The square root of a complex number is another complex number that, when multiplied by itself (squared), gives the original complex number. A complex number includes both a real part and an imaginary part, and it is usually written as z = x + iy, where x is the real part and iy is the imaginary part.

In mathematics, finding a square root is the opposite of squaring a number. So, to find the square root of a complex number, we are working backward from a known result. This process involves a specific formula and steps to separate the real and imaginary parts properly.

This concept helps in solving equations and analyzing signals in engineering and physics. In this article, you'll learn an easy-to-follow method for finding the square root of a complex number, along with the formula, explanation, and solved examples to guide you step by step.

Rectangular Form and Polar Form of a Complex Number

Rectangular Form of a Complex Number:

The rectangular form of a complex number is expressed as a + bi, where 'a' represents the real part and 'b' represents the imaginary part of the complex number. This form allows us to represent complex numbers in terms of their Cartesian coordinates on the complex plane.

Polar Form of a Complex Number:

The polar form of a complex number is expressed as r(cosθ + isinθ), where 'r' represents the modulus (magnitude) of the complex number, and 'θ' represents the argument (angle) of the complex number. This form represents complex numbers in terms of their distance from the origin (modulus) and the angle they make with the positive real axis (argument).

Formulas for Conversion:

Rectangular to Polar Form:

Modulus (r) = √(a^2 + b^2)

Argument (θ) = tan^(-1)(b/a)

Polar to Rectangular Form:

Real part (a) = r * cos(θ)

Imaginary part (b) = r * sin(θ)

The square root of a complex number always gives us two possible complex numbers, similar to how the square root of a real number gives both a positive and a negative value. For example, if we have the equation p = a², then the square root of p will be ±a (both +a and −a).

In the same way, when we find the square root of a complex number, we get a pair of complex numbers as the result. These two values, when squared, will both return the original complex number. This is because squaring either of the pair (positive or negative) gives the same result.

Understanding this concept is important in complex number calculations. In the next section, we will go deeper into how this works, including the formula used to find the square root of a complex number and the step-by-step method to solve it.

Formula for Square Root of Complex Number

Similar to the square root of a natural number, the square root of a complex number is obtained in pairs. When we square these pairs we get the initial complex number as the result. The formula for finding the square root of a complex number is as follows:

√(x + iy) = ±[√(√(x² + y²) + √x²) + i·y/|y|·√(√(x² + y²) − √x²)]

OR

√(x + iy) = ±[√(|z| + √x²) + i·y/|y|·√(|z| − √x²)]

Here, z = x + iy and y ≠ 0

Check out this article on the Modulus of a Complex Number.

How to Find the Square Root of Complex Numbers
Let us now understand how to derive or find the formula for the square root of a complex number system. Consider a complex number, z = x + iy for which we have to find the square root.

Step 1: Let p + iq be the square root of x + iy. We can write this as, √(x + iy) = p + iq

Step 2: Squaring on both sides we get:
x + iy = (p + iq)²
That is, x + iy = p² + (iq)² + i·2pq
⇒ x + iy = p² − q² + i·2pq

Step 3: Comparing and equating the real and imaginary part we get:
p² − q² = x and 2pq = y

Step 4: Using the algebraic formula we get,
(p² + q²)² = (p² − q²)² + 4p²q²

We can say that, p² − q² = x and 2pq = y
Therefore; (p² + q²)² = x² + y²

Step 5: Taking the square root on both sides we get,
(p² + q²) = √(x² + y²)

As p² + q² is always positive, the summation of squares of non-zero real numbers is always > 0.

Step 6: Substitute the value in the equation; here x = p² − q²

Step 7: Solving the equation we get;
p = ±√[(√(x² + y²) + √x²)] and q = ±√[(√(x² + y²) − √x²)]

Step 8: Since 2pq = y, therefore:
p and q will hold the identical sign if y > 0
p and q will hold opposing signs if y < 0

Therefore, the square root of z = x + iy is:
√(x + iy) = ±[√(√(x² + y²) + √x²) + i·y/|y|·√(√(x² + y²) − √x²)]

Also,
√(x + iy) = ±[√(|z| + √x²) + i·y/|y|·√(|z| − √x²)]

Learn about De Moivre’s Theorem

Polar Form of Square Root of Complex Numbers

In the previous header, you learned about the square root of a complex number direct formula with the definition and derivation approach. Let us now understand how to find the square root of a complex number in polar form. The roots of such a complex number are equal to: z^(1/n) or zⁿ. That is, we determine the nth root of the equation,
z = r(cosθ + i·sinθ)

Hence:
z^(1/n) = r^(1/n) [cos((θ + 2πk)/n) + i·sin((θ + 2πk)/n)], in radians.
Also,
z^(1/n) = r^(1/n) [cos((θ + 360°k)/n) + i·sin((θ + 360°k)/n)], in degrees.

