argY

argY, or the argument of a complex number Y, is the angle between the positive real axis and the line from the origin to the point representing Y in the complex plane. It is denoted arg(Y) and is defined for all nonzero Y. If Y is written in polar form as Y = r e^{iθ}, with r = |Y|, then θ is the argument of Y. In many contexts the principal value Arg(Y) is used, restricted to a specific interval such as (−π, π] or [0, 2π). The full set of possible arguments is { θ + 2πk : k ∈ Z }.

The argument is undefined for Y = 0, since the direction of the vector is not determined. The concept is central to expressing complex numbers in polar form, where Y = |Y| e^{i arg(Y)} and |Y| is the magnitude or modulus of Y. The principal value Arg(Y) provides a single representative angle for most practical purposes.

Examples illustrate the concept: Y = 3 + 4i has arg ≈ 0.9273 radians (53.13 degrees). Y = -2 has arg π (or −π, depending on convention). Y = 1 has arg 0. The argument describes the phase of Y and is directly related to polar coordinates and to the phase component in signal processing and phasor analysis.

History notes indicate that the idea originated with the Argand plane, named after Jean-Robert Argand, who helped formalize the geometric interpretation of complex numbers. See also magnitude, phase, polar form, principal value, and complex plane.