User:Aquatiki/Sandbox
Jump to navigation
Jump to search
- โ โ Natural Numbers โ 1,2,3,โฆ
- Numbers used for counting discrete objects. Equality is determined by direct inspection.
- ๐ โ Whole Numbers โ 0,1,2,3,โฆ
- Adds zero, the additive identity. Useful when โnoneโ must be distinguished from โdoes not existโ.
- โค โ Integers (German Zahlen) โ โฆ,โ3,โ2,โ1,0,1,2,3,โฆ
- Numbers closed under subtraction. Every number has an additive inverse. Often interpreted as magnitude with direction.
- โ โ 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.
- ๐ โ 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.
- ๐ โ Origami Numbers
- Lengths constructible with origami folds or neusis (marked ruler). Equivalent to extending the constructible toolkit to include cube roots.
- ๐ โ 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.
- โ โ 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 Newtonโs method. Earlier digits may occasionally require revision during computation.
- ๐พ โ 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.
- ฮฃ โ 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.
- ๐ โ 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.
- ๐ธ โ Arbitrary Algorithmic Numbers
- Numbers defined by any explicit algorithm generating digits or approximations, even if they have no particular geometric or analytic meaning.
- ๐ โ Uncomputable Numbers
- Quantities that can be defined logically but whose digits cannot be generated by any algorithm.
- ๐ โ Divergent Numbers
- Values assigned to divergent series by summation methods such as Cesร ro, Abel, Hรถlder, or Borel.
- Example: <math>1-1+1-1+\dots =_{\text{Cesร ro}} \tfrac12</math>.
- ฮถ โ 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.
- โโโ exhibitable
- ๐โ๐ geometric
- ๐โ๐ algorithmic approximation
- ๐ธโ๐ algorithm / logic
- ๐โฮถ regularization