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)
Famous quotes containing the word definition:
“... we all know the wag’s definition of a philanthropist: a man whose charity increases directly as the square of the distance.”
—George Eliot [Mary Ann (or Marian)
“Scientific method is the way to truth, but it affords, even in
principle, no unique definition of truth. Any so-called pragmatic
definition of truth is doomed to failure equally.”
—Willard Van Orman Quine (b. 1908)
“... if, as women, we accept a philosophy of history that asserts that women are by definition assimilated into the male universal, that we can understand our past through a male lens—if we are unaware that women even have a history—we live our lives similarly unanchored, drifting in response to a veering wind of myth and bias.”
—Adrienne Rich (b. 1929)