IMARGUMENT function

The IMARGUMENT function in Excel is used to return the argument (also called the phase angle or theta in polar coordinates) of a complex number. The argument is the angle in radians between the positive real axis and the line representing the complex number in the complex plane.

Syntax

IMARGUMENT(inumber)

Parameters

• inumber: The complex number (in the form of a string) for which you want to calculate the argument. The complex number should be entered in the form a + bi or a + bj, where a is the real part and b is the imaginary part. If the complex number is in polar form, it can be represented as r * (cos θ + i * sin θ).

How It Works

The IMARGUMENT function calculates the argument (angle) of the complex number using the formula:

$$\theta = \text{atan2}(b, a)$$

Where:

• a is the real part of the complex number.
• b is the imaginary part of the complex number.
• atan2 is a function that returns the angle (in radians) between the positive real axis and the point (a, b).

The argument is usually expressed in radians, and it ranges from -π to π. If you want the result in degrees, you can convert the radians using the DEGREES function.

Examples

1. Basic Example: To calculate the argument of the complex number 3 + 4i:

   =IMARGUMENT("3+4i")
   

   The result will be approximately 0.9273 radians (which is approximately 53.13 degrees). The formula used is:

   $$\text{atan2}(4, 3) \approx 0.9273 \, \text{radians}$$

2. Complex Number with Negative Real and Imaginary Part: To calculate the argument of the complex number -1 - 1i:

   =IMARGUMENT("-1-1i")
   

   The result will be approximately -2.3562 radians (which is approximately -135 degrees), because the complex number lies in the third quadrant of the complex plane.

3. Complex Number on the Real Axis: To calculate the argument of the complex number 5 + 0i (purely real number):

   =IMARGUMENT("5")
   

   The result will be 0 radians because the complex number lies on the positive real axis.

4. Complex Number on the Imaginary Axis: To calculate the argument of the complex number 0 + 5i (purely imaginary number):

   =IMARGUMENT("0+5i")
   

   The result will be 1.5708 radians (which is π/2, or 90 degrees), as it lies directly on the positive imaginary axis.

5. Using Cell Reference: If cell A1 contains the complex number "2+2i", you can calculate its argument:

   =IMARGUMENT(A1)
   

   The result will be approximately 0.7854 radians (which is approximately 45 degrees).

Convert Result to Degrees

To convert the result from radians to degrees, use the DEGREES function:

=DEGREES(IMARGUMENT("3+4i"))

This will return 53.13 degrees, which is the argument of 3 + 4i in degrees.

Important Notes

• The IMARGUMENT function returns the angle in radians, and if you need the result in degrees, you must manually convert it using the DEGREES function.
• The function works only with valid complex numbers. If the input is not in a valid complex number format, Excel will return a #VALUE! error.
• Complex numbers can be expressed in two forms:
  • Rectangular form: a + bi or a + bj (where a is the real part and b is the imaginary part).
  • Polar form: r * (cos θ + i * sin θ), but the IMARGUMENT function requires the rectangular form for input.

Summary

The IMARGUMENT function in Excel calculates the argument (phase angle) of a complex number in radians. It is particularly useful for converting complex numbers to polar coordinates and analyzing their angular components in mathematical, engineering, and scientific contexts.