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Imaginery Square Root Calculator

Reviewed by Calculator Editorial Team

This imaginary square root calculator helps you find the square roots of complex numbers. Complex numbers have both real and imaginary parts, and their square roots can be calculated using mathematical formulas. Understanding how to find the square roots of complex numbers is important in various fields of mathematics and engineering.

What is the imaginary square root?

The imaginary square root refers to the square roots of complex numbers that have an imaginary component. A complex number is typically written in the form a + bi, where a is the real part, b is the imaginary part, and i is the imaginary unit (√-1).

For a complex number z = a + bi, the square roots can be found using the following formula:

Square Root Formula for Complex Numbers

√(a + bi) = ±(√[(a + √(a² + b²))/2] + i √[(√(a² + b²) - a)/2])

This formula gives two square roots for any non-zero complex number, which are complex conjugates of each other.

How to calculate the imaginary square root

To calculate the square root of a complex number, follow these steps:

  1. Identify the real part (a) and the imaginary part (b) of the complex number.
  2. Calculate the magnitude of the complex number: √(a² + b²).
  3. Use the square root formula to find the two square roots.
  4. Simplify the expressions to get the final square roots.

Note

The square roots of a complex number are always complex numbers, even if the original number has no imaginary part (i.e., it's purely real).

Formula

The formula for finding the square roots of a complex number z = a + bi is:

Square Root Formula

√(a + bi) = ±(√[(a + √(a² + b²))/2] + i √[(√(a² + b²) - a)/2])

This formula provides both the positive and negative square roots of the complex number.

Example calculation

Let's find the square roots of the complex number 3 + 4i.

  1. Identify a = 3 and b = 4.
  2. Calculate the magnitude: √(3² + 4²) = √(9 + 16) = √25 = 5.
  3. Apply the square root formula:
    • First root: √[(3 + 5)/2] + i √[(5 - 3)/2] = √4 + i √1 = 2 + i
    • Second root: -√[(3 + 5)/2] - i √[(5 - 3)/2] = -2 - i

The square roots of 3 + 4i are 2 + i and -2 - i.

FAQ

What is the difference between real and imaginary square roots?

Real square roots are numbers that, when multiplied by themselves, give a positive real number. Imaginary square roots involve complex numbers and result in complex numbers as well.

Can a complex number have a real square root?

Yes, if the complex number is purely real (i.e., the imaginary part is zero), its square roots will also be real numbers.

How many square roots does a complex number have?

A non-zero complex number has exactly two square roots, which are complex conjugates of each other.