Pi

Also known as Archimedes' constant, π, pi number, ratio of circumference to diameter, 3.14..., tau over two, tau/2, 0.5tau

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Encyclopedic overview

48 sections

Contents

Fundamentals
Name
Definition
Irrationality and normality
Transcendence
Continued fractions
Approximate value and digits
Complex numbers and Euler's identity
History
Polygon approximation era
Infinite series
Rate of convergence
Irrationality and transcendence
Adoption of the symbol {{pi}}
Modern quest for more digits
Motives for computing <span class="texhtml mvar" style="font-style:italic;">π</span>
Computer era and iterative algorithms
Rapidly convergent series
Monte Carlo methods
Spigot algorithms
Role and characterizations in mathematics
Geometry and trigonometry
Unit of angle
Eigenvalues
Inequalities
Fourier transform and Heisenberg uncertainty principle
Gaussian integrals
Topology
Cauchy's integral formula
Vector calculus and physics
Total curvature
Gamma function and Stirling's approximation
Number theory and Riemann zeta function
Fourier series
Modular forms and theta functions
Cauchy distribution and potential theory
In the Mandelbrot set
Outside mathematics
Describing physical phenomena
Memorizing digits
In popular culture
See also
References
Explanatory notes
Citations
Sources
Further reading
External links