Class Cp

Represents the field of complex p-adic numbers, denoted ℂₚ in mathematics.

Inheritance
object
Cp
Inherited Members
Namespace: MathLib
Assembly: MathLib.dll
Syntax
public class Cp
Remarks

ℂₚ is the algebraic closure and completion of ℚₚ with respect to the p-adic norm. It contains all solutions to algebraic equations over ℚₚ, as well as transcendental numbers. Thus, ℂₚ generalizes ℚₚ by extending it to include both algebraic and transcendental elements.

ℂₚ contains the following sets of numbers:

  • ℚ: All rational numbers.
  • ℚₚ: All ultimately periodic p-adic numbers.
  • ℂ: All complex numbers, including imaginary numbers like √(-1).
  • All algebraic numbers over ℚₚ.
ℂₚ is closed under addition, subtraction, multiplication, division, exponentiation, logarithms, and square roots. It supports all operations on complex numbers within the p-adic framework.
Examples

The square root of -1 does not exist in ℚₚ but exists in ℂₚ, yielding the imaginary unit i.

A 5-adic expansion for the rational number i (√-1) is:

√(-1) = i = "…04240422331102414131141421404340423140223032431212"₅ (no repeating pattern)
This is the OEIS sequence A210850, the other OEIS sequences for i is A210851