# Documentation/Calc Functions/IMARGUMENT

IMARGUMENT

## Summary:

Calculates the principal value of the argument of a complex number.

## Syntax:

IMARGUMENT(ComplexNumber)

## Returns:

Returns a real number in the range -π < φ ≤ π which is the argument (in radians) of the specified complex number.

## Arguments:

ComplexNumber is a string (in quotation marks) representing a complex number in the form a+bi or a+bj, or a reference to a cell containing that string, whose argument is to be calculated. a and b are real numbers.

• If ComplexNumber does not have an imaginary part (b equal to 0), the real part can be passed to IMARGUMENT as a number rather than a string.
• If both real and imaginary parts of ComplexNumber are 0, IMARGUMENT reports an invalid argument error (Err:502).
• If ComplexNumber is not a valid complex number, IMARGUMENT reports an invalid argument error (Err:502).

### Details specific to IMARGUMENT function

A complex number of the form a+bi can be written in trigonometric form as a+bi = r(cosφ+isinφ), where r is the absolute value or modulus of the complex number and φ is its argument.

IMARGUMENT calculates the argument of the complex number using the formula:

and then if b < 0, φ = -φ.

### 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
=IMARGUMENT("3.4+4.5j") Here the function calculates the argument of a complex number that has non-zero real and imaginary parts. 0.923749
=IMARGUMENT("-i") Here the function calculates the argument of a complex number that has no real part. -1.570796
=IMARGUMENT(-1) Here the function calculates the argument of a complex number that has no imaginary part. 3.141593
=IMARGUMENT(F1) where cell F1 contains the complex number 2-3j. Here the function calculates the argument of a complex number that is entered in a cell. -0.982794

IMARGUMENT