Calculator Inputs
Choose one complex-plane inequality family, enter values, and plot the solution region directly on the Argand plane.
Plotly Graph
The blue shaded area marks the solution set. The red curve or line marks the boundary when one exists.
Formula Used
Let z = x + iy and c = h + ik.
|z - c| = √((x - h)² + (y - k)²)
So |z - c| <= r becomes (x - h)² + (y - k)² <= r² when r ≥ 0.
If α = p + qi and β = u + vi, then
Re((p + qi)z + (u + vi)) = px - qy + u.
If α = p + qi and β = u + vi, then
Im((p + qi)z + (u + vi)) = py + qx + v.
Modulus inequalities create circles, disks, or exteriors.
Real-part and imaginary-part inequalities create half-planes bounded by straight lines.
How to Use This Calculator
- Select the inequality family that matches your problem.
- Choose the operator: <, ≤, >, or ≥.
- Enter the center and radius for modulus mode, or α, β, and the threshold for linear modes.
- Set the graph window so the region is clearly visible.
- Press Solve Inequality to show the result above the form.
- Review the equivalent Cartesian form, boundary, and region description.
- Inspect the Plotly graph to confirm the shaded solution set.
- Download the result summary as CSV or PDF when needed.
Example Data Table
| Example | Input Form | Equivalent Cartesian Form | Region Type |
|---|---|---|---|
| Example 1 | |z - (2 - 3i)| ≤ 4 | (x - 2)² + (y + 3)² ≤ 16 | Closed disk |
| Example 2 | Re((1 + 2i)z + (3 - i)) ≥ 5 | x - 2y - 2 ≥ 0 | Closed half-plane |
| Example 3 | Im((1 + 2i)z + (3 - i)) < 4 | 2x + y - 5 < 0 | Open half-plane |
These samples help verify the solver, graph, and export features before entering custom values.
Frequently Asked Questions
1) What is a complex inequality?
A complex inequality describes a region in the complex plane rather than a simple interval on the real line. It usually compares modulus, real part, imaginary part, or distance from a complex center.
2) Why can’t complex numbers be ordered like real numbers?
Complex numbers do not have a natural total order compatible with addition and multiplication. That is why this solver interprets inequalities through geometry, modulus, real part, or imaginary part instead.
3) What does |z - c| ≤ r represent?
It represents every complex number whose distance from c is at most r. Geometrically, that is a closed disk centered at c with radius r.
4) What is the difference between open and closed regions?
Strict inequalities like < and > exclude the boundary. Non-strict inequalities like ≤ and ≥ include the boundary line or circle as part of the solution set.
5) What do α and β do in linear modes?
α controls the tilt and orientation of the boundary line. β shifts the line by changing the constant term in the real-part or imaginary-part expression.
6) What happens if the modulus radius is negative?
A modulus is always non-negative. So a negative radius makes some inequalities impossible and others automatically true. The calculator detects that case and reports the correct region.
7) How should I read the graph?
The blue shading is the solution region. The red line or circle is the boundary. If the operator is strict, the boundary is only a guide and is not included.
8) Can I use this for classroom checks and homework review?
Yes. It is useful for verifying region type, algebraic conversion, and graph shape. It should support study, revision, and quick visual confirmation.