DelSquared.Radicals 1.0.0-beta2

Radicals

Overview

.NET implementation of radical expressions, where the radicand is a rational number and the index is a positive integer. Enables arithmetic and string formatting of radical expressions, and can handle rational radicands of arbitrary precision (see dependencies). Seamlessly integrates with other numeric types such as int, long, and BigInteger.

NuGet package for this library is published here.

Usage

This library provides three structures to enable radical expression arithmetic:

• Radical
• RadicalSum
• RadicalSumRatio

Radical

The Radical structure encapsulates a single radical expression with rational radicand and positive integer index:

var sqrt2 = Radical.Sqrt(2);                        // Sqrt(2)
var sqrt_1_3 = Radical.Sqrt((Rational)1/3);         // Sqrt(1/3)    = (1/3)*Sqrt(3)
var root_3_1_2 = Radical.NthRoot((Rational)1/2, 3); // Root[3](1/2) = (1/2)*Root[3](4)

Note - see dependencies section for information on the Rational structure

The structure automatically simplifies radicals to simplest form, as described here.

Radicals can be multiplied and divided by other radicals, returning new Radicals:

var result1 = Radical.Sqrt(2) * Radical.Sqrt(3/4);  // Sqrt(2) * Sqrt(3/4)       = (1/2)*Sqrt(6)
var result2 = result1 * Radical.Sqrt(2);            // (1/2)*Sqrt(6)*Sqrt(2)     = Sqrt(3)
var result3 = result2 / result1;                    // Sqrt(3) / [(1/2)*Sqrt(6)] = Sqrt(2)
var result4 = result3 / Radical.Sqrt(5);            // Sqrt(2) / Sqrt(5)         = (1/5)*Sqrt(10)

They can also be multiplied and divided by most numeric types, returning new Radicals:

var result1 = Radical.Sqrt(2) * 3;                  // 3 * Sqrt(2)
var result2 = 4 * result1;                          // 4 * 3 * Sqrt(2)   = 12 * Sqrt(2)
var result3 = result2 / 3;                          // 12 * Sqrt(2) / 3  = 4 * Sqrt(2)
var result4 = 5 / result3;                          // 5 / (4 * Sqrt(2)) = (5/8)*Sqrt(2)

Additionally, Radicals of differing indices can be multiplied and divided:

var result1 = Radical.Sqrt(2) 
    * Radical.NthRoot(2,3);                         // Sqrt(2) * Root[3](2)     = Root[6](32)
var result2 = result1 / Root[3](5);                 // Root[6](32) / Root[3](5) = (1/5)*Root[6](20000)

RadicalSum

Radicals can also be added and subtracted from each other. However, unless they have identical indices and radicands, the simplest form result is not another radical, but instead is a sum of radicals. The RadicalSum structure encapsulates this notion and enables Radical addition and subtraction:

var result1 = Radical.Sqrt(2) + Radical.Sqrt(3);    // Sqrt(2) + Sqrt(3)
var result2 = result1 + Radical.Sqrt(2);            // [Sqrt(2) + Sqrt(3)] + Sqrt(2) = 2*Sqrt(2) + Sqrt(3)
var result3 = result1 - Radical.NthRoot(5,3);       // 2*Sqrt(2) + Sqrt(3) + (-1)*Root[3](5)
var result4 = result3 + 11;                         // 11 + 2*Sqrt(2) + Sqrt(3) + (-1)*Root[3](5)

Similar to Radicals, RadicalSums can be multiplied by most numeric types, Radicals, and other RadicalSums to return new RadicalSums...

var result1 = 
    (Radical.Sqrt(2) + Radical.Sqrt(3)) 
        * Radical.Sqrt(2);                          // [Sqrt(2) + Sqrt(3)] * Sqrt(2) = 2 + Sqrt(6)
var result2 = 
    result1
        * (Radical.Sqrt(5) + Radical.Sqrt(7));      // [2 + Sqrt(6)] * [Sqrt(5) + Sqrt(7)]
                                                    // = 2*Sqrt(5) + 2*Sqrt(7) + Sqrt(30) + Sqrt(35)
                                                    
var result3 = result2 * 3;                          // 6*Sqrt(5) + 6*Sqrt(7) + 3*Sqrt(30) + 3*Sqrt(35)

...as well as divided by most numeric types and Radicals to return new RadicalSums:

var result4 = result3 / 2;                          // 3*Sqrt(5) + 3*Sqrt(7) + (3/2)*Sqrt(30) + (3/2)*Sqrt(35)
var result5 = result4 / Radical.Sqrt(2);            // (3/2)*Sqrt(10) + (3/2)*Sqrt(14) + (3/2)*Sqrt(15) + (3/4)*Sqrt(70)

RadicalSumRatio

RadicalSums can also be divided by other RadicalSums. However the result is not another RadicalSum, but is instead a RadicalSumRatio where the numerator and denominator are both RadicalSums:

var result1 =
 (Radical.Sqrt(2) + Radical.Sqrt(3))
     / (Radical.Sqrt(2) - Radical.Sqrt(3));         // Sqrt(2) + Sqrt(3)
                                                    // -----------------
                                                    // Sqrt(2) - Sqrt(3)

All arithmetic operators between RadicalSumRatios and numeric types, Radicals, and RadicalSums will return new RadicalSumRatios.

Performance

This project was intended as a small fun side project. As such, I haven't put much effort or thought into optimization. While it suited my needs and should be okay for most casual implementations, don't expect optimal performance and use at your own risk! Of course, I'm always open to suggestions and improvements. 😃

Background

The original inspiration for this project came from working on a program that would recursively generate Clebsch-Gordan coefficients (https://en.wikipedia.org/wiki/Clebsch%E2%80%93Gordan_coefficients). While this succeeded in generating the coefficients in decimal form, I wanted it to present them in radical form similarly to how they're usually presented in tables (e.g., here). I created this library to enable radical expression arithmetic during the calculation of the coefficients, and to format the result as usually presented in tables.

Dependencies

This project is dependent on the following NuGet packages:

Rationals: Encapsulates rational numbers of theoretically arbitrary precision. Based on BigInteger.

Open.Numeric.Primes: Provides methods to get prime factorization of BigInteger values.