Excel COMPLEX Function
last modified October 5, 2026
The COMPLEX function converts real and imaginary coefficients into
a complex number in Excel. Complex numbers are used in engineering, physics,
and advanced mathematics. This tutorial provides a comprehensive guide to using
the COMPLEX function with detailed examples. You'll learn basic
syntax, how the result text is formatted, edge cases such as zero and unit
coefficients, and how to feed the result into Excel's other complex number
functions.
COMPLEX Function Basics
The COMPLEX function creates a complex number from real and
imaginary coefficients. Complex numbers are expressed in the form x + yi or
x + yj. Excel uses either "i" or "j" as the imaginary unit and returns the
result as text, not as a number.
| Component | Description |
|---|---|
| Function Name | COMPLEX |
| Syntax | =COMPLEX(real_num, i_num, [suffix]) |
| Arguments | real_num (required), i_num (required), suffix (optional) |
| Return Value | Text representation of a complex number |
This table breaks down the essential components of the COMPLEX
function. It shows the function name, syntax format, required and optional
arguments, and the type of value returned.
| Argument | Meaning |
|---|---|
| real_num | The real coefficient. Required. It can be a literal, a cell reference, or an expression that returns a number. |
| i_num | The imaginary coefficient. Required. It is the number that multiplies the imaginary unit in the result. |
| suffix | The imaginary unit: lowercase "i" or "j".
Optional; when omitted, Excel uses "i". |
The suffix is case sensitive. Only lowercase "i" and
"j" are accepted; uppercase "I" or "J"
returns #VALUE!. Functions that combine two or more complex
numbers, such as IMSUM, require every value to use the same
suffix.
Basic COMPLEX Example
This example demonstrates the simplest use of the COMPLEX function with basic numeric inputs.
=COMPLEX(3, 4)
This formula creates a complex number with 3 as the real part and 4 as the imaginary part. The result will be "3+4i". When the suffix is omitted, Excel defaults to using "i" as the imaginary unit.
COMPLEX with Different Suffix
This example shows how to specify a different suffix for the imaginary unit. Engineers often use "j" instead of "i".
=COMPLEX(5, 2, "j")
This formula creates a complex number with 5 as the real part and 2 as the imaginary part, using "j" as the suffix. The result will be "5+2j". The suffix must be enclosed in quotes and can only be "i" or "j".
COMPLEX with Negative Imaginary Part
This example demonstrates how COMPLEX handles negative imaginary coefficients. The function automatically formats the output correctly.
=COMPLEX(7, -3)
This formula creates a complex number with 7 as the real part and -3 as the imaginary part. The result will be "7-3i". The function automatically uses the correct operator (+ or -) based on the sign of the imaginary coefficient.
COMPLEX with Zero Real Part
This example shows how COMPLEX handles cases where the real part is zero. The output format changes to show only the imaginary part.
=COMPLEX(0, 5)
This formula creates a complex number with 0 as the real part and 5 as the imaginary part. The result will be "5i". When the real part is zero, Excel omits it from the display, showing only the imaginary component.
COMPLEX with Zero Imaginary Part
This example demonstrates how COMPLEX handles cases where the imaginary part is zero. The output shows only the real number.
=COMPLEX(4, 0)
This formula creates a complex number with 4 as the real part and 0 as the imaginary part. The result will be "4". When the imaginary part is zero, Excel returns just the real number as a text string.
COMPLEX with Unit and Negative Coefficients
Excel shortens the result whenever a coefficient is zero or the imaginary
coefficient is 1 or -1. Knowing these compact forms matters, because the text
that COMPLEX returns is exactly what the other complex number
functions parse.
| Formula | Result | Note |
|---|---|---|
| =COMPLEX(0,0) | 0 | Both parts are zero |
| =COMPLEX(0,1) | i | Unit imaginary part loses its digit |
| =COMPLEX(0,-1) | -i | The sign is kept |
| =COMPLEX(4,1) | 4+i | Real part plus a unit imaginary part |
| =COMPLEX(4,-1) | 4-i | Negative unit imaginary part |
| =COMPLEX(0,-5) | -5i | Zero real part with a negative imaginary |
| =COMPLEX(-3,4) | -3+4i | Negative real part |
| =COMPLEX(-3,-4) | -3-4i | Both parts negative |
Notice that the sign of the imaginary part is always written explicitly, so
COMPLEX(7,-3) becomes 7-3i rather than
7+-3i. The same rule applies to the real part:
COMPLEX(-3,4) returns -3+4i. These short forms are
valid input for the IM functions, so IMABS("i") returns 1 and
IMREAL("-3-4i") returns -3.
