COMPLEX in Google Sheets: Build Complex Values from Numeric Components
Create complex values in Google Sheets using real and imaginary coefficients, verify component arithmetic and use documented i or j suffix examples.
Verification status: See the evidence and test scope described in this guide. How formula examples are checked
Quick answer: COMPLEX(real_part,imaginary_part,[suffix]) constructs a complex value from its two coefficients. The imaginary suffix defaults to i; Google also documents a j-suffix example.
Availability and version history
- 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 COMPLEX reference. Documentation and independent models, with no native Sheets execution or wider client certification.
Syntax and the two coefficients
=COMPLEX(real_part,imaginary_part,[suffix])
Real_part is the real coefficient, and imaginary_part is the coefficient multiplying the imaginary unit. For a value such as 4+3i, the inputs are numeric 4 and 3. The second input is not already the text “3i”.
The optional suffix defaults to i. Google’s examples establish ordinary i output, a j suffix and simplification of a coefficient of negative one to −i. The sparse reference does not give a complete accepted-suffix list, case-sensitivity rule or invalid-input error table. This guide uses only documented lowercase i/j notation and does not import another spreadsheet’s rejection rules.
The notation i has the mathematical property i squared equals −1. A j suffix is another label used for that same imaginary unit in these examples; it does not change the coefficients.
Example 1: construct a list from numeric columns
Enter only the first two numeric columns in A1:B5. The third column below is the expected output for C2:C5, calculated with =COMPLEX(A2,B2) filled down.
| A, real coefficient | B, imaginary coefficient | C, expected value |
|---|---|---|
| 4 | 3 | 4+3i |
| -2 | 6 | -2+6i |
| 5 | -1 | 5-i |
| 0 | 2.5 | 2.5i |
The last row has no real component. Its imaginary coefficient is 2.5, so it is not the real number 2.5. In the third row, the coefficient −1 is represented by −i rather than a separate positive imaginary term.
In D2 use =COMPLEX(A2,B2,"j") and fill down. Expected values are 4+3j, −2+6j, 5−j and 2.5j. The change is notation; the real and imaginary input columns stay the same.
These are component-based expected labels, not claims about the native TYPE classification of every possible degenerate result.
Example 2: add and subtract using explicit components
Enter the real and imaginary parts of two values in F2:G3:
| F, real | G, imaginary |
|---|---|
| 4 | 3 |
| 1 | -2 |
The first value is 4+3i and the second is 1−2i. Addition combines corresponding components:
=COMPLEX(F2+F3,G2+G3)
Expected result: 5+i. Subtraction uses =COMPLEX(F2-F3,G2-G3), giving 3+5i.
This construction keeps ordinary arithmetic on the numeric coefficient cells. It does not assume that using a general plus or minus operator on the displayed complex labels implements complex arithmetic in Sheets.
Advanced example: multiply using the real and imaginary parts
For (a+bi)(c+di), distribute the terms and use i squared equals −1. The real coefficient is ac−bd, and the imaginary coefficient is ad+bc.
With the F2:G3 fixture, enter:
=COMPLEX(F2*F3-G2*G3,F2*G3+G2*F3)
The real calculation is 4×1−3×(−2)=10. The imaginary calculation is 4×(−2)+3×1=−5. Expected result: 10−5i.
The sign in the real coefficient matters. Adding bd instead of subtracting it would give −2 rather than 10, a different product. Retaining the component calculations in helper cells makes that mistake visible.
Advanced example: conjugates and a magnitude check
The conjugate of 4+3i is 4−3i, constructed with =COMPLEX(F2,-G2). Multiplying the two mathematically gives real coefficient 4×4−3×(−3)=25 and imaginary coefficient 4×(−3)+3×4=0.
The magnitude is =SQRT(F2^2+G2^2), yielding five. Its square equals 25, matching the real coefficient of the conjugate product. SQRT returns a real square root for this positive quantity; it is not being used to construct the imaginary unit.
The guide reports the numeric product components separately. It does not certify the exact display or return type of a zero-imaginary COMPLEX result, which the reviewed page does not fully specify.
Advanced example: a quarter-turn in component form
Multiplication by i transforms a+bi into −b+ai. For F2:G2, use =COMPLEX(-G2,F2), giving −3+4i. Applying the transformation again gives −4−3i, then 3−4i, then the original 4+3i.
All four states have squared magnitude 25. This independent geometric control checks the component signs and order without relying on unreviewed complex-function defaults.
Input and verification limits
Keep real and imaginary coefficients numeric, and use one consistent suffix in an exported dataset. The dedicated reference links to component-extraction functions, but their detailed contracts are separate from this construction guide.
Uppercase suffixes, arbitrary alternate labels, numeric-text coercion, invalid coefficient errors and zero-only output classification remain unverified here. Native complex rendering and locale-dependent punctuation were not executed. The worked arithmetic uses exact small coefficients and explicitly documented lowercase notation.
References and verification
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.