# Documentation/Calc Functions/IMPRODUCT

IMPRODUCT

## Summary:

Calculates the product of multiple complex numbers.

## Syntax:

IMPRODUCT(Complex 1 [; Complex 2 [; … [; Complex 255]]])

## Returns:

Returns a text string giving the complex number that results from multiplying together the complex numbers passed as arguments.

## Arguments:

Complex 1 is a string (in quotation marks) representing a complex number in the form a+bi or a+bj, or a reference to one or more cells containing such strings, which are the complex numbers to be multiplied together. a and b are real numbers.

Complex 2 ... Complex 255 are as Complex 1.

• If any of the complex numbers passed to IMPRODUCT does not have an imaginary part (b equal to 0), the real part can be passed to IMPRODUCT as a number rather than a string.
• If any of the strings passed to IMPRODUCT is not a valid complex number, or any of the referenced cells contain an invalid complex number, IMPRODUCT reports an invalid argument error (Err:502).
• It is possible for the arguments passed to IMPRODUCT to contain no complex numbers. In this case IMPRODUCT returns 0.
• If the complex numbers passed to IMPRODUCT use different imaginary units (i or j), IMPRODUCT returns the result using the imaginary unit of the first complex number processed.

### Details specific to IMPRODUCT function

For two complex numbers of the form a+bi and c+di, IMPRODUCT implements the following equation:

The equation is applied repeatedly depending how many complex numbers are specified in the arguments.

### General information about Calc's handling of complex numbers

 For convenience, the information in this subsection is repeated on all pages describing functions that manipulate complex numbers.

• A complex number is a number that can be expressed in the form a+bi, where a and b are real numbers and i is a symbol for the imaginary unit, such that i2=−1. a is known as the real part of the complex number and b is known as its imaginary part.
• In some disciplines using i to indicate the imaginary unit may be undesirable and, in such cases, the symbol j may be used instead. The functions provided by Calc to process complex numbers can use either i or j.
• For additional general background information, visit Wikipedia’s Complex number page.
• Within Calc, a complex number is represented as a string expression of the form "a+bi" or "a+bj", where a and b are real numbers. If the complex number happens to be a real number (b=0), then it can be represented as either a string expression or a numeric value. Complex number string expressions should not contain any space characters. Examples of valid Calc complex numbers include "2", -6, "2+3i", "3j", "12.34+56.78i", and "1e+23-1e-22j". Calc would not recognize the strings "1+j2", "i+1", and "i+j" as valid complex numbers.
• The representation of the complex numbers in strings can have up to 15 significant digits and scientific style is automatically used if needed (visit Wikipedia’s Scientific notation page for more information). The decimal delimiter in complex number string expressions is always a dot, irrespective of locale setting.

## Examples:

Formula Description Returns
=IMPRODUCT("3+4j"; "5-3i") Here the function multiplies two complex numbers and returns the result using the imaginary unit of the first complex number. 27+11j
=IMPRODUCT("3.4+5.6i"; 7.8) Here the function multiplies the complex number in the first argument by the real number in the second argument (which is passed as a number rather than a string). 26.52+43.68i
=IMPRODUCT(E1; E2; E3) where cells E1, E2, and E3 contain the complex numbers 1+2i, 3-4i, and -5+6i respectively. Here the function multiplies the three complex numbers in cells E1, E2, and E3 (identified as separate arguments). -67+56i
=IMPRODUCT(E1:E3) where cells E1, E2, and E3 contain the complex numbers 1+2i, 3-4i, and -5+6i respectively. Here the function again multiplies the three complex numbers in cells E1, E2, and E3 (identified as a cell range in a single argument). -67+56i

IMPRODUCT