AcademyComplex Arithmetic
Academy
Complex Addition
Level 1 - Math I (Physics) topic page in Complex Arithmetic.
Adding Complex Numbers
Complex addition is performed component-wise, adding real parts and imaginary parts separately.
Component-Wise Addition
For two complex numbers \(z_1 = a + bi\) and \(z_2 = c + di\):
Addition
\[z_1 + z_2 = (a + c) + (b + d)i\]
Geometric Interpretation
Adding complex numbers corresponds to vector addition in the complex plane:
- Each complex number is a vector from the origin
- Adding \(z_1 + z_2\) means placing the vectors tip-to-tail
- The resulting vector goes from the origin to the sum
VectorSum
\[\vec{z_1} + \vec{z_2} = \overrightarrow{\text{tip of } z_1 \text{ to tip of combined}}\]
Properties
Complex addition is:
- Commutative: \(z_1 + z_2 = z_2 + z_1\)
- Associative: \((z_1 + z_2) + z_3 = z_1 + (z_2 + z_3)\)
- Has identity \(0\): \(z + 0 = z\)
Example
Example
\[(3 + 2i) + (1 + 4i) = 4 + 6i\]
The result has real part 4 and imaginary part 6.