COMPLEX with Decimal Values
This example shows how COMPLEX handles decimal numbers for both real and imaginary parts. The function maintains the decimal precision in the output.
=COMPLEX(2.5, 1.75)
This formula creates a complex number with 2.5 as the real part and 1.75 as the imaginary part. The result will be "2.5+1.75i". The function preserves the decimal places from both input values in the output string.
COMPLEX in Array Formulas
COMPLEX expects scalar coefficients, so a bare range has to be
coerced before it can be processed element by element. Adding zero to each
range turns it into an array of numbers, and the formula then spills one
complex number per row in Excel 365, Excel 2021, and Excel 2024.
| A | B | C |
|---|---|---|
| 1 | 2 | =COMPLEX(A1:A3+0, B1:B3+0) |
| 3 | 4 | 3+4i |
| 5 | 6 | 5+6i |
The table shows real parts in column A and imaginary parts in column B. The
formula in C1 returns 1+2i and spills down to C3.
=COMPLEX(A1:A3+0, B1:B3+0)
This array formula creates three complex numbers: "1+2i", "3+4i", and "5+6i".
Without the +0 coercion the range is treated as a single value and the formula
returns one #VALUE! error. In Excel 2019 and earlier, confirm the
formula with Ctrl+Shift+Enter so that it is evaluated as an array.
COMPLEX with Other Complex Functions
COMPLEX produces text, but that text is understood by every
function in Excel's complex number family. This makes it easy to build values
from separate real and imaginary parts and then operate on them.
=IMSUM(COMPLEX(1,2), COMPLEX(3,4)) ' 4+6i =IMPRODUCT(COMPLEX(1,2), COMPLEX(3,4)) ' -5+10i =IMABS(COMPLEX(3,4)) ' 5 =IMREAL(COMPLEX(3,4)) ' 3 =IMAGINARY(COMPLEX(3,4)) ' 4
IMSUM adds the two complex numbers, and IMPRODUCT
multiplies them: (1+2i)(3+4i) = -5+10i. IMABS returns the
magnitude sqrt(3^2+4^2) = 5, while IMREAL and
IMAGINARY pull the two coefficients back out as ordinary numbers.
All of these functions accept either the text produced by COMPLEX
or complex-number text typed directly, such as "3+4i".
COMPLEX Errors and Troubleshooting
Most COMPLEX problems come from non-numeric coefficients or from a
suffix that Excel does not accept. The table below lists the common causes and
their fixes.
| Cause | Symptom | Fix |
|---|---|---|
| real_num is non-numeric text | #VALUE! error | Pass a number, or a cell that contains one |
| i_num is non-numeric text | #VALUE! error | Pass a number, or a cell that contains one |
| suffix is neither "i" nor "j" | #VALUE! error | Use exactly "i" or "j" |
| suffix is uppercase "I" or "J" | #VALUE! error | Switch to lowercase; the suffix is case sensitive |
| Mixed suffixes in IMSUM or IMPRODUCT | #VALUE! error | Use the same suffix for every complex number |
| A required argument is omitted | #VALUE! or a "too few arguments" warning | Supply both real_num and i_num |
| A coefficient cell is blank | The blank is treated as 0 | Fill the cell, or coerce it with +0 if a blank is not wanted |
| A whole range is passed to COMPLEX | A single #VALUE! error | Coerce the range with +0 so it spills element by element |
The COMPLEX function is essential for working with complex numbers
in Excel. It turns separate real and imaginary coefficients into the exact text
format that IMSUM, IMPRODUCT, IMABS,
IMREAL, and IMAGINARY expect. Remember the two
details that trip people up most: the result is text, so it cannot be used in
ordinary arithmetic, and the suffix is case sensitive and must match across
values that are combined. Master these rules, and COMPLEX becomes a
reliable foundation for engineering and mathematical work in spreadsheets.
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