How to use
Use this converter to rewrite a complex number for multiplication, rotation or interpreting its argument. Choose Rectangular for inputs a and b, or Polar for inputs r and θ. The same two fields change their labels with the input form. Choose degrees or radians, then calculate. After a successful conversion, changing only the angle unit recalculates; editing a number requires a new confirmation.
Enter each numeric component in its own field. Use a dot or comma as the decimal separator without thousands separators; scientific notation such as 2e-4 is accepted. Do not type i into a numeric field, and do not paste a symbolic expression. This tool evaluates numeric components rather than acting as a symbolic algebra system.
Examples appear in gray and work directly with Calculate. Focusing any numeric field clears all examples together. If you leave the numeric fields without entering anything, the examples return. Once you type or paste in any field, examples will not return, even if you delete that entry. Your own values use the normal dark text. Empty or invalid inputs leave an unfinished diagram with axes and no calculated points. Valid values update the diagram. Calculate confirms the detailed result for copying; editing invalidates that confirmed result. Clear empties the inputs without restoring examples.
Method and interpretation
From a + bi, r = √(a² + b²) and θ = atan2(b,a), which preserves the quadrant. From polar inputs, a = r cos θ and b = r sin θ. Output arguments use (-180°,180°] or (-π,π]. The equivalent exponential notation is r·e^(iθ), always with θ in radians. The diagram draws the radius from the origin and the signed angle from the positive real axis.
The plot uses equal scales on the real and imaginary axes, with the origin marked 0. Its range adjusts to the current result. A point can appear at the origin when its components are tiny relative to another point. The legend supplies numeric coordinates independently of marker placement; use those values when screen resolution cannot show the difference.
Examples
-1 + i becomes r = √2 ≈ 1.414214 and θ = 135° ≈ 2.356194 rad. Its exponential form is approximately 1.414214·e^(i·2.356194). Using atan(b/a) alone would select the wrong quadrant.
r = 2 and θ = -90° becomes 0 - 2i. Converting those rectangular components back gives radius 2 and argument -90°. Angles differing by a full turn describe the same complex number.
Limits and troubleshooting
Components may be zero or have absolute values from 1e-100 through 1e100. Results use floating-point arithmetic, with up to six decimals for this mathematical task and scientific notation for very small or large values. Rounded output is not an exact symbolic answer. Use full internal precision for subsequent work rather than repeatedly copying and rounding intermediate values.
Frequently asked questions
What is the argument of zero?
It is undefined because a zero-length radius has no direction. The result shows 0 with an undefined argument instead of claiming that its argument is zero.
Can I use a negative radius?
No. Use a nonnegative radius; a negative radius can be rewritten as a positive one with a half-turn added to the angle.
Why does the displayed angle change by a full turn?
The output is normalized to the stated principal range. For example, 270° is shown as -90° without changing the number.
Is this the general polar-coordinate tool?
This page focuses on complex-number notation, including i and the exponential form. It does not calculate distances between arbitrary coordinate pairs.