User:Aquatiki/Sandbox
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- ℕ – Natural Numbers – 1,2,3,…
- Numbers used for counting discrete objects. Equality is determined by direct inspection.
- 𝕎 – Whole Numbers – 0,1,2,3,…
- Adds zero, the additive identity. Useful when “none” must be distinguished from “does not exist”.
- ℤ – Integers (German Zahlen) – …,−3,−2,−1,0,1,2,3,…
- Numbers closed under subtraction. Every number has an additive inverse. Often interpreted as magnitude with direction.
- ℚ – Rational Numbers (quotients)
- Numbers of the form <math>\frac{a}{b}</math> with <math>a,b\in\mathbb{Z}</math> and <math>b\neq0</math>.
- Equality has a finite certificate: <math>\frac{a}{b}=\frac{c}{d} \iff ad=bc</math>.
- Most quantities in the world cannot actually be divided into arbitrary rational parts.
- 𝕂 – Constructible Numbers (German Konstruierbare)
- Lengths constructible with compass and straightedge. Equivalent to starting from 0 and 1 and allowing <math>+,-,\times,\div</math> and square roots.
- They cannot be exhaustively exhibited as decimals or directly compared.
- 𝕆 – Origami Numbers
- Lengths constructible with origami folds or neusis (marked ruler). Equivalent to extending the constructible toolkit to include cube roots.
- 𝕊 – Shifting Root Numbers
- Numbers whose digits can be generated sequentially by classical digit-extraction root algorithms. Each digit is determined exactly and never later revised.
- ℙ – Polynomial Numbers
- Roots of finite polynomials <math>a_nx^n+\dots+a_1x+a_0=0</math>.
- They can be approximated to arbitrary precision by iterative methods such as Newton’s method. Earlier digits may occasionally require revision during computation.
- 𝔾 – Geometric Numbers
- Numbers arising from geometric quantities that are visually meaningful but difficult to access numerically. Examples include <math>\pi</math>, <math>\ln 2</math>, and <math>\sin(1)</math>.
- The new additions at this stage are transcendental.
- Σ – Series-defined Numbers
- Numbers defined by infinite sums <math>\sum_{n=0}^{\infty} a_n</math>.
- Stopping after finitely many terms yields a predictable approximation.
- 𝕃 – Limit-defined Numbers
- Numbers defined as limits of sequences <math>\lim_{n\to\infty} a_n</math>.
- Each stage recomputes the value from a finite rule.
- 𝔸 – Arbitrary Algorithmic Numbers
- Numbers defined by any explicit algorithm generating digits or approximations, even if they have no particular geometric or analytic meaning.
- 𝕌 – Uncomputable Numbers
- Quantities that can be defined logically but whose digits cannot be generated by any algorithm.
- 𝔇 – Divergent Numbers
- Values assigned to divergent series by summation methods such as Cesàro, Abel, Hölder, or Borel.
- Example: <math>1-1+1-1+\dots =_{\text{Cesàro}} \tfrac12</math>.
- ζ – Zeta Numbers
- Values assigned to divergent sums using analytic continuation, especially through the Riemann zeta function.
- Example: <math>1+2+3+4+\dots =_{\zeta} -\tfrac{1}{12}</math>.
- Ramanujan is the most famous advocate of this interpretation.
- ℕ–ℚ exhibitable
- 𝕂–𝕆 geometric
- 𝕊–𝕃 algorithmic approximation
- 𝔸–𝕌 algorithm / logic
- 𝔇–ζ regularization