Input
Results
Complex arithmeticRadicand —
Root(s) —
Symbolic —
Formula used
For n < 0: let a = |n|.
- Principal root: √n =
i · √a - Negative root: −√n =
−i · √a - Both roots:
± i · √a - Polar form:
r = √a,θ = ± 90° (±π/2)
For n ≥ 0 (optional): √n = ± √n (real).
We also simplify integers: if a = k²·m, then √(−a) = k·i·√m.
How to use
- Enter the radicand n (negative for imaginary roots).
- Choose the branch: principal, negative, or both.
- Set decimal precision and optional displays.
- Press Calculate to see numeric, symbolic, and polar forms.
- Export your results as CSV or PDF, or share a link.
Example data table
| n | Principal root | Negative root | Symbolic | Polar (r, θ) |
|---|
Identities & Properties (Quick Guide)
- Conjugates: if z = i√a then
\u0305z= −i√a. - Magnitude: |√(−a)| = √a for a ≥ 0.
- Arguments: principal at 90°, negative branch at −90°.
- Squaring: (i√a)² = −a; (−i√a)² = −a.
- Product rule: √(−a) √(−b) = ±√(ab), depends on branches.
Use conjugates to rationalize denominators in complex fractions.
Integer Decomposition Table (a = k²·m)
| a | k | m (square-free) | √(−a) symbolic |
|---|---|---|---|
| 1 | 1 | 1 | i |
| 4 | 2 | 1 | 2i |
| 9 | 3 | 1 | 3i |
| 12 | 2 | 3 | 2i√3 |
| 18 | 3 | 2 | 3i√2 |
| 20 | 2 | 5 | 2i√5 |
| 45 | 3 | 5 | 3i√5 |
| 50 | 5 | 2 | 5i√2 |
| 72 | 6 | 2 | 6i√2 |
Use this to simplify integer radicands before numeric evaluation.
Polar Form Quick Reference
| n | r = |√n| | Principal θ | Negative θ | Rectangular (principal) | Rectangular (negative) |
|---|---|---|---|---|---|
| -1 | 1 | 90° | −90° | 0 + i·1 | 0 − i·1 |
| -3 | √3 | 90° | −90° | 0 + i·√3 | 0 − i·√3 |
| -5 | √5 | 90° | −90° | 0 + i·√5 | 0 − i·√5 |
| -12 | 2√3 | 90° | −90° | 0 + i·2√3 | 0 − i·2√3 |
Angles are measured in degrees; convert to radians as needed.
FAQs
It refers to the square root of a negative number, which is imaginary: √(−a) = ± i·√a. The principal branch is +i·√a, the other branch is −i·√a.
Because squaring either +i·√a or −i·√a gives −a. Quadratic equations generally have two roots, symmetric across the origin in the complex plane.
For n = −a, the magnitude is √a and the arguments are ±90° (±π/2) since the roots lie on the imaginary axis.
Yes. If enabled, the calculator returns real roots ±√n for non-negative n to keep workflows consistent across datasets containing mixed signs.
For integer a, we factor out the largest perfect square: if a = k²·m, then √(−a) = k·i·√m. For non-integers, we present numeric forms.