The complex components include six basic characteristics describing complex numbersabsolute value (modulus) , argument (phase) , real part , imaginary part , complex conjugate , and sign function (signum) . It is impossible to define real and imaginary parts of the complex number through other functions or complex characteristics. They are too basic, so their symbols can be described by simple sentences, for example, " gives the real part of the number ," and " gives the imaginary part of the number ."
All other complex components are defined by the following formulas:
Geometrically, the absolute value (or modulus) of a complex number is the Euclidean distance from to the origin, which can also be described by the formula:
Geometrically, the argument of a complex number is the phase angle (in radians) that the line from 0 to makes with the positive real axis. So, the complex number can be presented by the formulas:
Geometrically, the real part of a complex number is the projection of the complex point on the real axis. So, the real part of the complex number can be presented by the formulas:
Geometrically, the imaginary part of a complex number is the projection of complex point on the imaginary axis. So, the imaginary part of the complex number can be presented by the formulas:
Geometrically, the complex conjugate of a complex number is the complex point , which is symmetrical to with respect to the real axis. So, the conjugate value of the complex number can be presented by the formulas:
Geometrically, the sign function (signum) is the complex point that lays on the intersection of the unit circle and the line from 0 to (if ). So, the conjugate value of the complex number can be presented by the formulas:
