. Square root of complex number formula


Best Answer

square root of complex number a + ib is

√(a + ib) = x + iy,

(x + iy)2 = a + ib.

Derivation of Square Root of Complex Number Formula

Let us assume that the square root of complex number a + ib to be x + iy,

i.e. √(a + ib) = x + iy.

squaring both the sides, we have

[√(a + ib)]2 = (x + iy)2

⇒ a + ib = x2 + (iy)2 + i2xy

⇒ a + ib = x2 - y2 + i2xy [Because i2 = -1]

Comparing real and imaginary parts of the above equation, we have

a = x2 - y2, b = 2xy

We know that (x2 + y2)2 = (x2 - y2)2 + 4x2y2

⇒ (x2 + y2)2 = a2 + b2

⇒ x2 + y2 = √(a2 + b2)

Now, we have x2 + y2 = √(a2 + b2) and a = x2 - y2. Solving for these two values, we have

x = ± √{[√(a2 + b2) + a]/2} and y = ± √{[√(a2 + b2) - a]/2}

Hence, the square root of complex number a + ib (b ≠ 0) is given by

√(a + ib) = ± (√{[√(a2 + b2) + a]/2} + (ib/|b|) √{[√(a2 + b2) - a]/2})

Thus, the formula to determine the square root of complex number is:

Square root of complex number

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