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(complex_num)
  • complex_num - The complex number in the form "x+yi".

How to use 

The Excel IMSQRT function returns the square root of a complex number. For example, given "-1" the function returns "6.12323E-17 + i" as output.

=IMSQRT(COMPLEX(-1,0)) // returns 6.12323E-17 + i

Due to floating-point precision, the function returns a very small value for the real part of the output when, in reality, the output should just be "i".

Explanation

Taking a complex number's square root is the same as raising the complex number to 1/2 power.

Square root as an exponent.

In Excel, we can write a formula that is equivalent to the complex square root function like this, where z is a complex number.

=IMPOWER(z,COMPLEX(1/2,0)) // equivalent to IMSQRT(z)

The complex square root is defined by writing the complex number in its polar form and taking the square root of both sides.

Definition of the complex square root.

After simplifying, we get the expression equal to this formula in Excel, where "z" is a complex number.

=IMPRODUCT(
    COMPLEX(SQRT(IMABS(z)), 0),
    IMEXP(COMPLEX(0, IMARGUMENT(z)/2)))
) // equivalent to IMSQRT(z)

The real part of the function's output plotted over the complex plane looks like this.

Real part of the complex square root plotted over the complex plane.

The imaginary part of the function's output plotted over the complex plane looks like this.

Imaginary part of the complex square root plotted over the complex plane.

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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.