Real Part
Functions & Advanced AlgebraThe real part of a complex number a + bi is the real number a, written Re(z) = a.
Formula
Re(a + bi) = a
Definition
The real part of a complex number $a + bi$ is simply $a$, the non-imaginary portion. It is the number without the $i$ attached.
Example
For $5 + 3i$, the real part is $5$. For $-2 + 7i$, the real part is $-2$. For $4i$ (= $0 + 4i$), the real part is $0$.
Key Insight
Think of a complex number as a coordinate pair $(a, b)$ on a graph. The real part $a$ is the horizontal coordinate, and the imaginary part $b$ is the vertical coordinate.
Definition
For a complex number $z = a + bi$, the real part is $\text{Re}(z) = a$. Equivalently, $\text{Re}(z) = (z + \bar{z})/2$ where $\bar{z} = a - bi$ is the complex conjugate. The real part is the projection of $z$ onto the real axis.
Example
$\text{Re}(3 - 5i) = 3$. $\text{Re}((2+i)^2) = \text{Re}(4 + 4i + i^2) = \text{Re}(3 + 4i) = 3$. Note: $\text{Re}(zw) \neq \text{Re}(z)\text{Re}(w)$ in general.
Key Insight
In the complex plane, $\text{Re}(z)$ is the $x$-coordinate. Horizontal lines are sets of constant real part. The real axis is where $\text{Im}(z) = 0$, containing all real numbers as a subset of the complex plane.
Definition
The map $\text{Re}: \mathbb{C} \to \mathbb{R}$ is $\mathbb{R}$-linear but not $\mathbb{C}$-linear. For analytic functions $f = u + iv$ ($u, v$ real-valued), $\text{Re}(f) = u$ is harmonic: $u_{xx} + u_{yy} = 0$ (Laplace's equation). By the Cauchy-Riemann equations, $u$ and $v$ are conjugate harmonic functions, so $\text{Re}(f)$ determines $\text{Im}(f)$ up to a constant.
Example
For $f(z) = z^2 = (x+iy)^2 = x^2 - y^2 + 2xyi$, $\text{Re}(f) = x^2 - y^2$ is harmonic. Its conjugate harmonic function is $2xy = \text{Im}(f)$. Together they satisfy the Cauchy-Riemann equations.
Key Insight
Harmonic functions (real parts of analytic functions) satisfy the mean value property and the maximum principle. This makes them central to potential theory, electrostatics, and steady-state heat flow.