How to work an example 01 Type A and B as complex expressions — 3 + 4i, 1 - 2i, 2*e^(i*pi/3), (1+i)/(2-i) all parse. 02 Press an operator to see the result in rectangular form, followed by |z|, arg in radians and degrees, the conjugate and the polar form. 03 For a root question , set n and press All n-th roots of A: the list starts with the principal root and ascends by argument. 04 Use Rect ↔ Polar to convert both ways; the DEG/RAD toggle switches the angle unit and the converted value follows. 05 Read the result block and, when you need it elsewhere, write |z| or arg(A) to the workspace as a scalar. Worked readouts With A = 3 + 4i and B = 1 − 2i , as displayed:
Operation
Readout
A + B
4 + 2i
A − B
2 + 6i
A × B
11 − 2i
A ÷ B
−1 + 2i
A^B
−21.08313969 − 24.00021071i
abs(A)
|A| = 5 ; arg(A) = 0.927295218 rad = 53.13010235°
conj(A)
3 − 4i
inv(A)
1/A = 0.12 − 0.16i
sqrt(A)
√A = ±(2 + i ); z₀ = 2 + i = 2.23607·e^(0.463648i) , z₁ = −2 − i
exp(A)
−13.12878308 − 15.20078446i ; |z| = 20.08553692 ; arg = −2.283185307 rad = −130.8168819°
ln(A)
1.609437912 + 0.927295218i
z³ = 8, n = 3
z₀ = 2 , z₁ = −1 + 1.73205i = 2·e^(2.0944i), z₂ = −1 − 1.73205i = 2·e^(−2.0944i)
Rect → Polar
3 + 4i = 5 ∠ 53.1301° = 5·e^(0.927295i)
The two representations check each other: 3² + 4² = 25 gives |A| = 5 , and 5·e^(0.927295i) converts back to 3 + 4i. Division is the conjugate trick in action — (3+4i)/(1−2i) = (3+4i)(1+2i)/5 = −1 + 2i — and z³ = 8 returns the three cube roots spaced 120° apart.
Why every root is listed A complex number has n distinct n-th roots, not one. The panel computes |A|^(1/n) and rotates it by (arg A + 2πk)/n for k = 0 … n−1, so z³ = 8 gives 2 , −1 + 1.73205i and −1 − 1.73205i , not just the real 2. The same branch logic governs ln(A): the panel returns the principal value (arg A ∈ (−π, π]) and says so, so ln(A) = 1.609437912 + 0.927295218i ; other branches differ by 2πk·i. exp and ln are inverses up to that branch choice.
Limits
Root order 1–64. The n field accepts integers from 1 to 64 and rejects anything else; z = 0 returns the single root 0. Single expressions, scalar output. Inputs are evaluated by the same expression engine as the scientific calculator, but the panel does not graph complex functions, solve complex systems or handle matrices — complex matrices belong to matrix tools . What it doesn't do. It does not carry units, format results as fractions, or export LaTeX; take the result to step-by-step for a written derivation or to the scientific calculator for further numeric work. The branch and the sign Two readings surprise people. First, sqrt(A) prints ±(2 + i) — both roots, because the panel never hides the second one. Second, ln is the principal branch: ln(−1) returns 3.141592654i , not the −πi you may expect, and every other branch is a 2πi shift away. When comparing against a textbook that uses a different cut, check the arg row before deciding the panel is wrong.
Where it is useful AC impedance in rectangular and polar form Impedance is a complex number: type R + jX as A, convert to polar for magnitude and phase, and multiply or divide impedances with the operator buttons. The RLC impedance panel covers the series circuit directly; this panel is for the algebra around it.
Roots of unity and phasors Set n and read the evenly spaced roots; the polar strings show the rotation angles directly. For a phasor question, exp(A) and ln(A) move between the exponential and rectangular views without retyping.
Privacy Every complex operation is evaluated locally in your browser; nothing you type is uploaded.
References
Wikipedia, Complex number , en.wikipedia.org （访问日期：2026-10-07）— rectangular and polar forms, conjugation.
Wikipedia, nth root , en.wikipedia.org （访问日期：2026-10-07）— n distinct complex roots.
Wikipedia, Complex logarithm , en.wikipedia.org （访问日期：2026-10-07）— principal branch and branch cuts.
Wikipedia, Euler's formula , en.wikipedia.org （访问日期：2026-10-07）— r·e^(θi) notation.
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Updated 2026-10-07