Z₁ = 3 + 4i | |Z₁| = 5 | arg = 53.13°
Z₂ = 1 − 2i
Computes Z₁ⁿ using De Moivre's theorem
Quick Examples:
Result (Rectangular)
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Polar Form
Modulus |Z|
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Argument θ
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All Roots
format_list_numbered Step-by-Step
iⁿ cycles with period 4: use n mod 4
Enter the real and imaginary parts of the first complex number Z₁ = a + bi
For binary operations (add, subtract, multiply, divide), also enter Z₂ = c + di
Select from 11 operations: arithmetic, modulus, argument, conjugate, polar form, power, roots, or e^z
See the result in both rectangular (a+bi) and polar (r∠θ) forms with step-by-step working
A complex number z = a + bi has real part a and imaginary part b. The modulus is |z| = √(a²+b²). The argument θ = atan2(b, a). Multiplication is easiest in polar form: |z₁||z₂|∠(θ₁+θ₂). Division: (|z₁|/|z₂|)∠(θ₁−θ₂). Powers by De Moivre: z^n = rⁿ(cos nθ + i·sin nθ).
z = a + bi, |z| = √(a²+b²), z^n = rⁿ(cos nθ + i sin nθ)
Remember: i² = −1. This is the key identity for complex multiplication
Modulus and argument: |z| = √(a²+b²), θ = atan2(b, a)
Conjugate of (a+bi) is (a−bi). Multiplying z·z* = |z|² (always real)
For powers and roots, De Moivre's theorem is far easier than expanding
Polar multiplication: multiply moduli, add arguments
The n-th roots of z are equally spaced around a circle of radius ⁿ√|z|
Complex numbers extend the real number system by introducing the imaginary unit i, where i-squared equals negative one. Far from being a purely abstract concept, complex numbers are essential tools in electrical engineering for analyzing AC circuits using phasor notation, in signal processing for representing frequency components via the Fourier transform, in control systems for stability analysis, and in quantum mechanics for describing wave functions. A complex number z = a + bi has a real part a and imaginary part b, and can equivalently be expressed in polar form as r times e-to-the-i-theta, where r is the magnitude and theta is the argument. Converting between rectangular and polar forms, computing products and quotients, finding powers and roots — these operations follow specific rules that are straightforward but tedious to perform by hand, especially when precision matters. This complex number calculator handles all standard operations including addition, subtraction, multiplication, division, powers, nth roots, conjugates, and magnitude calculations, showing detailed step-by-step solutions for each operation.
A complex number calculator performs arithmetic and geometric operations on numbers of the form a + bi, where i is the imaginary unit (i² = −1). Our calculator supports all standard operations: addition, subtraction, multiplication, division, modulus, argument, conjugate, polar form conversion, De Moivre powers, nth roots, and complex exponential.
Complex numbers can be expressed in rectangular form (a + bi) or polar form (r∠θ). The modulus r = √(a²+b²) is the distance from the origin, and the argument θ = atan2(b,a) is the angle. Polar form makes multiplication, division, and powers much simpler: multiply moduli and add arguments.
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All formulas verified against official standards.