Complex Numbers in C#

A complex number in C# consists of two components: a real part and an imaginary part. For example, in the complex number 7+5i, the real part is 7 and the imaginary part is 5 (the coefficient of i).

C# allows you to create custom data structures to represent complex numbers using struct and implement mathematical operations using operator overloading.

Syntax

Following is the syntax for creating a complex number structure −

public struct Complex {
   public double real;
   public double imaginary;
   
   public Complex(double real, double imaginary) {
      this.real = real;
      this.imaginary = imaginary;
   }
}

Following is the syntax for overloading the addition operator −

public static Complex operator +(Complex one, Complex two) {
   return new Complex(one.real + two.real, one.imaginary + two.imaginary);
}

Understanding Complex Number Structure

Using Basic Complex Number Operations

Example

using System;

public struct Complex {
   public double real;
   public double imaginary;
   
   public Complex(double real, double imaginary) {
      this.real = real;
      this.imaginary = imaginary;
   }
   
   public static Complex operator +(Complex one, Complex two) {
      return new Complex(one.real + two.real, one.imaginary + two.imaginary);
   }
   
   public override string ToString() {
      if (imaginary >= 0)
         return String.Format("{0} + {1}i", real, imaginary);
      else
         return String.Format("{0} - {1}i", real, Math.Abs(imaginary));
   }
}

class Program {
   static void Main() {
      Complex val1 = new Complex(7, 1);
      Complex val2 = new Complex(2, 6);
      
      Complex result = val1 + val2;
      
      Console.WriteLine("First: {0}", val1);
      Console.WriteLine("Second: {0}", val2);
      Console.WriteLine("Result (Sum): {0}", result);
   }
}

The output of the above code is −

First: 7 + 1i
Second: 2 + 6i
Result (Sum): 9 + 7i

Using Complex Numbers with Multiple Operations

Example

using System;

public struct Complex {
   public double real;
   public double imaginary;
   
   public Complex(double real, double imaginary) {
      this.real = real;
      this.imaginary = imaginary;
   }
   
   public static Complex operator +(Complex a, Complex b) {
      return new Complex(a.real + b.real, a.imaginary + b.imaginary);
   }
   
   public static Complex operator -(Complex a, Complex b) {
      return new Complex(a.real - b.real, a.imaginary - b.imaginary);
   }
   
   public static Complex operator *(Complex a, Complex b) {
      return new Complex(
         a.real * b.real - a.imaginary * b.imaginary,
         a.real * b.imaginary + a.imaginary * b.real
      );
   }
   
   public double Magnitude() {
      return Math.Sqrt(real * real + imaginary * imaginary);
   }
   
   public override string ToString() {
      if (imaginary >= 0)
         return String.Format("{0} + {1}i", real, imaginary);
      else
         return String.Format("{0} - {1}i", real, Math.Abs(imaginary));
   }
}

class Program {
   static void Main() {
      Complex c1 = new Complex(3, 4);
      Complex c2 = new Complex(1, -2);
      
      Console.WriteLine("Complex 1: {0}", c1);
      Console.WriteLine("Complex 2: {0}", c2);
      Console.WriteLine("Addition: {0}", c1 + c2);
      Console.WriteLine("Subtraction: {0}", c1 - c2);
      Console.WriteLine("Multiplication: {0}", c1 * c2);
      Console.WriteLine("Magnitude of c1: {0:F2}", c1.Magnitude());
   }
}

The output of the above code is −

Complex 1: 3 + 4i
Complex 2: 1 - 2i
Addition: 4 + 2i
Subtraction: 2 + 6i
Multiplication: 11 - 2i
Magnitude of c1: 5.00

Using Built-in System.Numerics.Complex

C# also provides a built-in Complex class in the System.Numerics namespace −

Example

using System;
using System.Numerics;

class Program {
   static void Main() {
      Complex c1 = new Complex(3, 4);
      Complex c2 = new Complex(1, -2);
      
      Console.WriteLine("Complex 1: {0}", c1);
      Console.WriteLine("Complex 2: {0}", c2);
      Console.WriteLine("Addition: {0}", c1 + c2);
      Console.WriteLine("Magnitude: {0:F2}", c1.Magnitude);
      Console.WriteLine("Phase: {0:F2} radians", c1.Phase);
   }
}

The output of the above code is −

Complex 1: (3, 4)
Complex 2: (1, -2)
Addition: (4, 2)
Magnitude: 5.00
Phase: 0.93 radians

Conclusion

Complex numbers in C# can be implemented using custom structures with operator overloading or by using the built-in System.Numerics.Complex class. Both approaches allow you to perform mathematical operations like addition, subtraction, multiplication, and calculate properties like magnitude and phase.