Complex Square Root Calculator

Input

Entry mode

Precision (decimals)

Use Tab then Enter to compute

Real part (a)

Imag part (b)

Or as text

Accepted formats: a+bi, a-bi

Results

Principal root

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Second root

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Input (rect)

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Input (polar)

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Step-by-step

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Example data

#	Form	Value
1	Rect	3 + 4i
2	Rect	-5 + 12i
3	Rect	-7 - 24i
4	Rect	-9 + 0i
5	Rect	0 - 1i
6	Polar	10 ∠ 210°

Formula used

Let z = a + b i, modulus r = √(a² + b²), argument θ = atan2(b,a) (−π < θ ≤ π).

Principal square root in rectangular form:

√z = u + v i
u = √((r + a)/2)
v = s · √((r − a)/2),   s = 1 if b ≥ 0, else −1
Special case: if b = 0 and a < 0, take u = 0, v = √(−a)

In polar form: if z = r ∠ θ, then

√z (principal) = √r ∠ (θ/2)
Second root     = −√r ∠ (θ/2)

How to use this calculator

FAQs

1) Why are there two square roots?

If w is a square root of z, then −w is also a square root, because (−w)² = w² = z.

2) What is the principal root convention?

We take θ ∈ (−π, π]; the principal root has argument θ/2 ∈ (−π/2, π/2], giving non-negative imaginary part when b ≥ 0.

3) How are negative reals handled?

For z = a < 0, principal root is purely imaginary: √z = i √|a|; the other root is −i √|a|.

4) Can I enter angles in radians?

This version expects degrees for polar input. Convert radians via degrees = radians × 180/π.

5) Why do my results differ slightly elsewhere?

Differences usually come from rounding, argument range conventions, or alternate branch cut choices.

Special Cases & Quick Rules

z = 0 ⇒ √z = 0 (double root).
Purely real a > 0 ⇒ roots are ±√a (real).
Purely real a < 0 ⇒ roots are ± i √|a| (imaginary).
Purely imaginary bi ⇒ magnitude √|b|, arguments ±45° if b > 0, or ±135° if b < 0.
Conjugate symmetry: √(\\overline{z}) = \\overline{√z} for principal branch.
Scaling: √(c z) = √c √z for c ≥ 0.

Worked Examples (auto‑computed)

#	z (a+bi)	Principal root	Polar (r∠θ°)

Verification (w² ≈ z)

#	Principal root w	w² (computed)	Original z

Note: Small differences are due to rounding to the chosen precision.