Complex Root Calculator
This calculator finds the nth roots of complex numbers, showing all possible solutions in both rectangular and polar forms. It’s useful in advanced algebra, complex analysis, and engineering applications where root extraction from complex numbers is required.
Calculate the nth Roots of a Complex Number
Complex Root Formula (De Moivre’s Theorem)
Explanation:
To find the nth roots of a complex number $$z$$, convert it to polar form $$r(\cos \theta + i \sin \theta)$$, apply De Moivre’s Theorem, and calculate all $$n$$ distinct roots by rotating $$\theta$$ around the unit circle.
Complex Root – Calculation Example
Find the square roots of z = 1 + i
Step 1: Convert to polar form: r = √2, θ = π/4
Step 2: n = 2 → calculate two roots
Root 1: √√2 [cos(π/8) + i·sin(π/8)]
Root 2: √√2 [cos(π/8 + π) + i·sin(π/8 + π)]
Results (approx):
Root 1 ≈ 1.0987 + 0.4551i
Root 2 ≈ -1.0987 – 0.4551i
Complex roots are key in solving polynomial equations, analyzing AC circuits, and modeling oscillations. This calculator automates the complex root-finding process, providing both the modulus–argument form and standard (a + bi) form, making it easier to interpret and visualize results.