class Complex
    {
        private double a;
        private double b;
        public static Complex i = new Complex(0, 1);
        public Complex(double a, double b)
        {
            this.a = a;
            this.b = b;
        }
        public static Complex operator +(Complex z1, Complex z2)
        {
            return new Complex(z1.a + z2.a, z1.b + z2.b);
        }
        public static Complex operator -(Complex z1, Complex z2)
        {
            return new Complex(z1.a - z2.a, z1.b - z2.b);
        }
        public static Complex operator *(Complex z1, Complex z2)
        {
            double arg1 = z1.Arg(), arg2 = z2.Arg();
            double r1 = z1.Abs(), r2 = z2.Abs();
            double a = r1 * r2 * Math.Cos(radian(arg1 + arg2));
            double b = r1 * r2 * Math.Sin(radian(arg1 + arg2));
            return new Complex(a, b);
        }
        public static Complex operator /(Complex z1, Complex z2)
        {
            double r1 = z1.Abs(), r2 = z2.Abs();
            double arg1 = z1.Arg(), arg2 = z2.Arg();
            double a = r1 / r2 * Math.Cos(radian(arg1 - arg2));
            double b = r1 / r2 * Math.Sin(radian(arg1 - arg2));
            return new Complex(a, b);
        }
        public static Complex Pow(Complex z, uint n)
        {
            double arg = z.Arg(), r = z.Abs();
            double a = Math.Pow(r, n) * Math.Cos(radian(n * arg));
            double b = Math.Pow(r, n) * Math.Sin(radian(n * arg));
            return new Complex(a, b);
        }
        public static Complex[] Root(Complex z, uint n)
        {
            Complex[] roots = new Complex[n];
            double a, b, r = z.Abs(), arg = z.Arg();
            for (int k = 0; k <= n - 1; k++)
            {
                a = Math.Pow(r, 1.0 / n) * Math.Cos(radian((arg + 360 * k) / n));
                b = Math.Pow(r, 1.0 / n) * Math.Sin(radian((arg + 360 * k) / n));
                roots[k] = new Complex(a, b);
            }
            return roots;
        }
        public static implicit operator Complex(double d)
        {
            return new Complex(d, 0);
        }
        public static Complex Cis(double degrees)
        {
            double angle_r = radian(degrees);
            return new Complex(Math.Cos(angle_r), Math.Sin(angle_r));
        }
        public override string ToString()
        {
            string realPart = ((a != 0) ? a + "" : null);
            string imaginaryPart_i = ((a != 0 && b > 0) ? "+" : null) + b + "i";
            if (b == 1) imaginaryPart_i = ((a != 0) ? "+" : null) + "i";
            else if (b == -1) imaginaryPart_i = "-i";
            else if (b == 0) imaginaryPart_i = "";
            if (a == 0 && b == 0)
                return "0";
            return realPart + imaginaryPart_i;
        }
        public string Polar()
        {
            double arg = Arg();
            return Abs() + "(cos(" + arg + ")+i*sin(" + arg + "))";
        }
        public Complex Conjugate()
        {
            return new Complex(a, -b);
        }
        public double Abs()
        {
            return Math.Sqrt(a * a + b * b);
        }
        public double Arg()
        {
            double arg = Math.Atan(b / a) * 180 / Math.PI;
            if (a < 0 && b >= 0)
                arg += 180;
            else if (a < 0 && b < 0)
                arg -= 180;
            else if (a == 0 && b > 0)
                arg = 90;
            else if (a == 0 && b < 0)
                arg = -90;
            return arg;
        }
        public static double Re(Complex z)
        {
            return z.a;
        }
        public static double Im(Complex z)
        {
            return z.b;
        }
        private static double radian(double degrees)
        {
            return degrees * Math.PI / 180;
        }
    }