Excel IMSUM and IMPRODUCT Functions

last modified October 5, 2026

The IMSUM and IMPRODUCT functions are Excel's specialized tools for adding and multiplying complex numbers. A complex number is written as text in the form x+yi or x+yj, where x is the real part and y is the imaginary part. This tutorial provides a comprehensive guide to both functions with detailed examples. You'll learn the syntax, how Excel formats the result, the rule that the imaginary suffix must match across arguments, and how to combine the functions with COMPLEX and with cell references.

IMSUM/IMPRODUCT Function Basics

The IMSUM function adds complex numbers, while IMPRODUCT multiplies them. Both functions read complex numbers from text and return a complex number as text, so the result cannot be used directly in ordinary arithmetic.

Function	Description	Syntax
IMSUM	Returns the sum of complex numbers	=IMSUM(inumber1, [inumber2], ...)
IMPRODUCT	Returns the product of complex numbers	=IMPRODUCT(inumber1, [inumber2], ...)

Both functions accept 1 to 255 complex numbers and return a complex number as text. The table below describes the shared argument.

Argument	Meaning
inumber1	The first complex number to combine. Required. It can be text such as "3+4i", a cell reference, a value returned by COMPLEX, or a real number such as 30.
inumber2, ...	Additional complex numbers. Optional. Up to 254 more values, for a total of 255. They may be literals, cell references, ranges, or arrays.

Real numbers are accepted and treated as complex numbers with a zero imaginary part, so 5 is understood as 5+0i.

Adding and Multiplying Complex Numbers

Both functions follow the standard algebra of complex numbers. For two numbers a+bi and c+di, Excel adds the real parts and the imaginary parts separately, and for multiplication it applies the distributive rule and simplifies i squared to -1.

Operation	Rule	Example
Addition	(a+bi) + (c+di) = (a+c) + (b+d)i	(3+4i) + (5-3i) = 8+i
Multiplication	(a+bi)(c+di) = (ac-bd) + (ad+bc)i	(3+4i)(5-3i) = 27+11i

For the product, the real part is ac-bd because the two imaginary terms combine into -bd, and the imaginary part is ad+bc. Excel performs these steps internally and writes the answer back in x+yi form.

Basic IMSUM Example

This example demonstrates the simplest use of the IMSUM function with two complex numbers.

=IMSUM("3+4i", "1+2i")

This formula adds two complex numbers: 3+4i and 1+2i. The result will be "4+6i". The function adds the real parts (3+1) and imaginary parts (4+2) separately.

Basic IMPRODUCT Example

This example shows the basic multiplication of two complex numbers using IMPRODUCT.

=IMPRODUCT("2+3i", "1+4i")

This formula multiplies 2+3i by 1+4i. The result is "-10+11i" because: (2*1 - 3*4) + (2*4 + 3*1)i = (2-12) + (8+3)i = -10 + 11i.

IMSUM with Cell References

A more practical use involves adding complex numbers stored in cells. Here's an example with cell references.

A	B
5+2i
3+7i
	=IMSUM(A1, A2)

The table shows complex numbers in column A and an IMSUM formula in cell B3 that adds them. The result will be "8+9i" (5+3 real parts, 2+7 imaginary).

=IMSUM(A1, A2)

This formula sums complex numbers in cells A1 and A2. Using cell references makes the function more flexible for working with spreadsheet data.

IMPRODUCT with Multiple Numbers

IMPRODUCT can multiply several complex numbers at once. This example shows multiplication of three complex numbers.

=IMPRODUCT("1+i", "2+3i", "1-i")

This formula multiplies three complex numbers step by step. First 1+i and 2+3i give -1+5i, then multiplied by 1-i gives 4+6i. The final result is "4+6i".

IMSUM with Real Numbers

IMSUM can handle regular numbers by treating them as complex numbers with zero imaginary part. This example demonstrates this behavior.

=IMSUM("5", "3+2i")

The formula adds 5 (treated as 5+0i) and 3+2i. The result is "8+2i". This shows how IMSUM seamlessly handles mixing real and complex numbers.

IMPRODUCT accepts real numbers the same way. The next formula multiplies 1+2i by the real number 30, which is treated as 30+0i.

