Complex Number Calculator

A complex number is expressed in the standard Cartesian form z = a + bi, where 'a' is the real part, 'b' is the imaginary part, and 'i' represents the imaginary unit satisfying i² = -1. For any given complex number, this calculator determines its geometric properties and executes multi-number operations.

The magnitude (or absolute value) of a complex number z = a + bi is computed using the Pythagorean theorem: |z| = sqrt(a² + b²). The phase angle (argument) is determined via the two-argument arctangent function: θ = atan2(b, a). When combining two complex numbers z₁ = a₁ + b₁i and z₂ = a₂ + b₂i, addition is performed component-wise: (a₁ + a₂) + (b₁ + b₂)i. Multiplication distributes terms using the property i² = -1, yielding (a₁a₂ - b₁b₂) + (a₁b₂ + a₂b₁)i.

Let us calculate the sum and magnitude of two sample complex numbers: Z₁ = 3 + 4i and Z₂ = 1 - 2i.

Step 1: Calculate the magnitude of Z₁
Using the magnitude formula, |Z₁| = sqrt(3² + 4²) = sqrt(9 + 16) = sqrt(25) = 5.

Step 2: Calculate the phase angle of Z₁
Using the phase formula, θ₁ = atan2(4, 3) ≈ 0.927 radians (or approximately 53.13 degrees).

Step 3: Calculate the sum of Z₁ and Z₂
Combine the real parts and imaginary parts separately: Real sum = 3 + 1 = 4. Imaginary sum = 4 + (-2) = 2. Thus, the resulting complex sum is 4 + 2i.

Always double-check the signs of your real and imaginary components before entering them into the fields, as negative sign placement errors are the leading cause of incorrect phase angles. When working with trigonometric functions in engineering applications, ensure whether your final output requires degrees or radians. Utilize the adjustable significant figure settings to match the precision requirements of your specific academic or professional project.