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

Chapter: 12th Mathematics : UNIT 2 : Complex Numbers

Square roots of a complex number

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 = √[a+ib] = x + iy where x, y R

a + ib =  ( x + iy )2  = x2  - y2  + i2xy

Equating real and imaginary parts, we get 

x2 - y 2 = a and 2xy = b

(x2  + y2 )2 =  (x2 - y2 ) 2  + 4x2 y2  = a2  + b2

x2  + y2√[a2+b2], since x2 + y2 is positive 

Solving x2 + y2 = a  and  x2+ y2 =√[a2+b2], we get


Since 2xy = 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


 

Example 2.17

Find the square root of 6 - 8i.

Solution

We compute |6 - 8i| = √[62 + (-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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12th Mathematics : UNIT 2 : Complex Numbers


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