Calculator Inputs
Formula Used
For a complex number z = a + bi, the magnitude or modulus is:
|z| = √(a² + b²)
If the input is already in polar form z = r(cosθ + i sinθ), then the magnitude is simply:
|z| = r
The argument is found using arg(z) = atan2(b, a). The squared magnitude is |z|² = a² + b², which is useful in algebra, geometry, and signal calculations.
How to Use This Calculator
- Select Rectangular Form if your number is written as a + bi.
- Select Polar Form if your number is written as r ∠ θ.
- Enter the values in the visible fields only.
- Choose the angle unit as degrees or radians.
- Set your preferred decimal precision.
- Click Calculate Magnitude to show the result above the form.
- Use the CSV or PDF buttons to export the result summary.
- Review the graph, formula section, and example table for verification.
Example Data Table
| Complex Number | Real Part | Imaginary Part | Formula | Magnitude |
|---|---|---|---|---|
| 3 + 4i | 3 | 4 | √(3² + 4²) | 5 |
| 5 - 12i | 5 | -12 | √(5² + 12²) | 13 |
| -8 + 6i | -8 | 6 | √((-8)² + 6²) | 10 |
| 0 + 7i | 0 | 7 | √(0² + 7²) | 7 |
| -9 - 12i | -9 | -12 | √((-9)² + (-12)²) | 15 |
Frequently Asked Questions
1) What is the magnitude of a complex number?
The magnitude is the distance from the origin to the point representing the complex number on the complex plane. It shows the size of the number and is always zero or positive.
2) How do I calculate the magnitude from a + bi?
Use the formula |z| = √(a² + b²). Square the real part, square the imaginary part, add them, then take the square root. This gives the modulus.
3) Is magnitude the same as modulus?
Yes. In complex number work, magnitude and modulus mean the same thing. Both refer to the absolute size of the complex number.
4) What happens when the complex number is purely real?
If the imaginary part is zero, the magnitude becomes the absolute value of the real part. For example, the magnitude of -7 + 0i is 7.
5) What happens when the complex number is purely imaginary?
If the real part is zero, the magnitude becomes the absolute value of the imaginary part. For example, the magnitude of 0 - 9i is 9.
6) Can I find magnitude from polar form directly?
Yes. In polar form z = r(cosθ + i sinθ), the magnitude is simply r. No additional square root step is needed.
7) Why is the argument also shown?
The argument gives the angle of the complex number on the complex plane. Together, magnitude and argument fully describe the number in polar form.
8) Why is the magnitude never negative?
Magnitude represents geometric distance from the origin. Distance cannot be negative, so the modulus is always zero or a positive value.