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Square Root of Complex Number
The square root of complex number gives a pair of complex numbers whose square is the original complex number. The square root of a complex number can be determined using a formula. Just like the square root of a natural number comes in pairs (Square root of x^{2} is x and x), the square root of complex number a + ib is given by √(a + ib) = ±(x + iy), where x and y are real numbers.
In this article, let us go through the concept of finding the square root of a complex number using the assumption method and using the formula. We will also learn to determine the square root of a complex number in polar form along with some solved examples for a better understanding of the concept.
What is Square Root of Complex Number?
The square root of a complex number is another complex number whose square is the given complex number. For instance, if the square root of complex number a + ib is √(a + ib) = x + iy, then we have (x + iy)^{2} = a + ib. One of the simple ways to calculate the square root of a complex number a + ib is to compare the real and imaginary parts of the equation √(a + ib) = x + iy by squaring both sides and then finding the values of x and y. Let us see the formula for finding the square root of complex number.
Square Root of Complex Number Formula
Now, let us derive the formula to find the square root of a complex number a + ib. Assume the square root of complex number a + ib to be x + iy, that is, √(a + ib) = x + iy. Now squaring both sides of the equation, we have
[√(a + ib)]^{2} = (x + iy)^{2}
⇒ a + ib = x^{2} + (iy)^{2} + i2xy
⇒ a + ib = x^{2}  y^{2} + i2xy [Because i^{2} = 1]
Comparing real and imaginary parts of the above equation, we have
a = x^{2}  y^{2} , b = 2xy
We know that (x^{2} + y^{2})^{2} = (x^{2}  y^{2})^{2} + 4x^{2}y^{2}
⇒ (x^{2} + y^{2})^{2} = a^{2} + b^{2}
⇒ x^{2} + y^{2} = √(a^{2} + b^{2}) [Because x^{2} + y^{2} is always positive as sum of squares of nonzero real numbers is always greater than zero]
Now, we have x^{2} + y^{2} = √(a^{2} + b^{2}) and a = x^{2}  y^{2}. Solving for these two values, we have
x = ± √{[√(a^{2} + b^{2}) + a]/2} and y = ± √{[√(a^{2} + b^{2})  a]/2}
Since 2xy = b, therefore we have
 x and y have the same sign if b > 0
 x and y have opposite signs if b < 0
Therefore the square root of complex number a + ib (b ≠ 0) is given by √(a + ib) = ± (√{[√(a^{2} + b^{2}) + a]/2} + (ib/b) √{[√(a^{2} + b^{2})  a]/2})
Hence, the formula to determine the square root of complex number is:
Square Root of Complex Number in Polar Form
To determine the square root of a complex number in polar form, we use the n^{th} root theorem for complex numbers. The n^{th} Root Theorem states that for a complex number z = r(cosθ + i sinθ), the n^{th} root is given by z^{1/n} = r^{1/n }[cos [(θ + 2kπ)/n] + i sin [(θ + 2kπ)/n]], where k = 0, 1, 2, 3, ..., n1. In order to obtain the periodic roots of the complex number, we add 2kπ to θ. So, using the formula for n^{th} root, we can determine the formula to find the square root of complex number in polar form. The formula is,
z^{1/2} = r^{1/2 }[cos [(θ + 2kπ)/2] + i sin [(θ + 2kπ)/2]], where k = 0, 1
Tips on Square Root of Complex Number
Suppose the square root of z = a + ib is x + iy.Then, the easiest way to determine the square root of a complex number with rectangular coordinates is using the formula \(\sqrt{a + ib} = \pm\left ( \sqrt{\dfrac{z+a}{2}}+i\dfrac{b}{b}\sqrt{\dfrac{za}{2}}\right )\). We have,
 If b < 0, then b/b = 1, and x and y have opposite signs
 If b > 0, then b/b = 1, and x and y have same signs
Another way to determine the square root of complex number with rectangular coordinates is by determining its polar coordinates and then using the formula z^{1/2} = r^{1/2 }[cos [(θ + 2kπ)/2] + i sin [(θ + 2kπ)/2]], where k = 0, 1.
