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