Square Root of Complex Number
2026-02-28 01:30 Diff
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Last updated on October 30, 2025

A pair of complex numbers that, when squared, give the original complex number, is known as a square root of a complex number. The square root of a complex number(a + ib) is expressed as √(a + ib) = ±(x + iy), where x and y are real numbers. We will explore more about the square root of complex numbers in this article.

What is a Complex Number?

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  • a represent the real part,
  • b represent the imaginary part, and
  • i is the imaginary unit

 i2 = -1. 3 + 2i, -5 - 4i are examples of complex numbers.

What is the square Root of Complex Number?

The square root of a complex number is another complex number that, when squared, gives the original complex number. If the square root of a complex number is √(a + ib) = ±(x + iy), then (x + iy)2 = a + ib.
 

The simplest way to find the square root of a complex number in the form of x + iy, square it, and then compare the real and imaginary parts to determine the values of x and y. 

Rectangular Form and Polar Form of a Complex Number

  • Rectangular Form of a Complex Number:

    The rectangular form of a complex number shows it using Cartesian coordinates on the complex plane. It is written as a + bi, where a is the real part and b is the imaginary part. 

  • Polar Form of a Complex Number:

    The polar form represents a complex number based on its distance from the origin and the angle it makes with the positive real axis. It is written as r(cosθ + isinθ), where r is the modulus of the complex number, and θ is the argument of the complex number. 

  • Formulas for Conversion:

Rectangular to Polar Form:
Modulus (r) = √(a2 + b2)
Argument (θ) = tan-1ba

Polar to rectangular Form:
Real part (a) = r × cosθ
Imaginary part (b) = r × sinθ

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Formula for Square Root of Complex Number

The Square root of a complex number is found in a pair, similar to the square root of real numbers. When either of these values is squared, the result is the original complex number.

The general representation of the square root of (x + iy) is:

\(√(x + iy) = ±[√(√(x2 + y2) + √x2) + i.y/|y|. √(√(x2 + y2) - √x2)] \ or \ \\ √(x + iy) = ±[√(|z| + √x2) + i.y/|y|. √(|z| - √x2)]\)

Here, z = x + iy and y ≠ 0

Polar Form of Square Root of Complex Numbers

A complex number can be written in polar form as:
z = r(cosθ + isinθ)

Where

  • r is the modulus (distance from the origin),
  • Θ is the argument (angle with the positive real axis).

To find the nth root of z, we use:

\(z1/2 = r1/ncos θ + 3600kn + isin θ + 3600k n\\ \ Or\ \\ z1/2 = r1/ncos θ + 2kn + isin θ + 2k n\)

Here, k = 0, 1, 2, …, n - 1, which means n has different roots. 

For the square root (when n = 2), the formula becomes:

\(z1/2 = r1/2cos θ + 3600k2 + isin θ + 3600k 2\\ \text{Or in radians:} \\ z1/2 = r1/2cos θ + 2k2 + isin θ + 2k 2\)

Where k = 0 or 1 because a square root always has two values.

General Formula for nth Root of Complex Number

If a complex number is written in polar form as:
z = r(cosθ + isinθ)

Then the n-th roots of z are given by:

\(z^k = r^{1/n} [cos ({θ + 2\pi k \over n}) + isin({ θ + 2k \over n})]\)

Here,

  • r is the modulus of the complex number, found by r = |z|.
  • θ is the angle that the number makes with the real axis.
  • n is the type you want to find.
  • k is a whole number starting from 0 up to n - 1.
  • Each value of k gives a different root. 

Tips and tricks for Square root of Complex number

Finding the square root of a complex number z = a + ib can be done in two easy ways: using the direct formula or using the polar form. Given below are the tips to remember. 

1. Using the Direct Formula:

The direct formula is quicker if we already know the values of a and b.
Write z = a + ib, where a is the real part and b is the imaginary part.

The modulus is |z| = a2 + b2

Use the formula:
a + ib = |z| + a2  + i b|b||z| - a2

  • If b > 0, both x and y have the same sign.
  • If b < 0, x and y have opposite signs.

2. Using Polar Form:

Polar form is better when you know the modulus and angle, or want to see the geometric meaning.

Convert z = a + ib into a polar form:
z = r(cosθ + isinθ)
where, r = |z| and θ = tan-1(ba)

Use the formula: 
\(z^{1/2} = r^{1/2}[cos ((θ + 2k)/ 2) + isin ((θ + 2k)/ 2)]\)

With k = 0, 1, to get the two roots.

