Real Part
The 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, the horizontal coordinate if you think of the complex number as the point $(a, b)$ on a graph. Formally, $\text{Re}(z) = a$, equivalently $\text{Re}(z) = (z + \bar{z})/2$ where $\bar{z} = a - bi$ is the complex conjugate, and it is the projection of $z$ onto the real axis. The map $\text{Re}: \mathbb{C} \to \mathbb{R}$ is $\mathbb{R}$-linear but not $\mathbb{C}$-linear; for analytic functions $f = u + iv$, $\text{Re}(f) = u$ is harmonic ($u_{xx} + u_{yy} = 0$), and 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 $5 + 3i$, the real part is $5$; for $-2 + 7i$, it is $-2$; and for $4i$ (that is, $0 + 4i$), it is $0$. $\text{Re}(3 - 5i) = 3$, and $\text{Re}((2+i)^2) = \text{Re}(3 + 4i) = 3$, though note $\text{Re}(zw) \neq \text{Re}(z)\text{Re}(w)$ in general. For $f(z) = z^2 = x^2 - y^2 + 2xyi$, $\text{Re}(f) = x^2 - y^2$ is harmonic, with conjugate harmonic function $2xy = \text{Im}(f)$, together satisfying the Cauchy-Riemann equations.
Key Insight
Think of a complex number as a coordinate pair $(a, b)$: the real part $a$ is the horizontal coordinate and the imaginary part $b$ is the vertical, so in the complex plane, horizontal lines are sets of constant real part and the real axis (where $\text{Im}(z) = 0$) contains all real numbers as a subset. Harmonic functions, real parts of analytic functions, satisfy the mean value property and the maximum principle, which makes them central to potential theory, electrostatics, and steady-state heat flow.