Square Root of an Imaginary Number
Square Root of an Imaginary Number
The square root of an imaginary number involves complex numbers. The imaginary unit i satisfies i² = −1.
√i (square root of i)
√i = (1 + i) / √2 ≈ 0.7071 + 0.7071i
This can be derived using polar form: i = e^(iπ/2), so √i = e^(iπ/4) = cos(45°) + i·sin(45°) = (√2/2) + (√2/2)i.
General formula: √(bi)
√(bi) = √(b/2) · (1 + i) where b > 0
Example: √(8i) = √4 · (1+i) = 2 + 2i
Imaginary and complex square roots are used in electrical engineering (AC circuits), quantum mechanics, and signal processing.