IMSQRT in Google Sheets: Take Complex Square Roots and Check Both Components

• Introduced: first introduction date is not established by the reviewed official history; no desktop release number is inferred.
• Availability: documented in the rolling Google Sheets service; these examples use the web worksheet workflow.
• Checked: October 14, 2026; official IMSQRT reference. Documentation and independent models, with no native Sheets execution or wider client certification.

=IMSQRT(complex_number)

The input is required. Google illustrates numeric four, complex text and a COMPLEX result. Its j-suffix example 3−4j returns 2−j.

Unlike ordinary SQRT, IMSQRT can operate on a complex number with a negative real component. However, the reviewed body does not specify a complete branch interval or negative-axis endpoint convention. Those details must not be borrowed silently from another product.

The worked output models below use positive-real-component inputs with the root having positive real coefficient, consistent with the explicit documented examples. Negative-axis selection and a complete native error table remain outside the verified scope.

Enter only the first column as Plain text in A2:A5. Put =IMSQRT(A2) in B2 and fill down.

The first result checks because (3+2i) squared has real coefficient 9−4=5 and imaginary coefficient 2×3×2=12. Reversing the imaginary sign gives 5−12i.

For the third row, 3 squared minus 1 squared is eight, and twice 3×1 is six. The fourth row preserves the positive real root coefficient and reverses the imaginary sign.

For direct documentation controls, =IMSQRT(4) returns two, =IMSQRT("3+4i") returns 2+i and =IMSQRT("3-4j") returns 2−j. These examples do not establish every other parsing or branch case.

In C2 use =IMPRODUCT(B2,B2) and fill down. Expected squared quantities are exactly the original input quantities in A2:A5 under the mathematical model.

Check the real and imaginary components separately:

=IMREAL(C2)-IMREAL(A2)
=IMAGINARY(C2)-IMAGINARY(A2)

All eight expected differences are zero for these simple fixtures. A complex result can have the correct magnitude and still have the wrong direction, so checking only its size is insufficient.

IMPRODUCT carries out the complex square. Alternatively, extract root coefficients u and v and check u squared minus v squared against the input real coefficient, and 2uv against its imaginary coefficient.

For =IMSQRT("2+i"), an independent positive-real-root model has real coefficient approximately 1.455346690225 and imaginary coefficient approximately 0.343560749723.

The magnitude of the original input is the square root of five. The modeled root coefficients can be derived as the square roots of (square root of five plus two)/2 and (square root of five minus two)/2, respectively.

Their squared difference is two and their doubled product is one. The root magnitude is the fourth root of five, approximately 1.495348781, rather than the original magnitude of approximately 2.236067977.

To check an actual workbook result stored in D2, use =ABS(IMREAL(D2)^2-IMAGINARY(D2)^2-2)<0.000000001 and =ABS(2*IMREAL(D2)*IMAGINARY(D2)-1)<0.000000001. These are explicit tolerances for this modest example, not guarantees at every scale.

Both 3+2i and −3−2i square to 5+12i. A single IMSQRT result cannot return both quantities in one scalar cell. Negating the modeled root produces the second mathematical root.

Consequently, taking a square root after squaring does not generally recover the original signed complex value. Starting with −3−2i, squaring produces 5+12i; the positive-real-root model used above selects 3+2i instead.

This loss of sign information is a mathematical limitation, not proof that the original negative value was invalid. Preserve the original input if subsequent work depends on which root was intended.

Google explicitly permits complex inputs with negative real components. For example, the independently calculated quantities 2+3i and −2−3i both square to −5+12i.

That identity proves they are mathematical roots, but the sparse dedicated reference does not by itself establish which native representative is returned for every such input or at a negative-axis boundary. No expected native output is assigned to those disputed boundary cases here.

The documented equivalence with a complex power of 0.5 is useful, but it does not supply a missing branch specification. Exact invalid-text errors, zero-boundary formatting and extreme precision behavior remain unverified. No native Sheets execution was performed.

• Google IMSQRT reference.
• Editorial testing standards.

Examples use fictional data, English function names and comma separators. Expected outputs are independent mathematical calculations or explicitly documented rules, not native Sheets captures. Check the spreadsheet locale when reproducing punctuation-sensitive formulas.