modulus a prime.
* ℓ-adic - 'el-adic'. - [https://www.youtube.com/watch?v=aj4FozCSg8g New Theories Reveal the Nature of Numbers] - [https://en.wikipedia.org/wiki/%C3%89tale_cohomology Étale cohomology]
* [https://en.wikipedia.org/wiki/Non-standard_positional_numeral_systems Wikipedia: Non-standard positional numeral systems]
* 𝒢ₙ, 𝒢(p,q) - The geometric algebra generated by the vector space of signature (p,q) is 𝒢(p,q). 𝒢ₙ refers to all of them.  𝒢(2,0) / 𝒢(3,0) is a 2D/3D Euclidean algebra. "Geometric Algebra for Physicists".
* [https://en.wikipedia.org/wiki/Golden_ratio_base phinary] in base-φ
* [https://en.wikipedia.org/wiki/Non-integer_representation Non-integer representation]