For the above formulas, k = 0, 1, 2, 3, …, n − 1

In the similar manner, the formula for square root is:
z^(1/2) = r^(1/2) [cos((θ + 360°k)/2) + i·sin((θ + 360°k)/2)], in degrees.
OR
z^(1/2) = r^(1/2) [cos((θ + 2πk)/2) + i·sin((θ + 2πk)/2)], in radians.

For the above equation, the value of k = 0, 1

Solved Examples on Square Root of Complex Number

With all the knowledge of the square root of complex number shortcut formulas, definition and how to find the same. Let us practice some solved examples to understand these concepts from an exam viewpoint.

Solved Example 1: Find the square root of the complex number 2 + 3i.

Solution:

We are given the complex number:
z = 2 + 3i

We use the formula for the square root of a complex number:
√(x + iy) = ±[ √((|z| + x)/2) + i·(sign of y)·√((|z| − x)/2) ]

Step 1: Calculate the modulus of z
|z| = √(x² + y²) = √(2² + 3²) = √(4 + 9) = √13 ≈ 3.6056

Step 2: Apply the values to the formula

Let’s compute:

A = √((|z| + x)/2) = √((3.6056 + 2)/2) = √(5.6056/2) = √2.8028 ≈ 1.673
B = √((|z| − x)/2) = √((3.6056 − 2)/2) = √(1.6056/2) = √0.8028 ≈ 0.896

Since y = 3 (positive), we keep B as positive.

Step 3: Write the final result

√(2 + 3i) = ±(1.673 + 0.896i)

Learn the different Properties of Complex Numbers.

Solved Example 2: Find the square root of complex number 5 + 12i.

Solution: To specify the square root of 5 + 12i, the formula used is:

√(x + iy) = ±[√(|z| + √x²) + i·y/|y|·√(|z| − √x²)]

Here, z = 5 + 12i

Obtain the modulus value of z:

|z| = |5 + 12i| = √(5² + 12²) = √169 = 13

Substituting the values in the equation:

√(x + iy) = ±[√(13 + √25) + i·12/12·√(13 − √25)]

√(x + iy) = ±[√18 + i·√8]

√(x + iy) = ±[√9 + i·√4]

√(x + iy) = ±[3 + i·2]

Hence, √(5 + 12i) = ±[3 + i·2]

Learn more about the Polar Form of Complex Numbers.

Solved Example 3: Obtain the square root of z = 3[cos(π/3) + i·sin(π/3)]

Solution:

To calculate the square root of a complex number in polar form the formula used is:

z^(1/2) = r^(1/2)[cos((θ + 360°k)/2) + i·sin((θ + 360°k)/2)], in degrees.
OR
z^(1/2) = r^(1/2)[cos((θ + 2πk)/2) + i·sin((θ + 2πk)/2)], in radians.

Here the value of k = 0, 1.

Also, we have r = 3, θ = π/3. The roots of z are:

For k = 0:

zₐ = √3 [cos((π/3 + 2π·0)/2) + i·sin((π/3 + 2π·0)/2)]
zₐ = √3 [cos(π/6) + i·sin(π/6)]

For k = 1:

z_b = √3 [cos((π/3 + 2π·1)/2) + i·sin((π/3 + 2π·1)/2)]
z_b = √3 [cos(7π/6) + i·sin(7π/6)]

Solved Example 4: Find the square root of complex number 9 + 40i

To find the square root of a complex number in rectangular form, we can use the following steps:

Step 1: Express the complex number in the form a + bi.

In this case, we have the complex number 9 + 40i, which is already in the desired form.

Step 2: Split the complex number into its real and imaginary parts.

For 9 + 40i, the real part (a) is 9 and the imaginary part (b) is 40.

Step 3: Find the modulus (magnitude) of the complex number.

The modulus of a complex number a + bi is given by |z| = √(a^2 + b^2).

For our complex number 9 + 40i, the modulus is |9 + 40i| = √(9^2 + 40^2) = √(81 + 1600) = √1681 = 41.

Step 4: Find the principal argument (angle) of the complex number.

The principal argument of a complex number a + bi is given by arg(z) = tan^(-1)(b/a).

For our complex number 9 + 40i, the principal argument is arg(9 + 40i) = tan^(-1)(40/9) ≈ 1.32 radians (in the range of -π to π).

Step 5: Calculate the square root in rectangular form.

To find the square root of a complex number in rectangular form, we can use the formula:

√(a + bi) = ±(√((√(a^2 + b^2) + a) / 2) + (√(√(a^2 + b^2) - a) / 2)i).

Applying this formula to our complex number 9 + 40i, we get:

√(9 + 40i) = ±(√((√(1681) + 9) / 2) + (√(√(1681) - 9) / 2)i).

Simplifying further:

√(9 + 40i) = ±(√((41 + 9) / 2) + (√(41 - 9) / 2)i).

√(9 + 40i) = ±(√(50/2) + √(32/2)i).

√(9 + 40i) = ±(√25 + √16i).

√(9 + 40i) = ±(5 + 4i).

Therefore, the square root of the complex number 9 + 40i is ±(5 + 4i).

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