Complex Negation¶

Definition¶

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

Interpretation¶

Negating a complex number changes the sign of both its real and its imaginary part: real negation is applied separately to $a$ and $b$. Then $z+(-z)=(a-a)+\mathrm{i}(b-b)=0$$0$, so $-z$ is the additive inverse of $z$.

The arrows to $z$ and $-z$ have the same length and point in opposite directions: $-z$ is the mirror image of $z$ through the origin, reached by the half-turn marked $+\pi$. This matches complex multiplication: $-z=(-1)z$$(-1)z$, and $-1$ has length $1$ and angle $\pi$, so multiplying by it rotates by $\pi$ without stretching.

Negation should not be confused with the complex conjugate $\bar z=a-\mathrm{i}b$$\mathrm{i}b$, which changes only the sign of the imaginary part and reflects $z$ in the real axis. For $z=1-\mathrm{i}$ one has $-z=-1+\mathrm{i}$ but $\bar z=1+\mathrm{i}$$\mathrm{i}$.

Key Results¶

• $z$ and $-z$ are symmetric about the origin in the complex plane.

Examples¶

• $-(0+2\mathrm{i}) = 0-2\mathrm{i}$$2\mathrm{i}$.
• $-(1-\mathrm{i}) = -1+\mathrm{i}$$\mathrm{i}$.

Prerequisites¶

• Complex Numbers
• Negation

Related Concepts¶

• Complex Addition
• Complex Subtraction
• Complex Conjugate

Sources¶

• Math II: Linear Algebra · Week 1

Aliases: negative of a complex number Symbols: element of, pi, real numbers

Last updated: 20 October 2026 03:02