DelSquared.Radicals 1.1.0

dotnet add package DelSquared.Radicals --version 1.1.0                
NuGet\Install-Package DelSquared.Radicals -Version 1.1.0                
This command is intended to be used within the Package Manager Console in Visual Studio, as it uses the NuGet module's version of Install-Package.
<PackageReference Include="DelSquared.Radicals" Version="1.1.0" />                
For projects that support PackageReference, copy this XML node into the project file to reference the package.
paket add DelSquared.Radicals --version 1.1.0                
#r "nuget: DelSquared.Radicals, 1.1.0"                
#r directive can be used in F# Interactive and Polyglot Notebooks. Copy this into the interactive tool or source code of the script to reference the package.
// Install DelSquared.Radicals as a Cake Addin
#addin nuget:?package=DelSquared.Radicals&version=1.1.0

// Install DelSquared.Radicals as a Cake Tool
#tool nuget:?package=DelSquared.Radicals&version=1.1.0                



.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.


This library provides three structures to enable radical expression arithmetic:

  • Radical
  • RadicalSum
  • RadicalSumRatio


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)


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 = 
        * (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) 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)


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.


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. 😃


The original inspiration for this project came from working on a program that recursively calculates Clebsch-Gordan 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.

My Clebsch-Gordan coefficient calculator utilizing the Radicals package can be found here.


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.

Product Compatible and additional computed target framework versions.
.NET net5.0 is compatible.  net5.0-windows was computed.  net6.0 was computed.  net6.0-android was computed.  net6.0-ios was computed.  net6.0-maccatalyst was computed.  net6.0-macos was computed.  net6.0-tvos was computed.  net6.0-windows was computed.  net7.0 was computed.  net7.0-android was computed.  net7.0-ios was computed.  net7.0-maccatalyst was computed.  net7.0-macos was computed.  net7.0-tvos was computed.  net7.0-windows was computed.  net8.0 was computed.  net8.0-android was computed.  net8.0-browser was computed.  net8.0-ios was computed.  net8.0-maccatalyst was computed.  net8.0-macos was computed.  net8.0-tvos was computed.  net8.0-windows was computed. 
.NET Core netcoreapp2.0 was computed.  netcoreapp2.1 was computed.  netcoreapp2.2 was computed.  netcoreapp3.0 was computed.  netcoreapp3.1 was computed. 
.NET Standard netstandard2.0 is compatible.  netstandard2.1 is compatible. 
.NET Framework net461 was computed.  net462 was computed.  net463 was computed.  net47 was computed.  net471 was computed.  net472 was computed.  net48 was computed.  net481 was computed. 
MonoAndroid monoandroid was computed. 
MonoMac monomac was computed. 
MonoTouch monotouch was computed. 
Tizen tizen40 was computed.  tizen60 was computed. 
Xamarin.iOS xamarinios was computed. 
Xamarin.Mac xamarinmac was computed. 
Xamarin.TVOS xamarintvos was computed. 
Xamarin.WatchOS xamarinwatchos was computed. 
Compatible target framework(s)
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Version Downloads Last updated
1.1.0 439 1/18/2021
1.0.0 474 2/1/2020
1.0.0-beta3 404 12/26/2019
1.0.0-beta2 426 5/27/2019
1.0.0-beta1 422 4/4/2019