Mathematics : Complex Numbers: Square roots of a complex number

**Square roots of a complex number**

Let the square root of *a *+ *ib *be *x *+ *iy*

That is = âˆš[*a*+*ib*] = *x + iy *where x, y âˆˆ
**R**

*a + ib *= ( *x *+ *iy *)* ^{2 }* =

Equating real and imaginary parts, we get

*x** ^{2}* -

(*x** ^{2}* +

*x** ^{2}* +

Solving *x** ^{2}* +

Since 2*xy *= *b* it is clear that both *x *and
*y *will have the same sign when *b *is positive, and *x *and *y
*have different signs when *b *is negative.

Formula for finding
square root of a complex number

Note

Find the square root of 6 - 8*i*.

We compute |6 - 8i| = âˆš[6^{2} + (-8)^{2}] = 10

and applying the formula for square root, we get

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12th Mathematics : UNIT 2 : Complex Numbers : Square roots of a complex number | Definition, Formulas, Solved Example Problems

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