IMCONJUGATE in Google Sheets: Reverse Imaginary Signs and Check Division

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

=IMCONJUGATE(number)

Google’s equation image defines conjugation as a+bi becoming a−bi. There is one required input. Its examples include ordinary i notation and a j-suffixed value constructed with COMPLEX.

The sign change acts on the numeric imaginary coefficient. If that coefficient is already negative, its conjugate coefficient is positive. Avoid a general text replacement that changes every minus or plus character, because the real coefficient must stay unchanged.

The examples below check numeric components rather than requiring a particular spelling for zero or a coefficient of one. Google’s dedicated page does not supply a complete error, whitespace or import-coercion table.

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

The second record retains its negative real coefficient. The third and fourth records reverse a negative imaginary coefficient, so their conjugates have positive imaginary coefficients.

Use =IMREAL(B2) and =IMAGINARY(B2) in C2 and D2, filling down. Expected real coefficients are 2, −3, 6 and 0. Expected imaginary coefficients are −5, −4, 2 and 4.

For a separate documented notation control, =IMCONJUGATE(COMPLEX(-2,-5,"j")) returns the quantity −2+5j. Keep one suffix convention within later multi-input calculations.

In E2 enter =IMCONJUGATE(B2) and fill down. Each result has the original real and imaginary coefficients.

Check both coordinates explicitly:

=IMREAL(E2)=IMREAL(A2)
=IMAGINARY(E2)=IMAGINARY(A2)

Both comparisons are TRUE for each of these exact small-integer fixtures. They establish quantity equality in the model, without requiring a returned pure-imaginary value to include an explicit zero real term.

IMREAL and IMAGINARY are useful when a displayed complex string hides which part changed. Keeping these helper columns also makes incorrect input signs easier to locate.

Put =IMPRODUCT(A2,B2) in F2 and fill through F5. The expected real coefficients are 29, 25, 40 and 16, and every imaginary coefficient is zero.

The identity follows by expanding the product: (a+bi)(a−bi)=a squared plus b squared. For the first record, 2 squared plus 5 squared is 29.

Compare =IMREAL(F2) with =IMABS(A2)^2. They represent the same mathematical quantity. IMABS involves a square root, so a tolerance is more appropriate than a universal exact binary-equality claim for arbitrary data.

A product with zero imaginary coefficient may display as a real-looking result. This guide does not infer the native storage TYPE of every degenerate complex result; the numeric component check is the intended test.

To divide 2+5i by 3−i, multiply both numerator and denominator by the conjugate 3+i. The new numerator is 1+17i and the new denominator is ten.

Reproduce the intermediate values:

=IMPRODUCT("2+5i",IMCONJUGATE("3-i"))
=IMPRODUCT("3-i",IMCONJUGATE("3-i"))
=IMDIV("2+5i","3-i")

Expected quantities are 1+17i, a value with real coefficient ten and imaginary coefficient zero, and 0.1+1.7i. The direct IMDIV result agrees with dividing both numerator coefficients by ten.

This derivation explains the role of conjugation. It does not make division by zero valid: a zero denominator still has magnitude squared zero.

For 2+5i, the conjugate is 2−5i, the negative is −2−5i, and the reciprocal has real coefficient 2/29 and imaginary coefficient −5/29. Only the first operation leaves the real coefficient unchanged.

Adding a value to its conjugate produces twice its real coefficient: =IMSUM("2+5i",IMCONJUGATE("2+5i")) has real coefficient four and imaginary coefficient zero.

Use the operation that matches the question. Conjugation is a reflection across the real axis, not subtraction from zero or inversion. Invalid-string behavior, extreme numeric precision and unlisted coercions remain unverified. No native Sheets execution was performed.

• Google IMCONJUGATE 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.