Summary

The Excel IMSQRT function returns the square root of a complex number.

Purpose 

Get square root of a complex number

Return value 

The square root of the complex number.

Syntax

=IMSQRT(inumber)
  • inumber - A string representing a complex number.

How to use 

The Excel IMSQRT function returns the square root of a complex number. For example:

=IMSQRT("3+4i") // returns "2+i"

Excel handles complex numbers as strings formatted like "x+yi" or "x+yj". Use the COMPLEX function to get the string representing a complex number.

Basic Example

The IMSQRT function returns the complex number's principal square root (the one with a non-negative real part). For example:

=IMSQRT("2i") // returns "1+i"

The two possible square roots of "2i" are "1+i" and "-1-i", so "1+i" is returned because it has a non-negative real part.

Square Root of Negative One Example

A fundamental property of complex numbers is that the square root of "-1" is "i". However, when you take the square root of "-1", the result contains a tiny error due to how Excel handles floating-point arithmetic.

=IMSQRT("-1") // returns "6.12323399573677E-17+i"

The real part of the return value is a very small number close to zero. For practical purposes, it can be interpreted as just "i".

Explanation

The square root of a complex number is given by:

Definition of complex square root.

Where "r" is equal to the radius of the complex number and "θ" is equal to the angle of the complex number for the branch cut extending from -π to π (excluding -π).

In Excel, we can write the formula equivalent to the complex square root function like this, where B6 contains a string representing a complex number:

=LET(
r, IMABS(B6),
t, IMARGUMENT(B6),
COMPLEX(SQRT(r)*COS(t/2),SQRT(r)*SIN(t/2))
)

Notes

  • Excel's IMSQRT function always returns the principal root with a non-negative real part.

  • If the input is not a valid complex number, IMSQRT will return the #NUM! Error.

Dave Bruns Profile Picture

AuthorMicrosoft Most Valuable Professional Award

Dave Bruns

Hi - I'm Dave Bruns, and I run Exceljet with my wife, Lisa. Our goal is to help you work faster in Excel. We create short videos, and clear examples of formulas, functions, pivot tables, conditional formatting, and charts.