Definition
Arguments are defined in two equivalent ways:
- Geometrically, in relation to an Argand diagram, arg z is the angle φ from the positive real axis to the vector representing z. The numeric value is given by the angle in radians and is positive if measured counter-clockwise.
- Algebraically, an argument of the complex number z = x + iy is any real quantity such that
-
- for some positive real r. The quantity r is the modulus of z, written
The names amplitude or phase are sometimes used equivalently.
Under both definitions, it can be seen that the argument of any (non-zero) complex number has many possible values: firstly, as a geometrical angle, it is clear that whole circle rotations do not change the point, so angles differing by an integer multiple of 2π radians (a complete circle) are the same. Similarly, from the periodicity of sin and cos, the second definition also has this property.
Read more about this topic: Argument (complex Analysis)
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