Complex Conjugate¶

Definition¶

Given a complex number $z=a+\mathrm{i}b$ where $a,b\in\mathbb{R}$$\mathbb{R}$, the conjugate of the complex number $z$ is defined as $\overline{z}=a-\mathrm{i}b.$

Interpretation¶

Conjugation keeps the real part of $z$ and changes the sign of its imaginary part. Geometrically, it reflects $z$ in the real axis.

The point $\overline{z}=a-b\mathrm{i}$ lies straight below $z=a+b\mathrm{i}$$b\mathrm{i}$, at the same distance from the real axis. A reflection keeps lengths, so $|\overline{z}|=|z|$ for the modulus, and reverses angles, so the argument $\varphi$ becomes $-\varphi$.

Conjugation is useful because it produces real numbers: $z+\overline{z}=2a$ and $z\overline{z}=a^2+b^2=|z|^2$$|z{|}^2$. The second identity is behind division: multiplying numerator and denominator by $\overline{w}$ makes the denominator the real number $|w|^2$. Also $z$ is real if and only if $\overline{z}=z$$z$, and conjugating twice gives $z$ back. Because conjugation respects sums and products (Key Results), it sends a root of a polynomial with real coefficients to another root: the complex conjugate root theorem.

Key Results¶

• We have $\overline{z + w} = \overline{z} + \overline{w},$ $\overline{-z} = -\overline{z},$ and $\overline{zw} = \overline{z} \cdot \overline{w}.$
• $z$ and $\overline z$ are symmetric about the real axis.

Examples¶

• $\overline{1+\mathrm{i}}=1-\mathrm{i}$$\mathrm{i}$.
• $\overline{-2-3\mathrm{i}}=-2+3\mathrm{i}$$3\mathrm{i}$.

Prerequisites¶

• Complex Numbers

Related Concepts¶

• Complex Absolute Value
• Complex Division
• Fundamental Theorem of Algebra
• Negation

Sources¶

• Math II: Linear Algebra · Week 1

Aliases: complex conjugation, conjugate of a complex number Symbols: element of, phi, real numbers

Last updated: 20 October 2026 03:02