Imaginary Part
The imaginary part of a complex number a + bi is the real number b (the coefficient of i), written Im(z) = b.
Formula
Im(a + bi) = b
Definition
The imaginary part of a complex number $a + bi$ is the real number $b$, the coefficient in front of $i$, not the term $bi$ itself, a subtlety that surprises many students: the "imaginary part" is a label for $b$, which is itself a perfectly ordinary real number. Formally, $\text{Im}(z) = b$, equivalently $\text{Im}(z) = (z - \bar{z})/(2i)$, the projection onto the imaginary axis, and a number is purely imaginary if $\text{Re}(z) = 0$ and $\text{Im}(z) \neq 0$. $\text{Im}: \mathbb{C} \to \mathbb{R}$ satisfies $\text{Im}(z + w) = \text{Im}(z) + \text{Im}(w)$ and $\text{Im}(rz) = r\,\text{Im}(z)$ for real $r$, but $\text{Im}(iz) = \text{Re}(z)$ rather than $i\,\text{Im}(z)$; for analytic $f = u + iv$, $\text{Im}(f) = v$ is harmonic and conjugate to $u = \text{Re}(f)$, and the Hilbert transform relates them on the real line, $v = H[u]$.
Example
For $5 + 3i$, the imaginary part is $3$ (not $3i$); for $-2 + 7i$, it is $7$; and for $4$ (that is, $4 + 0i$), it is $0$. $\text{Im}(6 - 2i) = -2$, and $\text{Im}((1+i)^3) = \text{Im}(-2 + 2i) = 2$. In quantum mechanics, the wave function $\psi$ is complex-valued, and $|\psi|^2 = \text{Re}(\psi)^2 + \text{Im}(\psi)^2$ gives the probability density, with the phase (the ratio $\text{Im}/\text{Re}$) carrying physical information about interference; Euler's formula, $\text{Im}(e^{i\theta}) = \sin(\theta)$, shows the imaginary part is the oscillatory component of a complex exponential.
Key Insight
In the complex plane, $\text{Im}(z)$ is the vertical ($y$-) coordinate, the imaginary axis is where $\text{Re}(z) = 0$, and complex conjugation reflects a point across the real axis, changing the sign of $\text{Im}(z)$. The imaginary part carries phase information in wave phenomena, making complex exponentials the natural language for wave equations, both real and imaginary parts are needed, and their ratio encodes physically meaningful interference.