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Complex Addition

Definition

Given two complex numbers z=a+ibz=a+\mathrm{i}b and w=c+idw=c+\mathrm{i}d where a,b,c,d∈Ra,b,c,d\in\mathbb{R}, the sum of complex numbers zz and ww is defined as z+w=(a+c)+i(b+d).z+w=(a+c)+\mathrm{i}(b+d).

Interpretation

Complex numbers are added part by part: real parts with real parts and imaginary parts with imaginary parts. Nothing beyond the addition of real numbers is needed.

Arrows for z and w from the origin and their sum z + w as the diagonal of the parallelogram they span.

Draw zz and ww as arrows from 00. A copy of the arrow ww placed at the tip of zz ends at z+wz+w, since real parts and imaginary parts add separately. The two arrows and their copies form the dashed parallelogram, and z+wz+w is its diagonal from 00.

Because the addition works coordinate by coordinate, it is commutative and associative, 00 is the identity and the negation −z-z is the inverse of zz, so (C,+)(\mathbb{C},+) is an Abelian group. Adding a fixed ww shifts every point of the plane by the same arrow; multiplication instead rotates and stretches. The parallelogram rule returns later for the addition of vectors.

Key Results

  • The additive identity for complex numbers is 0C=0+0i,0_\mathbb{C} = 0 + 0\mathrm{i}, which is the real number 0.0.
  • Complex addition has commutativity: z1+z2=z2+z1.z_1 + z_2 = z_2 + z_1.
  • Complex addition has associativity: (z1+z2)+z3=z1+(z2+z3)(z_1 + z_2) + z_3 = z_1 + (z_2 + z_3).

Examples

  • (2+i)+(3+4i)=5+5i(2+\mathrm{i}) + (3+4\mathrm{i}) = 5+5\mathrm{i}.
  • (1−i)+(−1+i)=0C=0(1-\mathrm{i})+(-1+\mathrm{i}) = 0_{\mathbb{C}}=0.

Prerequisites

Sources

Aliases: addition of complex numbers, sum of complex numbers Symbols: complex numbers, element of, real numbers

Last updated: 20 October 2026 03:02