IMCONJUGATE in Excel: Reverse the Imaginary Sign and Verify Products

• Introduced: the first introduction version is not established by the reviewed official history.
• Availability: official help lists Excel for Microsoft 365, Excel for Microsoft 365 for Mac, Excel 2024, Excel 2024 for Mac, Excel 2021, Excel 2021 for Mac, Excel 2019, Excel 2016.
• Checked: October 14, 2026; official IMCONJUGATE reference. Documentation and independent models; no native Excel execution or wider platform certification.

=IMCONJUGATE(inumber)

Inumber is required and uses a complex value in i or j notation. Microsoft’s equation image explicitly changes a+bi to a−bi. COMPLEX can construct that input from numeric component cells.

Conjugation reflects a point across the real axis. It leaves the real coordinate unchanged and changes the sign of the imaginary coordinate. It is not ordinary negation, which changes both signs, and it is not generally a reciprocal.

The dedicated page does not provide a complete malformed-input or coercion/error table. Work with explicit valid complex values before extending the procedure to imported text with unknown formatting.

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

The last value has no nonzero real coefficient. Its imaginary coefficient −2 becomes positive two. A simplified output need not retain an explicit zero real term from the input.

For a j-notation control, =IMCONJUGATE("4+3j") has expected value 4−3j. Use a consistent suffix convention when combining several complex values in later formulas.

For A2:A5, extract the real coefficients with =IMREAL(A2) and the imaginary coefficients with =IMAGINARY(A2). Expected sequences are 4, −2, 5, 0 and 3, 6, −1, −2.

Apply those extractors to the conjugates in B2:B5. The real sequence remains 4, −2, 5, 0; the imaginary sequence becomes −3, −6, 1, 2.

This checks the defining transformation directly. IMREAL and IMAGINARY return numeric coefficients, so the sign check does not depend on how redundant zero terms are displayed.

In C2 use =IMCONJUGATE(B2) and fill down. The reconstructed quantities are the original 4+3i, −2+6i, 5−i and −2i.

Check =IMREAL(C2)=IMREAL(A2) and =IMAGINARY(C2)=IMAGINARY(A2), filling down. Both comparisons are TRUE for each exact small record.

This is a quantity-level identity. It does not promise byte-for-byte reproduction of the original text, which may have included an explicit zero real term or a coefficient written as one before i.

For a+bi, multiplying by a−bi gives a squared plus b squared, with zero imaginary coefficient. For the four records, those real values are 25, 40, 26 and 4.

Use =IMPRODUCT(A2,B2) and fill down, then inspect both components. Expected real coefficients are 25,40,26,4 and imaginary coefficients are all zero. The guide does not infer a native storage TYPE from the simplified displayed product.

For the first record, 4×4−3×(−3)=25 and 4×(−3)+3×4=0. The result agrees with =IMABS(A2)^2. IMPRODUCT performs the complete complex multiplication; IMABS supplies the magnitude.

Consider (4+3i)/(1−2i). The denominator’s conjugate is 1+2i. Multiplying numerator and denominator by it gives a numerator −2+11i and a real denominator five.

The quotient therefore has components −2/5 and 11/5, or −0.4 and 2.2. An independent Excel control is:

=IMDIV("4+3i","1-2i")

Check its components against those fractions rather than relying on a specific full-precision text rendering. IMDIV returns the quotient; the conjugate calculation explains why the denominator becomes real.

This derivation requires a nonzero denominator. A zero complex divisor has no reciprocal, and the dedicated conjugate reference does not establish a native division-error code.

For 4+3i, the conjugate is 4−3i. Negation gives −4−3i. The reciprocal is the conjugate divided by 25, with components 0.16 and −0.12.

These three operations answer different questions. They coincide only in special cases, so do not label a conjugated column Inverse or Negative unless that meaning is justified separately.

The sum of a value and its conjugate has twice its real coefficient and zero imaginary coefficient. For 4+3i that sum has real component eight; this is another exact check of the sign reversal.

The examples use moderate exact coefficients, with fractions retained where possible. Extreme-value behavior, malformed strings, mixed suffix errors and general blank coercion remain unverified here.

The displayed outputs are independently calculated from the documented conjugate equation. Native Excel parsing, rendering and error precedence were not executed.

• Microsoft IMCONJUGATE reference.
• Editorial testing standards.

Expected outputs are independently calculated examples or documented rules, not captured native Excel results. Formulas use English function names and comma argument separators. Locale, rendering and boundary restrictions are stated where relevant.