Arithmetic/Complex

Arithmetic/Complex
You are encouraged to solve this task according to the task description, using any language you may know.

A complex number is a number which can be written as "a + b*i" (sometimes shown as "b + a*i") where a and b are real numbers and i is the square root of -1. Typically, complex numbers are represented as a pair of real numbers called the "imaginary part" and "real part", where the imaginary part is the number to be multiplied by i.

Show addition, multiplication, negation, and inversion of complex numbers in separate functions (subtraction and division operations can be made with pairs of these operations).

Some languages have complex number libraries available. If your language does, show the operations. If your language does not, also show the definition of this type (print functions are not necessary).

Ada

<ada>with Ada.Numerics.Generic_Complex_Types;

procedure Complex_Operations is

  -- Ada provides a pre-defined generic package for complex types
  -- That package contains definitions for composition,
  -- negation, addition, subtraction, multiplication, division,
  -- conjugation, exponentiation, and absolute value, as well as
  -- basic comparison operations.
  -- Ada provides a second pre-defined package for sin, cos, tan, cot,
  -- arcsin, arccos, arctan, arccot, and the hyperbolic versions of 
  -- those trigonometric functions.
  
  -- The package Ada.Numerics.Generic_Complex_Types requires definition
  -- with the real type to be used in the complex type definition.
  
  package Complex_Type is new Ada.Numerics.Generic_Complex_Types(Long_Float);
  use Complex_Type;
  
  A : Complex := Compose_From_Cartesian(Re => 1.0, Im => 1.0);
  B : Complex := Compose_From_Cartesian(Re => 3.14159, Im => 1.25);
  C : Complex;
 

begin

  -- Addition
  C := A + B;
  -- Multiplication
  C := A * B;
  -- Inversion
  C := 1.0 / A;

end Complex_Operations;</ada>

Java

<java>public class Complex{

  public final double real;
  public final double imag;

  public Complex(){this(0,0)}//default values to 0...force of habit
  public Complex(double r, double i){real = r; imag = i;}

  public Complex add(Complex b){
     return new Complex(this.real + b.real, this.imag + b.imag);
  }

  public Complex mult(Complex b){
     //FOIL of (a+bi)(c+di) with i*i = -1
     return new Complex(this.real * b.real - this.imag * b.imag, this.real * b.imag + this.imag * b.real);
  }

  public Complex invert(){
     //1/(a+bi) * (a-bi)/(a-bi) = 1/(a+bi) but it's more workable
     double denom = real * real - imag * imag;
     return new Complex(real/denom, imag/denom);
  }

  public Complex neg(){
     return new Complex(-real, -imag);
  }

}</java>