Excel COMPLEX Function
last modified April 4, 2025
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, practical applications, and how to work with complex numbers in Excel.
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.
| 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.
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 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
This example demonstrates using COMPLEX with array inputs to create multiple complex numbers at once. This requires Excel 365 or Excel 2021.
| A | B | C |
|---|---|---|
| 1 | 2 | |
| 3 | 4 | |
| 5 | 6 | |
| =COMPLEX(A1:A3, B1:B3) |
The table shows real parts in column A and imaginary parts in column B. The array formula in C1 creates three complex numbers at once, spilling the results down the column.
=COMPLEX(A1:A3, B1:B3)
This array formula creates three complex numbers: "1+2i", "3+4i", and "5+6i". The results spill down starting from the cell where the formula is entered. This is a powerful feature for batch processing complex numbers.
COMPLEX with Other Engineering Functions
This example shows COMPLEX being used with other engineering functions like IMSUM to perform operations on complex numbers.
=IMSUM(COMPLEX(1,2), COMPLEX(3,4))
This formula first creates two complex numbers (1+2i and 3+4i), then sums them using IMSUM. The result is "4+6i". This demonstrates how COMPLEX can be used with Excel's other engineering functions for complex number calculations.
Error Handling in COMPLEX
This example shows what happens when invalid arguments are provided to the COMPLEX function. Excel returns specific error messages.
| Formula | Result | Reason |
|---|---|---|
| =COMPLEX("a", 1) | #VALUE! | Non-numeric real part |
| =COMPLEX(1, "b") | #VALUE! | Non-numeric imaginary part |
| =COMPLEX(1, 2, "k") | #VALUE! | Invalid suffix (not "i" or "j") |
The table demonstrates various error scenarios with the COMPLEX function. Excel returns #VALUE! when inputs are non-numeric or when an invalid suffix is used. Proper error handling is important when working with complex numbers.
The COMPLEX function is essential for working with complex numbers
in Excel. From basic number creation to advanced engineering calculations,
COMPLEX provides the foundation for complex number operations.
Mastering its use enables sophisticated mathematical and engineering
calculations directly in spreadsheets.
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