IM

Im is the standard notation for the imaginary part of a complex number in mathematics. If z is written in rectangular form as z = a + ib, where a and b are real and i is the imaginary unit with i^2 = -1, then Im(z) = b and the real part is Re(z) = a. Equivalently, z can be expressed as z = Re(z) + i Im(z). The imaginary part defines a real-valued function on the complex plane, corresponding to the vertical coordinate.

Im is linear over the real numbers: Im(z1 + z2) = Im(z1) + Im(z2), and Im(α z) = α Im(z) for any real α. It is not, in general, linear with respect to complex scalar multiplication. The relations z = Re(z) + i Im(z) and z̄ = Re(z) − i Im(z) show how the real and imaginary parts reconstruct z and its complex conjugate. In polar form z = r e^{iθ}, the imaginary part is given by Im(z) = r sin θ.

Common alternatives and conventions include using x + iy for z = x + i y, with Im(z) = y, or using lowercase im or the Imag function in some software libraries. The imaginary part plays a central role in complex analysis, signal processing, and many applied fields, where the decomposition z = Re(z) + i Im(z) separates magnitude and phase information and interacts with tools such as the Cauchy–Riemann equations and analytic function theory.