AcademyComplex Arithmetic
Academy
Real Part
Level 1 - Math I (Physics) topic page in Complex Arithmetic.
Real Part Function
The real part function extracts the real component of a complex number.
Definition
For \(z = a + bi\):
Definition
\[\text{Re}(z) = a\]
Properties
Addition
\[\text{Re}(z_1 + z_2) = \text{Re}(z_1) + \text{Re}(z_2)\]
Scalar
\[\text{Re}(cz) = c\text{Re}(z) \quad \text{for } c \in \mathbb{R}\]
NotProduct
\[\text{Re}(z_1 z_2) \neq \text{Re}(z_1)\text{Re}(z_2)\]
Expressing Re(z) with Conjugate
ConjugateExpr
\[\text{Re}(z) = \frac{z + z^*}{2}\]
In Polar Form
If \(z = re^{i\theta}\):
PolarRe
\[\text{Re}(z) = r\cos\theta\]
The real part is the projection of the vector onto the real axis.
Example
Example
\[\text{Re}(3 + 4i) = 3\]
PolarExample
\[\text{Re}(5e^{i\pi/3}) = 5\cos\frac{\pi}{3} = 5 \cdot \frac{1}{2} = 2.5\]