Calculator Inputs
Example Data Table
| Case | z | w | Operation | Expected Focus |
|---|---|---|---|---|
| Basic vector | 3 + 4i | -2 + 1.5i | z + w | Modulus, angle, and point location |
| Rotation effect | 1 + 1i | 0 + 2i | z × w | Stretching and rotation |
| Division check | 5 - 3i | 2 + 1i | z ÷ w | Reciprocal and quotient behavior |
Formula Used
For a complex number z = a + bi, the plotted point is (a, b).
Modulus: |z| = √(a² + b²).
Argument: θ = atan2(b, a).
Polar form: z = r(cos θ + i sin θ), where r = |z|.
Exponential form: z = re^(iθ).
Conjugate: z̄ = a - bi.
Reciprocal: 1 / z = (a - bi) / (a² + b²), when z is not zero.
Multiplication: (a + bi)(c + di) = (ac - bd) + (ad + bc)i.
Division: (a + bi) / (c + di) = [(ac + bd) + (bc - ad)i] / (c² + d²).
Power rule: zⁿ = rⁿ[cos(nθ) + i sin(nθ)].
Root rule: z^(1/n) = r^(1/n)[cos((θ + 2πk) / n) + i sin((θ + 2πk) / n)].
How to Use This Calculator
- Enter the real and imaginary parts of z.
- Enter a second complex number w for comparison or operations.
- Select addition, subtraction, multiplication, or division.
- Choose a power and root count for deeper analysis.
- Set decimal places and a minimum axis limit.
- Press Calculate to see the graph above the form.
- Use CSV or PDF buttons to save the current result.
Graph Complex Numbers Guide
Understanding the Complex Plane
A complex number has a real part and an imaginary part. The graph places the real part on the horizontal axis. It places the imaginary part on the vertical axis. This view turns z equals a plus bi into a clear point. It also shows a vector from the origin. That vector helps explain size, direction, and rotation.
Why This Calculator Helps
This calculator is built for students, teachers, engineers, and analysts. It does more than plot one point. It compares two complex numbers. It shows the result of addition, subtraction, multiplication, or division. It also reports modulus, argument, conjugate, reciprocal, integer power, and roots. These outputs make complex arithmetic easier to inspect.
Reading the Graph
The graph uses a standard Argand diagram. Values to the right have positive real parts. Values to the left have negative real parts. Values above the origin have positive imaginary parts. Values below the origin have negative imaginary parts. The distance from the origin is the modulus. The angle from the positive real axis is the argument.
Advanced Uses
Complex graphs are useful in trigonometry, signals, circuits, controls, roots of equations, and transformations. Multiplication rotates and stretches a point. Division reverses part of that effect. Powers repeat rotation and scale. Roots divide the angle into equal directions around a circle. The root markers help show this symmetry clearly.
Practical Workflow
Enter the real and imaginary parts of z. Add a second value when you need comparison. Choose an operation. Select a power and root count. Adjust the axis limit if the plotted points look crowded. Use higher decimal precision for technical reports. After calculation, review the graph first. Then inspect the table and formula notes. Download CSV data for spreadsheets. Download the PDF summary for sharing. The example table gives sample cases before you calculate.
Checking Results
Always compare the rectangular and polar forms. They describe the same location. Small rounding differences are normal. A negative angle may also match a positive coterminal angle. For roots, each answer should have the same modulus. Their angles should be evenly spaced. Use the operation result row to confirm algebraic work step by step. This habit catches sign errors quickly during final review.
FAQs
What does this calculator graph?
It graphs complex numbers as points and vectors on the complex plane. The horizontal axis shows the real part. The vertical axis shows the imaginary part.
Can it compare two complex numbers?
Yes. Enter z and w. The graph shows both values. It also plots the selected operation result when the result is defined.
What is the modulus?
The modulus is the distance from the origin to the complex point. It is calculated with √(a² + b²).
What is the argument?
The argument is the angle from the positive real axis to the point vector. The calculator shows it in degrees and radians.
Why are roots shown around a circle?
Complex roots divide the full rotation evenly. They share the same root modulus and have equally spaced angles around the origin.
Can I divide by zero?
No. Division by a zero complex number is undefined. The calculator shows an undefined result when w equals 0 + 0i.
What does the conjugate mean?
The conjugate keeps the real part and changes the sign of the imaginary part. It reflects the point across the real axis.
How should I use the downloads?
Use CSV for spreadsheet work and data storage. Use the PDF summary when you need a quick report of the current calculation.