Zoho Sheet IMSQRT: Square Roots of Complex Numbers

• Introduced: The original release and date are not established by the checked official sources.
• Availability: Currently documented in Zoho Sheet’s rolling web-service help; there is no verified final supported-version endpoint.
• Scope: The English cloud reference does not establish identical behavior in every mobile client or imported workbook.
• Checked: October 14, 2026, against the official IMSQRT reference. Documentation and independent arithmetic checks; no native Zoho Sheet test.

=IMSQRT(complex_number)

All arguments are required; no optional parameter or default is documented in the official IMSQRT reference.

Use a fresh worksheet for each function so the results in column C do not overwrite another exercise. Enter the following in A1:B10, or import the two-column practice CSV at A1. Keep column A as text and column B as numbers. The literal text does not include enclosing quotation marks in cells.

Rows 7, 8 and 10 are optional boundary inputs, not part of the worked result checks. In particular, do not fill IMLN through the zero row and assume it has a logarithm.

A complex number has a real part a and an imaginary part b. If w=u+vi, then w²=(u²−v²)+2uvi. To solve w²=a+bi, both u²−v²=a and 2uv=b must hold. These two equations explain why a complex square root is not found by separately taking square roots of the real and imaginary components.

Let r=√(a²+b²). The mathematical principal square root can be written with u=√((r+a)/2) and signed v=sign(b)·√((r−a)/2) for nonzero b. For b=0 on a negative real axis, the usual principal root has nonnegative imaginary part. The other mathematical root is −u−vi. The official function page does not present a formal branch-cut policy, so the principal-root language here describes mathematics and the listed examples, not a universal tested engine guarantee.

These results come from the official example table; the checks are separate complex-number calculations.

With the practice inputs above in columns A:B, leave them unchanged. Type the following into column C:

A5 is 3+4j and A9 is -4i in the practice data. For C5, 2+j is a root because (2+j)²=4+4j−1=3+4j. For C9, squaring √2−√2i produces −4i.

One possible mathematical audit is to square the computed root: =IMPRODUCT(C5;C5). A more direct comparison is =IMPOWER(C5;2). The mathematical expected result is 3+4j for both formulas. Floating-point rounding may produce tiny residuals rather than exact zeros, so compare real and imaginary components numerically if you implement an automated audit.

You can also compare magnitudes: for any mathematical square root w of z, |w|²=|z|. With A5=3+4j, |z|=5 and |2+j|²=5. In a sheet, a proposed check is =IMABS(C5)^2 against =IMABS(A5). Zoho documents IMABS, but both expressions have mathematical expectation 5 for this input.

AC power and phasor algebra training

Complex square roots occur when rearranging relationships involving impedances and frequency-dependent terms. In a teaching workbook, store the complex expression of interest in A2 and calculate =IMSQRT(A2) in C2. Record whether the model expects the mathematical principal solution or the alternative negative root; selecting a physically meaningful branch depends on the real system, not simply a spreadsheet’s default.

Polynomial equations with complex discriminants

For a quadratic ax²+bx+c=0, the discriminant D=b²−4ac may be negative. Complex square roots allow the quadratic formula to be interpreted in complex arithmetic. Prepare the discriminant as a complex-number text with a documented notation, then use IMSQRT on that single value. This does not establish that Zoho’s scalar arithmetic operators automatically produce complex text; construct and validate the appropriate representation first, for example with documented COMPLEX.

Reciprocal relation to powers

IMSQRT(z) and IMPOWER(z;0.5) both express a square root mathematically. Branches matter for negative real inputs, and the IMPOWER documentation contains a separate suffix inconsistency. Test the two on positive, negative and mixed-complex inputs rather than treating identical names as a guarantee of the same string result. IMPOWER reference.

Quality control using symmetry

A nonzero complex number has two square roots: w and −w. Their squared values are equal, and their magnitudes match. If you use square roots in an engineering model, store assumptions about which branch was chosen and avoid silently switching signs between time steps. Neither the vendor example set nor these independent equations prove a particular branch-handling algorithm in every version of Zoho Sheet.

• Negative real input: The mathematical principal square root of −9 is +3i, but Zoho’s published examples do not identify a supported literal representation for −9 or assert exactly how this boundary is serialized. Test "-9+0i" and "-9" separately.
• Zero input: The mathematical root is zero, but the exact accepted string forms of zero and any display simplification have not been confirmed through native use.
• Scientific notation and uppercase units: The reference says text representing a complex number; it does not provide a complete grammar, case-conversion rules, or exponent-notation rules.
• Sign and branch boundary: At b=0, the sign(b) formula needs a branch convention. Treat values near the negative real axis carefully; floating-point noise can change which side of a branch cut the input occupies.
• Arrays: IMSQRT’s documented argument is singular. Whole-range execution and spill behaviour are not established by the cited page.
• Integer/decimal formatting: Zoho’s locale controls affect decimal separators and format conventions. Reproduce its English examples first; verify any locale-dependent text parsing before automating bulk calculations.
• Very large coefficients: The numeric range, number of significant digits and overflow/underflow thresholds are not established on the reference page.

Why are there two square roots but one returned value? The mathematical equation w²=z has two roots for nonzero z; a square-root function conventionally chooses one branch. Zoho’s examples show one answer each but do not fully document branch selection outside them.

Can I replace this with SQRT? Zoho’s SQRT documentation concerns the positive square root of a nonnegative real number. IMSQRT is the documented choice for text-form complex inputs.

Is the result a number or text? The input is documented as text representing a complex number, and the examples display complex-number strings. Do not assume a native numeric complex data type or downstream type coercion without application tests.

How can I check the answer? Square the predicted root by ordinary complex arithmetic; in a live workbook, independently test the suggested IMPRODUCT or IMPOWER comparison, allowing a floating-point tolerance.

Quote complex literals, such as "2+3j", but do not quote a reference such as A2. Multi-argument formulas here use the semicolons printed by Zoho’s English help; follow your workbook’s formula editor if its syntax differs.

The official page lists broad error families: inspect spelling and quotation marks for #NAME!, input types for #VALUE!, and cell references for #REF!. Its generic list does not establish the exact error returned for each malformed complex input. Avoid applying unverified blank-to-zero or uppercase-suffix conversions.

• COMPLEX constructs a complex representation.
• IMABS extracts magnitude.
• IMREAL and IMAGINARY let you inspect the two coefficients.

Compare independently expected coefficients within a suitable numerical tolerance. Printed precision and simplified zero components can vary; the examples do not establish a universal string-serialization rule.