Common Mistakes and How to Avoid Them in Square Root of Complex Numbers

When finding the square root of complex numbers, students often make an error that leads to incorrect answers. Here are some of the common mistakes and the ways to avoid them helps them.
 

Real Life Applications of Square Root of Complex Number

The concept of square roots of complex numbers is widely used in various scientific and engineering fields. It often appears when solving differential equations, calculating impedances in electrical circuits, or when analyzing wave functions. Given below are some of the real-life applications and how they are used.

  1. Signal Processing: In signal processing, complex numbers are fundamental for representing signals in the frequency domain. During operations such as the Fourier transform, square roots of complex numbers frequently arise when calculating the magnitude of frequency components, determining power, or adjusting phase. 
  2. Communication Systems: Square roots are used in computing the magnitude of these signals, which is essential in determining power, signal strength, and error correction. Complex numbers represent modulated signals in telecommunication. 
  3. Computer Graphics and 3D modeling: When doing certain geometric transformations or generating fractals, the square root of complex numbers is part of the calculations. Rotations, wave patterns, and wave phenomena are often modeled using complex numbers because they can represent both magnitude and phase in a single expression. 
  4. Aerodynamics: In analyzing airflow over wings or fluid movement, complex potential functions are used to simplify real-world fluid equations. The square root of complex numbers appears when calculating velocity potentials or solving flow equations around objects.
  5. Quantum Mechanics: Complex numbers represent the quantum states and wave functions. The probability amplitudes in quantum mechanics involve square roots when calculating probabilities or normalizing wave functions. 
     

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Problem 1

Find the square roots of -9.

Okay, lets begin

-9 = 3i
 

Explanation

We know -9 is a negative real number.
Since i = -1 

The square root of -9 is: 
-9 = 9 × -1 = 3i

Include both roots: + 3i and - 3i

Well explained 👍

Problem 2

Find the square root of 2i

Okay, lets begin

2i = (1 + i)
 

Explanation

Let z = 0 + 2i
Here, a = 0 and b = 2
 

  • The modulus r is: 
    r = a2 + b2 = 02 + 22 = 4 = 2
     
  • The argument :
      = tan -1 ba = tan -1 20= π2


So, 2i = 2cos π2 + i sin π2
 

  • To find the square root: 

z = 2i = r[cos π4 + isinπ4
= 2 12 + i12
= 1 + i 

Well explained 👍

Problem 3

Find the square root of -4

Okay, lets begin

-4 = 2i
 

Explanation

We know that -4 is negative, so it will have an imaginary root. 
-4 = 4 × -1 = 2i

The answers are: ± 2i.

Well explained 👍

Problem 4

Find the square root of 4i

Okay, lets begin

4i = (2+ 2i)
 

Explanation

  •  r = |4i| = 02 + 42 = 4  = 900
  • Use the formula: 

4i = 4[cos 450 + isin450] 
4 = 2, cos 450 = sin 450 = 2 2

  • Multiply: 2 2 2 + i 2 2 = 2+ 2i
     

Well explained 👍

Problem 5

Find the square root of -1.

Okay, lets begin

 -1 = i
 

Explanation

We know -1 = -1 × 1
By definition, i = -1
The square root of -1 is defined as i.

So, the two roots are + i and -1.

Well explained 👍

FAQs on Square Root of Complex Number

1.My child says square root of a complex number are unique. Is this correct?

No, every non-zero complex number has exactly two distinct square roots. If w is one square root of z, then the other square root is -w.

2.Can my child get complex square roots for real number.?

Yes, negative real numbers have complex square roots. For example, √-4 = ±2i.
 

3.Where should my child locate square roots of a complex number on the complex plane?

The square roots of a complex number are two points on a circle centered at the origin, and they are directly opposite to each other. 
 

4.How can my child identify if the square root is purely real or purely imaginary?

  • If the imaginary part is 0 and the real part is positive, the square root is purely real.
  • If the real part is 0 and the imaginary part is non-zero, the square root is purely complex.
     

5.How to define the principal square root of a complex number?

The principal square root is the one with the positive real part, or the one that lies in the right half of the complex plane. 

Jaskaran Singh Saluja

About the Author

Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.

Fun Fact

: He loves to play the quiz with kids through algebra to make kids love it.