AcademyComplex Arithmetic
Academy
Imaginary Part
Level 1 - Math I (Physics) topic page in Complex Arithmetic.
Imaginary Part Function
The imaginary part function extracts the imaginary coefficient of a complex number.
Definition
For \(z = a + bi\):
Definition
\[\text{Im}(z) = b\]
Note: \(\text{Im}(z)\) is a real number, not \(bi\).
Properties
Addition
\[\text{Im}(z_1 + z_2) = \text{Im}(z_1) + \text{Im}(z_2)\]
Scalar
\[\text{Im}(cz) = c\text{Im}(z) \quad \text{for } c \in \mathbb{R}\]
NotProduct
\[\text{Im}(z_1 z_2) \neq \text{Im}(z_1)\text{Im}(z_2)\]
Expressing Im(z) with Conjugate
ConjugateExpr
\[\text{Im}(z) = \frac{z - z^*}{2i}\]
In Polar Form
If \(z = re^{i\theta}\):
PolarIm
\[\text{Im}(z) = r\sin\theta\]
The imaginary part is the projection of the vector onto the imaginary axis.
Example
Example
\[\text{Im}(3 + 4i) = 4\]
PolarExample
\[\text{Im}(5e^{i\pi/3}) = 5\sin\frac{\pi}{3} = 5 \cdot \frac{\sqrt{3}}{2} = \frac{5\sqrt{3}}{2}\]