Complex Modulus Calculator
Modulus, argument, and conjugate of a+bi.
What it does
Complex numbers pair real with imaginary parts. Modulus is the straight distance from the origin, argument is the direction, and conjugate mirrors across the real axis. Together they power division, phasors, and fractals.
The tool above runs the trio. With 3 plus 4i, modulus is 5 at 53.13 degrees with conjugate 3 - 4i. With 5 plus 12i, modulus is 13 at 67.38 degrees. Both are 3-4-5 style triples, which is why classrooms favor them.
Quadrants decide argument signs. Atan2 places negative real parts past 90 degrees correctly, while plain arctangent would fold them. Conjugates keep the same modulus with flipped angle, so multiplying a number by its conjugate always returns a real sum of squares.
This page takes rectangular form only. It does not parse polar input or symbolic i. For division, multiply top and bottom by the conjugate shown.
How to use
- Enter the real part a.
- Enter the imaginary part b.
- Read modulus first.
- Read argument for direction.
- Read the conjugate for division work.
- Check quadrants when a is negative.
Formula
- a
- Real part a.
- b
- Imaginary part b.
- arg
- Angle from positive real axis in degrees.
Rectangular form a plus bi with real inputs. At the origin the modulus is zero and the coded argument is zero. Large values near a trillion need care with squares.
Worked example
Example: 3 plus 4i
Defaults: 3 plus 4i. Argument uses atan2 so quadrants land correctly.
- Inputs: real 3 with imaginary 4.
- Modulus: root of 9 plus 16 is 5.
- Argument: atan2(4, 3) is 53.13 degrees.
- Conjugate: 3 - 4i.
- A 5-12 case gives 13 at 67.38 deg. Tool shows the same.
Reference table
| Number | Modulus | Argument |
|---|---|---|
| 3 + 4i | 5 | 53.13 degrees |
| 5 + 12i | 13 | 67.38 degrees |
| 1 + 0i | 1 | 0 degrees |
| 0 + 5i | 5 | 90 degrees |
| -3 + 4i | 5 | 126.87 degrees |
Limits
- Rectangular input only. No polar parsing.
- Finite inputs. Huge squares overflow.
- Principal argument. Branch cuts ignored.
- Algebra only. No circuit modeling.
Common mistakes
- Using arctan instead of atan2 in two quadrants.
- Dropping the sign on the conjugate.
- Mixing radians with degrees.
- Reading modulus as argument.
- Squaring without rooting.
Try it yourself
Quick checks. Answers sit tucked under each question.
5 plus 12i?
Modulus is root of 25 plus 144 is 13 at 67.38 degrees. Conjugate 5 - 12i.
1 plus 0i?
Modulus is root of 1 plus 0 is 1 at 0 degrees. Conjugate 1 - 0i reads 1.
Questions
How is modulus figured?
Root of a squared plus b squared. Here 5.
How is argument figured?
atan2(b, a) in degrees. Here 53.13 degrees.
What is the conjugate?
Flip the sign of b. Here 3 - 4i.
Does modulus relate to conjugate?
Yes. Modulus times its conjugate equals a squared plus b squared.
What happens at 0 plus 0i?
Zero gives modulus zero with argument zero by convention here.
Does this take polar input?
No. This is algebra form only. Polar needs r and angle inputs.