Complex Subtraction¶

Definition¶

Given two complex numbers $z$ and $w$, the difference of complex numbers $z$ and $w$ is defined as $z-w=z+(-w).$

Interpretation¶

Subtracting $w$ means adding its negative: first take the complex negation $-w$, then use complex addition to add it to $z$. Both steps work part by part, so the real parts and the imaginary parts are subtracted separately.

For $z=a+b\mathrm{i}$ and $w=c+d\mathrm{i}$ this gives $z-w=(a-c)+(b-d)\mathrm{i}$$d)\mathrm{i}$. In the figure it is the green arrow from the tip of $w$ to the tip of $z$, since $w+(z-w)=z$$z$; the dashed arrow is the same arrow started at $0$, so its tip is the point $z-w$$w$. Its length $|z-w|$ is the distance between the points $z$ and $w$ (see modulus).

Subtraction is not commutative: $w-z=-(z-w)$ is the arrow from $z$ back to $w$, with the same length and the opposite direction. For example, $(1+2\mathrm{i})-(2-\mathrm{i})=-1+3\mathrm{i}$$3\mathrm{i}$, while $(2-\mathrm{i})-(1+2\mathrm{i})=1-3\mathrm{i}$$3\mathrm{i}$. The same arrow picture returns later for the subtraction of vectors.

Key Results¶

• We have $z-z=0_{\mathbb{C}}$ for any $z\in\mathbb{C}$$\mathbb{C}$.

Examples¶

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

Prerequisites¶

• Complex Negation
• Complex Addition
• Complex Number

Related Concepts¶

• Cartesian Representation of Complex Numbers

Sources¶

• Math II: Linear Algebra · Week 1

Aliases: difference of complex numbers, subtraction of complex numbers Symbols: complex numbers, element of

Last updated: 20 October 2026 03:02