How to use the IMARGUMENT() function in Excel

This function calculates the argument (phase angle Φ) in radians of a complex number given in rectangular form (x + yi or x + yj).

Syntax
IMARGUMENT(complex_number)

Argument

• complex_number (required)
  A text string representing a complex number in either:
  • « x+yi » format (mathematical convention)
  • « x+yj » format (engineering convention)

Technical Background
For a complex number z = x + yi:

• The argument Φ is the angle in the complex plane
• Calculated as: Φ = atan2(y, x)
• Range: -π < Φ ≤ π radians
• Relates to trigonometric form: |z|·(cosΦ + i·sinΦ)

Example

=IMARGUMENT(« 3-4i »)  // Returns -0.927295218 radians

Additional Examples

=IMARGUMENT(« 1+i »)     // Returns 0.785398163 (π/4 radians)

=IMARGUMENT(« -1-i »)    // Returns -2.35619449 (-3π/4 radians)

=IMARGUMENT(« 0+1j »)    // Returns 1.570796327 (π/2 radians)

=IMARGUMENT(« 5 »)       // Returns 0 (pure real number)

Key Features

• Returns angle in radians (-π to π)
• Supports both « i » and « j » notation
• Handles all four quadrants of complex plane
• Returns 0 for positive real numbers
• Returns π for negative real numbers

Error Conditions

• Returns #NUM! for:
  • Invalid complex number format
  • Non-numeric components
  • Empty string

Usage Notes

1. To convert result to degrees, use DEGREES() function
2. For zero (0+0i), returns 0 by convention
3. Angle sign follows standard mathematical convention:
   • Positive for upper half-plane
   • Negative for lower half-plane