Important Notes on Square Root of Complex Number
 The square root of complex number with rectangular coordinates is √(a + ib) = ± (√{[√(a^{2} + b^{2}) + a]/2} + ib/b √{[√(a^{2} + b^{2})  a]/2})
 The square root of a complex number with polar coordinates is z^{1/2} = r^{1/2 }[cos [(θ + 2kπ)/2] + i sin [(θ + 2kπ)/2]], where k = 0, 1.
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Square Root of Complex Number Examples

Example 1: Determine the square root of complex number 3 + 4i.
Solution: To determine the square root of 3 + 4i, we will determine its magnitude and compare with a + ib.
3 + 4i = √(3^{2} + 4^{2}) = √(9 + 16) = √25 = 5; a = 3 and b = 4 > 0
Therefore, using formula for square root of complex numbers, we have
√(3 + 4i) = ± (√[√(5 + 3)/2] + i√[√(5  3)/2] )
= ± (√(8/2) + i √(2/2))
= ±(√4 + i√1)
= ± (2 + i)
Answer: Hence, √(3 + 4i) = ± (2 + i)

Example 2: Determine the square root of z = 2[cos(π/4) + i sin(π/4)]
Solution: To determine the square root of complex number z = 2[cos(π/4) + i sin(π/4)] in polar form, we will use the formula z^{1/2} = r^{1/2 }[cos [(θ + 2kπ)/2] + i sin [(θ + 2kπ)/2]], where k = 0, 1
We have r = 2, θ = π/4. The roots of z are:
When k = 0, z_{1} = 2^{1/2 }[cos [(π/4 + 2(0)π)/2] + i sin [(π/4 + 2(0)π))/2]]
z_{1} = √2 [cos(π/8) + i sin(π/8)]When k = 1, z_{2} = 2^{1/2 }[cos [(π/4 + 2(1)π)/2] + i sin [(π/4 + 2(1)π))/2]]
z_{2} = √2 [cos(9π/8) + i sin(9π/8)]Answer: The square root of z = 2[cos(π/4) + i sin(π/4)] are z_{1} = √2 [cos(π/8) + i sin(π/8)] and z_{2} = √2 [cos(9π/8) + i sin(9π/8)]
FAQs on Square Root of Complex Number
What is the Square Root of Complex Number in Math?
The square root of a complex number gives a pair of complex numbers whose square is the original complex number. Like the square root of real numbers, on squaring the square root of complex number, we get the given complex number.
What is the Formula to Find the Square Root of Complex Number?
The formulas to determine the square root of complex numbers are:
 The square root of complex number with rectangular coordinates is √(a + ib) = ± (√{[√(a^{2} + b^{2}) + a]/2} + ib/b √{[√(a^{2} + b^{2})  a]/2})
 The square root of a complex number with polar coordinates is z^{1/2} = r^{1/2 }[cos [(θ + 2kπ)/2] + i sin [(θ + 2kπ)/2]], where k = 0, 1.
How to Find the Square Root of a Complex Number?
The square root of complex number can be calculated by substituting the values in formula. The square root of a + ib is ± (√{[√(a^{2} + b^{2}) + a]/2} + ib/b √{[√(a^{2} + b^{2})  a]/2}) and the square root of re^{i}^{θ} is r^{1/2 }[cos [(θ + 2kπ)/2] + i sin [(θ + 2kπ)/2]], where k = 0, 1.
Is Square Root of 53 a Complex Number?
The square root of 53 is not a complex number. 53 is a number that does not have a natural number as its square root. Square root of 53 cannot be expressed as a fraction in the form p/q which tells us that the square root of 53 is an irrational number.
What is the Square Root of Complex Number 7  24i?
The square root of 7  24i is ± (√{[√(7^{2} + (24)^{2}) + 7]/2}  i √{[√(7^{2} + (24)^{2})  7]/2}) = ± (√32/2  i √18/2) = ± (4  3i).
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