IMDIV in Google Sheets: Divide Complex Values and Verify Quotients

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

=IMDIV(dividend,divisor)

Both inputs are required. Google documents matching i or j suffixes and gives a separate example with a numeric real divisor. Complex operands with different suffixes cannot be divided together.

The equation image defines division of a+bi by c+di using denominator c squared plus d squared. The real numerator is ac+bd, and the imaginary numerator is bc−ad.

A zero divisor makes that denominator zero, so the mathematical quotient is undefined. The reviewed reference does not identify an exact native zero-divisor error code; this guide does not import one from another spreadsheet product.

Enter only the first two columns below as Plain text in A2:B4. Use =IMDIV(A2,B2) in C2 and fill down.

The first and second rows share a divisor but have different dividends. Reusing the same denominator calculation does not mean their quotient components will match.

As a documented numeric-divisor control, =IMDIV("4+2j",2) gives the quantity 2+j. A purely real divisor scales both coefficients by its reciprocal; it does not alter only the real coefficient.

For the second data row, A3 is 2+5i and B3 is 3−i. Their denominator is 3 squared plus (−1) squared, which equals ten.

Use these formulas in D3, E3 and F3:

=IMREAL(B3)^2+IMAGINARY(B3)^2
=(IMREAL(A3)*IMREAL(B3)+IMAGINARY(A3)*IMAGINARY(B3))/D3
=(IMAGINARY(A3)*IMREAL(B3)-IMREAL(A3)*IMAGINARY(B3))/D3

Expected results are ten, 0.1 and 1.7. The numerator checks are 2×3+5×(−1)=1 and 5×3−2×(−1)=17.

IMREAL and IMAGINARY provide numeric coefficients for this audit. The helper formula is a transparent model for these small values, not a guarantee that squaring huge coefficients is numerically safe.

The conjugate of 3−i is 3+i. Multiplying the second dividend 2+5i by 3+i gives 1+17i. Multiplying 3−i by 3+i gives a value with real coefficient ten and imaginary coefficient zero.

Reproduce those intermediates:

=IMPRODUCT(A3,IMCONJUGATE(B3))
=IMPRODUCT(B3,IMCONJUGATE(B3))

Dividing the new numerator’s coefficients by ten explains the result 0.1+1.7i. IMCONJUGATE removes the denominator’s imaginary term through this multiplication identity.

This derivation helps distinguish the correct plus sign in ac+bd from the minus sign used in ordinary complex multiplication’s real coefficient.

Reversing the second row, =IMDIV(B3,A3), has real coefficient 1/29 and imaginary coefficient −17/29, approximately 0.034482759 and −0.586206897.

That is not the original quotient with its sign reversed. Multiplying the two reciprocal quotients gives the mathematical value one, because both original operands are nonzero.

For every row in the first table, =IMPRODUCT(C2,B2) and the corresponding filled formulas recover A2:A4. Expected recovered component pairs are (11,13), (2,5) and (1,7).

Use a tolerance for noninteger recovery checks. The exact rational model establishes the identity, while displayed decimal strings may be shortened and native floating arithmetic has not been executed here.

Multiplying both operands of the second row by two gives 4+10i divided by 6−2i. The expected quotient is still 0.1+1.7i.

The component equation confirms why: every numerator product and the squared denominator gain a factor of four, which cancels. Scaling only the dividend or only the divisor would change the quotient.

For a reciprocal, construct the real dividend explicitly as =IMDIV(COMPLEX(1,0),"2+5i"). Expected coefficients are 2/29 and −5/29. This uses the dedicated reference’s constructed-complex workflow without assuming every numeric-dividend coercion rule from its separate numeric-divisor example.

A zero denominator has no reciprocal. Keep it distinguishable from a legitimate zero dividend divided by a nonzero value. Do not silently replace all failed divisions with zero, because malformed complex text and invalid source records are separate problems.

Suffix mismatches are documented as invalid, but exact error displays, general blank/coercion behavior, and very large or tiny coefficient stability remain unverified. No native Sheets execution was performed.

• Google IMDIV 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.