Try an Example
Pick a scenario to see how the calculator works, then adjust the values
Signal Addition
Add two phasor signals in complex form to find the resultant.
Key values: z₁ = 3 + 4i · z₂ = 1 − 2i · Addition
Circuit Impedance
Divide two impedances to compute the transfer ratio.
Key values: z₁ = 3 + 4i · z₂ = 1 + 2i · Division
De Moivre Power
Raise a complex number to the 10th power using De Moivre's theorem.
Key values: z₁ = 1 + i · n = 10 · Power
Cube Roots of −8
Find all three cube roots of −8, visualized as equidistant points on a circle.
Key values: z₁ = −8 + 0i · n = 3 · nth Root
This calculator is also known as Complex Number Argument Calculator.
Read the complete guidePrincipal Argument
The principal argument Arg(z) is the unique angle in (−π, π] such that z = |z|·(cosθ + i·sinθ). The function atan2(b, a) is used instead of arctan(b/a) because it correctly distinguishes all four quadrants and handles the cases where a = 0.
Examples
Argument of −1 + i
Find the principal argument of z = −1 + i.
arg(−1 + i) = atan2(1, −1) = 3π/4 radians = 135°. The point is in the second quadrant.
Key takeaway: Always use atan2 to get the correct quadrant. Plain arctan(b/a) would give −π/4, which is in the wrong quadrant.
Computing Arguments Correctly
Avoid common pitfalls with the argument:
- Always use atan2(b, a), not arctan(b/a), to handle all quadrants
- The principal argument is in (−π, π] or equivalently (−180°, 180°]
- arg(z₁·z₂) = arg(z₁) + arg(z₂) (up to a multiple of 2π)
Frequently Asked Questions about Complex Number Argument Calculator
Why is the argument of zero undefined?
The point z = 0 is at the origin, so there is no unique direction from the origin to itself. Every angle is equally valid, making arg(0) undefined.
What is the difference between arg(z) and Arg(z)?
arg(z) is multi-valued (differs by multiples of 2π). Arg(z) is the principal value, restricted to (−π, π]. This calculator returns the principal value.
Specialized Calculators
Choose from 11 specialized versions of this calculator, each optimized for specific use cases and calculation methods.
Operation
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