Step by Step solution about How to resolve the algorithm Euler's identity step by step in the Java programming language

The provided Java code calculates and prints the result of Euler's identity, which states that e^(iπ) + 1 = 0. Here's how the code works:

1. Complex Class:
   • The code defines a Complex class to represent complex numbers with real and imaginary parts(x and y).
   • It has constructors to initialize complex numbers and methods to perform operations like exponentiation (exp()) and addition (add()).
2. Main Method:
   • In the main method, two Complex objects are created:
     • new Complex(0, Math.PI) represents the imaginary number iπ.
     • new Complex(1, 0) represents the real number 1.
3. Exponentiation (exp()):
   • The exp() method of the Complex class calculates the exponential of a complex number using the formula e^(x + yi) = e^x * (cos(y) + i*sin(y)).
   • It takes the exponential of the real part (x) and multiplies it by the complex exponential of the imaginary part (y).
4. Addition (add()):
   • The add() method adds two Complex objects by combining their real and imaginary parts.
5. Euler's Identity Calculation:
   • The equation e^(i*Pi) + 1 is calculated by first exponentiating new Complex(0, Math.PI) (which represents iπ) using exp(), and then adding it to new Complex(1, 0) (which represents 1) using add().
6. Output:
   • The result of the calculation, which should be close to zero due to floating-point precision, is printed to the console.

In summary, this code demonstrates the calculation of Euler's identity using complex numbers and illustrates the basic operations of complex number manipulation.

public class EulerIdentity {

    public static void main(String[] args) {
        System.out.println("e ^ (i*Pi) + 1 = " + (new Complex(0, Math.PI).exp()).add(new Complex(1, 0)));
    }

    public static class Complex {

        private double x, y;
        
        public Complex(double re, double im) {
            x = re;
            y = im;
        }
        
        public Complex exp() {
            double exp = Math.exp(x);
            return new Complex(exp * Math.cos(y), exp * Math.sin(y));
        }
        
        public Complex add(Complex a) {
            return new Complex(x + a.x, y + a.y);
        }
        
        @Override
        public String toString() {
            return x + " + " + y + "i";
        }
    }
}