Complex numbers in C++ | Set 1

The complex library implements the complex class to contain complex numbers in cartesian form and several functions and overloads to operate with them.

• real() – It returns the real part of the complex number.
• imag() – It returns the imaginary part of the complex number.

  // Program illustrating the use of real() and  // imag() function #include <iostream>         // for std::complex, std::real, std::imag #include <complex>       using namespace std;    // driver function int main() {       // defines the complex number: (10 + 2i)   std::complex<double> mycomplex(10.0, 2.0);      // prints the real part using the real function   cout << "Real part: " << real(mycomplex) << endl;   cout << "Imaginary part: " << imag(mycomplex) << endl;   return 0; }

  Output:

  Real part: 10
  Imaginary part: 2
  

• abs() – It returns the absolute of the complex number.
• arg() – It returns the argument of the complex number.

  // Program illustrating the use of arg() and abs() #include <iostream>         // for std::complex, std::abs, std::atg #include <complex>  using namespace std;    // driver function int main () {       // defines the complex number: (3.0+4.0i)   std::complex<double> mycomplex (3.0, 4.0);      // prints the absolute value of the complex number   cout << "The absolute value of " << mycomplex << " is: ";   cout << abs(mycomplex) << endl;        // prints the argument of the complex number   cout << "The argument of " << mycomplex << " is: ";   cout << arg(mycomplex) << endl;      return 0; }

  Output:

  The absolute value of (3,4) is: 5
  The argument of (3,4) is: 0.927295
  

• polar() – It constructs a complex number from magnitude and phase angle.

  real = magnitude*cosine(phase angle)
  imaginary = magnitude*sine(phase angle)

  // Program illustrating the use of polar() #include <iostream>         // std::complex, std::polar #include <complex> using namespace std;    // driver function int main () {   cout << "The complex whose magnitude is " << 2.0;   cout << " and phase angle is " << 0.5;        // use of polar()   cout << " is " << polar (2.0, 0.5) << endl;      return 0; }

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  Output:

  
  
  

  The complex whose magnitude is 2 and phase angle is 0.5 is (1.75517,0.958851)
  

• norm() – It is used to find the norm(absolute value) of the complex number. If z = x + iy is a complex number with real part x and imaginary part y, the complex conjugate of z is defined as z'(z bar) = x – iy, and the absolute value, also called the norm, of z is defined as :
  

  // example to illustrate the use of norm() #include <iostream>         // for std::complex, std::norm #include <complex>  using namespace std;    // driver function int main () {       // initializing the complex: (3.0+4.0i)   std::complex<double> mycomplex (3.0, 4.0);      // use of norm()   cout << "The norm of " << mycomplex << " is "         << norm(mycomplex) <<endl;      return 0; }

  Output:

  The norm of (3,4) is 25.
  

• conj() – It returns the conjugate of the complex number x. The conjugate of a complex number (real,imag) is (real,-imag).

  // Illustrating the use of conj() #include <iostream>  using namespace std;    // std::complex, std::conj #include <complex>          // driver program int main () {   std::complex<double> mycomplex (10.0,2.0);      cout << "The conjugate of " << mycomplex << " is: ";        // use of conj()   cout << conj(mycomplex) << endl;   return 0; }

  Output:

   The conjugate of (10,2) is (10,-2)
  

• proj() – It returns the projection of z(complex number) onto the Riemann sphere. The projection of z is z, except for complex infinities, which are mapped to the complex value with a real component of INFINITY and an imaginary component of 0.0 or -0.0 (where supported), depending on the sign of the imaginary component of z.

  // Illustrating the use of proj()    #include <iostream> using namespace std;    // For std::complex, std::proj #include <complex>     // driver program int main() {     std::complex<double> c1(1, 2);     cout << "proj" << c1 << " = " << proj(c1) << endl;         std::complex<double> c2(INFINITY, -1);     cout << "proj" << c2 << " = " << proj(c2) << endl;         std::complex<double> c3(0, -INFINITY);     cout << "proj" << c3 << " = " << proj(c3) << endl; }

  Output:

  proj(1,2) = (1,2)
  proj(inf,-1) = (inf,-0)
  proj(0,-inf) = (inf,-0)
  

• sqrt() – Returns the square root of x using the principal branch, whose cuts are along the negative real axis.

  // Illustrating the use of sqrt() #include <iostream> using namespace std;    // For std::ccomplex, stdc::sqrt #include <complex>     // driver program int main() {         // use of sqrt()     cout << "Square root of -4 is "          << sqrt(std::complex<double>(-4, 0)) << endl          << "Square root of (-4,-0), the other side of the cut, is "          << sqrt(std::complex<double>(-4, -0.0)) << endl; }

  Output:

  Square root of -4 is (0,2)
  Square root of (-4,-0), the other side of the cut, is (0,-2)
  

Next article: Complex numbers in C++ | Set 2

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