=IMPRODUCT("1+2i", 30)

The result is "30+60i", because 30*(1+2i) scales both coefficients. This matches the example given in the Microsoft documentation for IMPRODUCT.

IMPRODUCT with COMPLEX

COMPLEX builds a complex number from separate real and imaginary coefficients and returns exactly the text that IMPRODUCT and IMSUM expect. This is convenient when the coefficients are stored in cells or computed by other formulas.

=IMPRODUCT(COMPLEX(3,4), COMPLEX(1,2))

COMPLEX(3,4) produces "3+4i" and COMPLEX(1,2) produces "1+2i", so the formula multiplies the same values as direct text input. The result is "-5+10i", because (3+4i)(1+2i) = 3+6i+4i+8i squared = 3+10i-8 = -5+10i.

IMSUM with Array of Numbers

IMSUM can work with arrays of complex numbers. This example demonstrates summing multiple complex values.

=IMSUM({"1+2i","3+4i","5+6i"})

This formula sums three complex numbers in an array. The result is "9+12i" (1+3+5 real parts, 2+4+6 imaginary). Array input makes bulk operations convenient.

The Matching Suffix Rule

The imaginary unit may be written as i or j, but every complex number passed to a single call must use the same suffix. Mixing them returns #VALUE!. The suffix is also case sensitive, so only lowercase i and j are accepted.

Formula	Result
=IMSUM("3+4i", "5-3i")	8+i
=IMSUM("3+4j", "5-3j")	8+j
=IMSUM("3+4i", "5-3j")	#VALUE! (suffixes differ)

The result uses the common suffix of the arguments. When the suffixes do not match, there is no single imaginary unit to report, so Excel returns #VALUE! instead.

How Excel Formats the Result

The result is always text in x+yi form, and Excel shortens it in a few cases. A coefficient of 1 on the imaginary part loses its digit, a zero real part is dropped, and a zero imaginary part leaves only the real number. Signs are written explicitly, so an imaginary part of 1 appears as +i and an imaginary part of -1 as -i.

Formula	Result	Note
=IMSUM("2i", "3i")	5i	Zero real part is dropped
=IMSUM("3+4i", "5-3i")	8+i	A unit imaginary part loses its 1
=IMPRODUCT("3+4i", "5-3i")	27+11i	Both parts are non-zero
=IMPRODUCT("1+2i", "3+4i")	-5+10i	Negative real part keeps its sign
=IMSUM("3+4i", "-3-4i")	0	Both parts cancel to zero

Error Handling

Both functions return errors for invalid complex number formats. This example shows error handling.

=IMSUM("3+4i", "invalid")

This formula returns the #VALUE! error because the second argument is not a valid complex number. Proper formatting like "x+yi" or "x+yj" is required for both functions to work correctly. Note that the error is #VALUE!, not #NUM!.

Errors and Troubleshooting

Cause	Symptom	Fix
An argument is not valid complex text	#VALUE! error	Use x+yi or x+yj, or build the value with COMPLEX
Suffixes differ within one call	#VALUE! error	Use the same suffix for every argument
Suffix is uppercase "I" or "J"	#VALUE! error	Switch to lowercase; the suffix is case sensitive
A referenced cell is blank	The blank is treated as 0	Fill the cell, or guard it with IF or IFERROR
A real number is supplied	Accepted as a complex number	No fix needed; the real number is treated as x+0i
The result is used in ordinary arithmetic	#VALUE! error	The result is text; use IMREAL and IMAGINARY to extract numbers

The IMSUM and IMPRODUCT functions are essential for engineering and scientific calculations involving complex numbers. They add and multiply values written in x+yi or x+yj form, accept real numbers as a special case, and work directly with text produced by COMPLEX. Remember the two details that cause most mistakes: the imaginary suffix must match across all arguments, and the result is text that must be fed back into the IM functions or split with IMREAL and IMAGINARY before it can be used in arithmetic. Master these rules, and IMSUM and IMPRODUCT become reliable building blocks for complex-number work in spreadsheets.

Author

My name is Jan Bodnar, and I am a passionate programmer with extensive programming experience. I have been writing programming articles since 2007. To date, I have authored over 1,400 articles and 8 e-books. I possess more than ten years of experience in teaching programming.

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