IMABS function

The IMABS function in Excel is used to return the absolute value (magnitude) of a complex number. The absolute value of a complex number is its distance from the origin in the complex plane and is always a non-negative number.

Syntax

IMABS(inumber)

Parameters

• inumber: This is the complex number (in the form of a string) for which you want to calculate the absolute value. 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, use r * (cos θ + i * sin θ).

How It Works

The IMABS function calculates the magnitude (or absolute value) of a complex number using the formula:

$$|a + bi| = \sqrt{a^2 + b^2}$$

Where:

• a is the real part of the complex number.
• b is the imaginary part of the complex number.
• i is the imaginary unit.

Examples

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

   =IMABS("3+4i")
   

   The result will be 5 because:

   $$\sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$$

2. Complex Number with Negative Imaginary Part: To calculate the absolute value of the complex number 5 - 12i:

   =IMABS("5-12i")
   

   The result will be 13 because:

   $$\sqrt{5^2 + (-12)^2} = \sqrt{25 + 144} = \sqrt{169} = 13$$

3. Complex Number with Decimal Parts: To calculate the absolute value of the complex number 2.5 + 3.5i:

   =IMABS("2.5+3.5i")
   

   The result will be 4.301 because:

   $$\sqrt{2.5^2 + 3.5^2} = \sqrt{6.25 + 12.25} = \sqrt{18.5} \approx 4.301$$

4. Using Cell Reference: If cell A1 contains the complex number "7+24i", you can calculate its absolute value:

   =IMABS(A1)
   

   The result will be 25 because:

   $$\sqrt{7^2 + 24^2} = \sqrt{49 + 576} = \sqrt{625} = 25$$

Important Notes

• The IMABS function works only with complex numbers. If the input is not in a valid complex number format, Excel will return a #VALUE! error.
• The result is always a real number, representing the magnitude of the complex number.
• Complex numbers can be expressed in two forms in Excel:
  • Rectangular form: "a + bi" or "a + bj" (where a is the real part and b is the imaginary part).
  • Polar form: In Excel, complex numbers are typically used in the rectangular form.

Summary

The IMABS function in Excel is used to calculate the magnitude or absolute value of a complex number. It is useful for applications in mathematics, engineering, and physics, where the magnitude of a complex number is required.