Calculator Inputs
Plotly Graph
The chart places your complex number on the Argand plane and draws a circle with radius equal to the modulus.
Formula Used
For a complex number: z = a + bi
Absolute value formula: |z| = √(a² + b²)
Squared modulus: |z|² = a² + b²
This works because the real part and imaginary part form a right triangle on the complex plane. The modulus is the straight-line distance from the origin to the point (a, b).
How to Use This Calculator
- Enter the real part of your complex number.
- Enter the imaginary part, including negative values when needed.
- Choose your preferred precision, graph padding, and circle detail level.
- Press the calculate button to view the modulus, argument, conjugate, normalized form, exports, and graph.
Example Data Table
| Complex Number | Real Part | Imaginary Part | Calculation | Absolute Value |
|---|---|---|---|---|
| 3 + 4i | 3 | 4 | √(3² + 4²) | 5.0000 |
| -5 + 12i | -5 | 12 | √(-5² + 12²) | 13.0000 |
| 0 - 7i | 0 | -7 | √(0² + -7²) | 7.0000 |
| 1.5 + 2.5i | 1.5 | 2.5 | √(1.5² + 2.5²) | 2.9155 |
Frequently Asked Questions
1. What is the absolute value of a complex number?
It is the modulus of the complex number. For z = a + bi, the absolute value measures the distance from the origin to the point (a, b) on the complex plane.
2. Why is the answer always non-negative?
The modulus represents distance, and distance cannot be negative. Even when the real or imaginary part is negative, the final absolute value remains zero or positive.
3. What happens when both parts are zero?
If the complex number is 0 + 0i, the absolute value is 0. The direction or argument is not defined, because the point sits exactly at the origin.
4. Is modulus the same as distance from the origin?
Yes. On the Argand plane, the modulus is the straight-line distance from the origin to the plotted complex number. That is why the Pythagorean formula is used.
5. Why does this calculator show the conjugate too?
The conjugate helps you understand related complex operations. It reflects the point across the real axis and is often used in division, simplification, and modulus identities.
6. Does changing precision affect the true result?
No. Precision only changes how many decimal places are displayed. The underlying calculation remains the same, but the shown value becomes more or less detailed.
7. Why is the graph useful for this topic?
The graph makes the modulus easier to understand visually. You can see the complex point, the radius from the origin, and the circle whose radius equals the absolute value.
8. Can I export the calculated values?
Yes. After calculation, you can download a CSV summary or a PDF report. This is helpful for worksheets, notes, audit trails, or sharing results.