Case 1
- Real part a
- 3
- Imaginary part b
- 4
5.00000
Open with these valuesComplex Number Modulus Calculator
5.00000
Result: 5.00000The modulus is how far the point a + bi sits from the origin: |z| = √(a² + b²). For 3 + 4i that is √25 = 5. It is never negative, and all four sign combinations of 3 and 4 give the same 5 — the modulus knows distance, not direction.
The numbers at a glance
Held fixed: Real part a 3.0000.
| Imaginary part b | Result |
|---|---|
| 0.0000 | 3.00000 |
| 1.0000 | 3.16228 |
| 2.0000 | 3.60555 |
| 3.0000 | 4.24264 |
| 4.0000Your value | 5.00000 |
| 5.0000 | 5.83095 |
| 6.0000 | 6.70820 |
| 7.0000 | 7.61577 |
| 8.0000 | 8.54400 |
Worked examples
Case 2
- Real part a
- 1
- Imaginary part b
- 1
1.41421
Open with these valuesCase 3
- Real part a
- -3
- Imaginary part b
- -4
5.00000
Open with these valuesHow it's calculated
|z| = √(a² + b²)
- StepEnter the real part a — the horizontal coordinate of the point.
- StepEnter the imaginary part b, the number in front of i.
- ResultRead the distance from the origin; the table gives the angle too.
Reference table
| a, b | Argument | Modulus |z| |
|---|---|---|
| 0, 0 | 0° (undefined, shown as zero) | 0.00000 |
| 0, -1 | -90° | 1.00000 |
| -1, 0 | 180° | 1.00000 |
| 1, 1 | 45° | 1.41421 |
| 3, 4 | 53.130° | 5.00000 |
| 3, -4 | -53.130° | 5.00000 |
| -3, 4 | 126.870° | 5.00000 |
| 0, 5 | 90° | 5.00000 |
| 6, 8 | 53.130° | 10.00000 |
Questions
How do I calculate the modulus of a complex number?
Square the real and imaginary parts, add them, take the square root: |z| = √(a² + b²). For z = 3 + 4i that is √(9 + 16) = √25 = 5. The modulus is never negative — it is the distance of the point a + bi from the origin of the complex plane.
Is the modulus the same as the absolute value?
Yes. The modulus is the absolute value of a complex number, written |z|, and it generalises the absolute value of a real number. A real number has b = 0, so |z| = √(a²) = |a|, exactly the familiar case.
How is this related to the length of a vector?
It is the same calculation. The point a + bi is the vector (a, b), and √(a² + b²) is its length — the Pythagorean theorem. The vector magnitude calculator does the same arithmetic with a third axis added.
What is the argument of a complex number?
The argument is the angle between the positive real axis and the line from the origin to a + bi, computed as atan2(b, a) and shown here in degrees. For 3 + 4i it is about 53.130°. The table above lists it beside the modulus of each row.
Why do all four sign combinations give 5?
Both parts are squared before they are added, so a minus sign disappears. That makes 3 + 4i, −3 + 4i, −3 − 4i and 3 − 4i share the modulus 5. What does change is the argument, which tracks the quadrant: 53.130°, 126.870°, −126.870° and −53.130°.
What are the modulus and argument of zero?
For z = 0 + 0i the modulus is 0, since the point sits at the origin. The argument is mathematically undefined there — no direction exists — and the table reports it as 0° by convention rather than leaving a gap.
Sources and last check
Information, not professional advice.