How to find Square root of a complex number
A number of the form where and is called a complex number where is called as real part and is called imaginary part of complex number. To find the square root of a complex number, we will assume that . Then square both the sides and compare real and imaginary part to find the value of and , which will give us the square root.
Formula for finding square root of a complex number:
The square root of is
Proof. Let square root of . That is where and in . Now square both the sides we get
Equating real and imaginary parts we get
Solving (i) and (iii) we get
Similarly . Since 2ab=y , it is clear that both and have the same sign when is positive and and have different sign when is negative. Therefore
Example : Find the square root of (-5-12i) .
Applying the formula for square root we get
( is negative)
Trick to find the square root of a complex number:
To find , follow the following steps:
- First find the number .
- Now factorise the given number in such a way that difference of square of these factors is equal to the real number .
Ex. Find the square root of 7+4i.
Sol. Find the number which is Equal to 12. Now factor 12 in such a way that difference of square of these factors is equal to the real number .
12= 4×3 and . Therefore .
Ex. Find the square root of .
Sol. Here and such that
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