Complex Number Calculator
Add, subtract, multiply, divide complex numbers; conjugate, magnitude |z|, argument θ and polar form.
Input Data
Results
At a glance:A complex z=x+yi has real part x, imag part y, i²=−1. Add/sub component-wise; z₁z₂=(ac−bd)+(ad+bc)i; z₁/z₂=(z₁·z̄₂)/|z₂|². Conjugate z̄=a−bi. Magnitude |z|=√(x²+y²). Argument θ=atan2(y,x). Polar z=r(cosθ+i sinθ), r=|z|. Argand diagram plots (x,y).
Formula
Add/sub: combine real and imag parts.
Mul: (ac−bd)+(ad+bc)i.
Div: multiply by z̄₂: [(ac+bd)+(bc−ad)i]/(c²+d²).
Conj: a−bi; mag: √(a²+b²); arg: atan2(b,a).
Polar: r(cosθ+i sinθ); Euler: r·e^{iθ}.
$$z_1z_2 = (ac-bd) + (ad+bc)i$$$$z = r(\cos\theta + i\sin\theta) = re^{i\theta}$$How to Use
- Pick an op (add/sub/mul/div/conjugate/magnitude/argument/polar).
- Enter z₁=a+bi; for +−×÷ also enter z₂=c+di.
- The tool gives result, magnitude, argument and polar form.
- Compare with the case studies for hand-working.
Complex operations
| Op | Result | Geometry |
|---|---|---|
| z₁+z₂ | (a+c)+(b+d)i | vector add |
| z₁z₂ | (ac−bd)+(ad+bc)i | magnitudes multiply, angles add |
| z₁/z₂ | [(ac+bd)+(bc−ad)i]/(c²+d²) | magnitudes divide, angles subtract |
| z̄ | a−bi | mirror over x-axis |
| |z| | √(a²+b²) | distance to origin |
Multiplication adds arguments: arg(z₁z₂)=arg(z₁)+arg(z₂).
Case Studies
(3+2i)+(1+4i)=4+6i
Add: real 3+1=4, imag 2+4=6 → 4+6i.
|z|=√(16+36)=√52≈7.2111, θ≈56.31°.
(3+2i)(1+4i)
=(3·1−2·4)+(3·4+2·1)i = −5+14i.
|z₁|=√13, |z₂|=√17, |product|=√221≈14.8661.
(3+2i)/(1+4i)
Multiply num & den by (1−4i): [(3+2i)(1−4i)]/17.
=(11−10i)/17 ≈ 0.6471−0.5882i.
Conjugate 3+2i → 3−2i
z̄=3−2i; |z̄|=|z|=√13≈3.6056.
z·z̄ = a²+b² = 13 (real).
Argument & polar: 1+i
|1+i|=√2≈1.4142, θ=45° (π/4).
Polar: √2(cos45°+i sin45°)=√2·e^{iπ/4}.
Pure imaginary i²=−1
i·i = −1; |i|=1, θ=90°.
Polar of i: 1·e^{iπ/2}.
FAQ
What is i?
i is the imaginary unit with i²=−1. A complex is a+bi, a real, b imaginary part.
How to divide complexes?
Multiply numerator and denominator by the conjugate of the denominator z̄₂=c−di, making the denominator real c²+d², then split real/imag parts.
What are magnitude and argument?
Magnitude |z|=√(a²+b²) is distance to origin; argument θ=atan2(b,a) is the angle from the positive real axis (rad or deg).
Why polar form?
Polar r(cosθ+i sinθ) makes multiplication (multiply magnitudes, add angles) and powers/roots (De Moivre) easy; engineering uses e^{iθ}.
Properties of conjugate?
z·z̄=a²+b²=|z|² is real; conjugate is used in division and to get real part (z+z̄)/2 and imag part (z−z̄)/(2i).
Why the − sign in results?
A negative imaginary part is shown as 'minus + value i' (e.g. 3−2i); that is standard notation.
Relation to AC circuits?
Impedance and phase are often complex; in EE i is written j to avoid confusion with current, but it means the same.
Is there a principal argument?
Argument is multi-valued (differs by 2π); this tool returns the principal value in (−180°,180°] or [0,360°); add/subtract 360° as needed.
Related Tools
References
Content review: Calculatorism Science Team. Complex add/sub/mul/div, conjugate, magnitude, argument and polar logic verified. Results are for reference only.