IMAGINARY in Google Sheets: Read Signed Imaginary Coefficients

• 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 IMAGINARY reference. Documentation and independent models, with no native Sheets execution or wider client certification.

=IMAGINARY(complex_number)

Google describes a+bi or a+bj notation. Its examples include a mixed complex value, a pure-imaginary j value and the quoted real string “45”, whose imaginary coefficient is zero.

That last example establishes the documented numeric-text control. It is not a complete policy for every Boolean, blank, malformed string or locale-dependent numeric input. Keep these distinctions visible instead of assuming all spreadsheet coercions are interchangeable.

A coefficient is a signed real number. For a value written 6−2i, the imaginary coefficient is −2; the imaginary term is that coefficient multiplied by the imaginary unit.

Enter only the first column as Plain text in A2:A5. Put =IMAGINARY(A2) in B2 and fill through B5.

=SUM(B2:B5) returns three. The negative coefficients reduce the total; using their absolute values instead would produce fifteen and answer a different question.

The last record is purely imaginary. Its missing explicit real term does not change its imaginary coefficient. For Google’s separate j control, =IMAGINARY("4j") returns four.

=IMAGINARY("45") has Google’s documented result zero. It does not mean that the source is empty or invalid; it is a real-only quantity with no imaginary contribution.

For numeric component cells, enter seven in D2 and zero in E2. The formula =IMAGINARY(COMPLEX(D2,E2)) has independently expected coefficient zero. COMPLEX makes the component construction explicit.

A genuinely missing cell or formula-produced empty text is a different input. This guide does not assign those cases the same behavior as a real-only complex value without function-specific evidence.

For the first fixture, use =IMREAL(A2) in C2 and fill down. Expected real coefficients are 2, −3, 6 and 0.

In F2 enter =COMPLEX(C2,B2) and fill down. The reconstructed quantities are 2+5i, −3+4i, 6−2i and −4i. Check their real and imaginary coefficients against C and B rather than expecting arbitrary original spacing or redundant zero terms to be reproduced.

IMREAL supplies the other coordinate. Values 2+5i and −3+5i share imaginary coefficient five but have different real components, so B alone cannot identify the full record.

Apply =IMAGINARY(IMCONJUGATE(A2)) to the four records. Expected results are −5, −4, 2 and 4.

The original and conjugated imaginary coefficients sum to zero for every row. Their real coefficients remain unchanged. IMCONJUGATE reverses the sign of one component rather than making that component universally positive.

This check also distinguishes conjugation from ordinary negation, which would reverse both components. Keep the original source when later calculations need to know which transformation was applied.

For a+bi, multiplication by i gives −b+ai. Therefore =IMPRODUCT("2+5i","i") has expected quantity −5+2i, and its imaginary coefficient is two.

Applying the rotation again gives −2−5i, whose imaginary coefficient is −5. The operation exchanges the roles of the components and changes a sign; it does not merely retain or multiply the original imaginary coefficient by one.

IMPRODUCT performs the complete multiplication. The example is a transparent coordinate transformation, not a claim that all complex operations act independently on the two columns.

For (2+5i)(3−i), the imaginary coefficient is 2×(−1)+5×3=13. The real coefficient is eleven, so the complete product is 11+13i.

=IMAGINARY(IMPRODUCT("2+5i","3-i")) has expected result thirteen. Multiplying just the two imaginary coefficients would give −5, which is not the imaginary coefficient of the product.

The dedicated source does not provide a general invalid-string error table, array-expansion contract or precision ceiling. The examples use exact small coefficients, explicit lowercase notation and independently calculated outputs. Native Sheets parsing, display and error precedence were not executed.

• Google IMAGINARY 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.