Enter Complex Number Values
Use real and imaginary parts. The graph updates after calculation.
Example Data Table
Use these sample values to test common patterns on the complex plane.
| Case | z₁ | z₂ | Suggested Operation | Expected Idea |
|---|---|---|---|---|
| Vector addition | 3 + 4i | 2 - 1i | Addition | Parts combine separately. |
| Rotation effect | 1 + 1i | 0 + 1i | Multiplication | Multiplying by i rotates 90 degrees. |
| Mirror point | -2 + 5i | 1 + 0i | Conjugate | Point reflects across the real axis. |
| Fourth roots | 16 + 0i | 0 + 0i | Roots | Roots spread evenly around the origin. |
Formula Used
Rectangular Form
z = a + bi
The real part is a. The imaginary part is b.
Modulus
|z| = √(a² + b²)
This is the distance from the origin.
Argument
θ = atan2(b, a)
This is the directed angle from the positive real axis.
Addition
(a + bi) + (c + di) = (a + c) + (b + d)i
Multiplication
(a + bi)(c + di) = (ac - bd) + (ad + bc)i
Roots
zₖ = r^(1/n)[cos((θ + 2πk)/n) + i sin((θ + 2πk)/n)]
How to Use This Calculator
- Enter the real and imaginary parts of z₁.
- Enter z₂ if your selected operation needs two complex numbers.
- Select addition, subtraction, multiplication, division, power, roots, conjugate, reciprocal, or rotation.
- Set power, root count, angle unit, graph range, and decimal places.
- Choose extra graph options, such as roots, conjugates, or the unit circle.
- Press the calculate button. Results will appear below the header and above the form.
- Use the CSV or PDF button to save your result.
Complex Number Graphing Guide
Complex numbers are easier to read when they are drawn. A number such as a + bi becomes a point on the plane. The real part moves the point left or right. The imaginary part moves it up or down. This calculator turns those values into a visual model.
Why the Complex Plane Matters
The complex plane gives each number a location. Distance from the origin is the modulus. Direction from the positive real axis is the argument. Together, these values create polar form. Polar form is useful for multiplication, division, powers, and roots.
Operations on the Graph
Addition shifts a point by another point. Subtraction measures the difference between two points. Multiplication changes both size and angle. Division reverses that change. Powers repeat the angle and raise the distance. Roots split the angle into equal paths around a circle.
Learning With Rectangular and Polar Forms
Rectangular form is best for direct arithmetic. Polar form is best for rotation and scaling. Exponential form is compact and powerful. The calculator shows all forms together. This helps you check answers from several angles.
Advanced Uses
Students can test textbook exercises. Teachers can create quick visual examples. Engineers can compare signals, impedance, and phase. Math learners can see why conjugates mirror across the real axis. They can also see why roots appear around the origin.
Graph Reading Tips
Start by checking the sign of each part. Positive real values move right. Negative real values move left. Positive imaginary values move up. Negative imaginary values move down. Then inspect the distance and angle. A large modulus means a longer vector. A larger angle means more counterclockwise rotation.
Better Problem Solving
Visual feedback reduces small arithmetic errors. It also makes abstract rules feel concrete. Use the graph beside your manual work. Compare the plotted result with the formula. When both match, your answer is stronger. Use the example table before entering custom values. It shows common patterns on the plane. Change one value at a time. Watch how the vector moves. This habit builds intuition. It also helps you explain answers in exams, reports, homework, and lessons with more confidence every time during complex number practice.
FAQs
1. What is a complex number?
A complex number has a real part and an imaginary part. It is written as a + bi, where i represents the square root of negative one.
2. What does the complex plane show?
The complex plane shows the real part on the horizontal axis and the imaginary part on the vertical axis. Each complex number becomes one point.
3. What is modulus?
Modulus is the distance from the origin to the complex number point. It is calculated with √(a² + b²).
4. What is argument?
Argument is the angle made with the positive real axis. It shows the direction of the complex number from the origin.
5. Why are roots shown around a circle?
Complex roots share equal angular spacing. They often form a circular pattern because each root has the same modulus but a different direction.
6. Why does multiplication rotate points?
In polar form, multiplying complex numbers multiplies their moduli and adds their arguments. That angle addition creates rotation on the graph.
7. What does the conjugate do on the graph?
The conjugate changes a + bi into a - bi. On the graph, it reflects the point across the real axis.
8. Can I export the calculated result?
Yes. Use the CSV button for spreadsheet data. Use the PDF button for a clean report of the selected result and key values.