Enter Rotation Inputs
Use Cartesian coordinates for the point and center. Choose angle units, direction, repeat count, and display precision.
Ready to calculate
Submit the form to display the rotated complex values above the form, generate the graph, and unlock CSV and PDF downloads.
Example Data Table
| Original z | Center c | Angle | Turns | Result zn |
|---|---|---|---|---|
| 5 + 2i | 0 + 0i | 45° | 1 | 2.1213 + 4.9497i |
| 3 + 4i | 1 + 1i | 90° | 1 | -2 + 3i |
| 2 - 1i | -1 + 2i | 180° | 1 | -4 + 5i |
| 4 + 0i | 0 + 0i | 30° | 3 | 0 + 4i |
Formula Used
Single rotation about center c:
z′ = c + eiθ(z − c)
Euler form of the rotation factor:
eiθ = cos(θ) + i sin(θ)
Repeated rotation after n turns:
zn = c + einθ(z − c)
Cartesian expansion:
If z − c = x + iy, then after rotation:
x′ = x cos(θ) − y sin(θ)
y′ = x sin(θ) + y cos(θ)
Distance preservation:
|z′ − c| = |z − c|, so rotation changes direction but keeps the radius from the center constant.
How to Use This Calculator
- Enter the real and imaginary parts of the complex point you want to rotate.
- Enter the real and imaginary parts of the rotation center. Use 0 and 0 for origin-centered rotation.
- Type the angle and choose degrees or radians.
- Select counterclockwise or clockwise direction.
- Choose how many times the same rotation should repeat.
- Set the decimal precision for displayed results.
- Press Calculate Rotation to show the result above the form and update the graph.
- Use the export buttons to download the result summary as CSV or PDF.
FAQs
1. What does complex rotation mean?
It means turning a complex point in the plane by a chosen angle. The point moves on a circle centered at the rotation center, while its distance from that center stays unchanged.
2. What happens when the center is 0 + 0i?
The rotation happens around the origin. In that case, the formula becomes z′ = eiθz, which is the standard complex-plane rotation rule taught in algebra and geometry.
3. Why does the calculator ask for direction?
Counterclockwise uses a positive angle, while clockwise uses a negative angle. The direction changes the sign of the rotation and therefore changes the final coordinates.
4. What does repeated rotation show?
Repeated rotation applies the same turn multiple times. This is useful for studying symmetry, polygon vertices, periodic motion, and how angles accumulate in the complex plane.
5. Does rotation change the modulus?
A rotation around the origin preserves the modulus of z. A rotation around another center preserves the distance from that center, though the modulus relative to the origin may change.
6. Can I use radians instead of degrees?
Yes. Choose radians in the angle unit field. The calculator converts and applies the value directly, then reports both effective and normalized angular values.
7. Why is the normalized angle useful?
Normalized angles place results within a standard range, usually 0° to 360°. That makes repeated rotations easier to interpret and compare on tables and graphs.
8. When would I use this calculator?
Use it for geometry practice, signal rotation, phasor analysis, transformations, symmetry problems, visual learning, and checking manual complex-number calculations quickly.