Online Square Root of Complex Number Tutor & Homework Help
Solution Square Root of Complex Number Tutoring Online, Tutor Help
TO FIND THE SQUARE ROOT OF THE COMPLEX NUMBER a + ib
WHERE a AND b (≠0) ARE REAL
WHERE a AND b (≠0) ARE REAL
Let √a + ib = x + iy where x and y are real.
Then, a + ib = (x + iy)2 = x2 + i2y2
+ 2x. iy = x2 - y2 + i. 2xy
Equating real and imaginary parts we get,
x2 - y2 = a (1)
2xy = b (2)
Now (x2 + y2)2 = (x2 - y2)2 + 4x2y2
= a2 + b2
Hence x2 + y2 = +√a2 + b2 (Since x, y both are real)
(3)
Now adding equations (1) and (3) we get,
Now equation (3) - equation (1) gives,
From equation (2) it is clear that both x and y will have the same signs (either both positive or
both negative) when b is positive; and x and y will have different signs (one positive and the
other negative) when b is negative.
Hence if b > 0 then the square roots of (a + ib) are:
And if b < 0 then the square roots of (a + ib) are:
Example 1:
Find the square root of
Solution:
Now let √1 + i = x + iy (where x and y are real)
Hence, 1 + i = (x + iy)2 = x2 - y2 + i. 2xy
Hence equating real and imaginary parts we get,
x2 - y2 = 1 (1)
2xy = 1 (2)
Now (x2 + y2)2 = (x2 - y2)2
+ 4x2y2 = 12 + 12 = 2
Or x2 + y2 = √2 (3)
Adding equations (1) and (3) we get,
Now equation (3) - equation (1) gives,
From (2) it is clear that xy > 0, hence x and y are both positive or both negative.
Hence the required square roots are:
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