Calculator Inputs
Example Data Table
| Example | z₁ | z₂ | Power | Root Order | Key Output |
|---|---|---|---|---|---|
| Case 1 | 3 + 4i | 1 - 2i | 3 | 4 | z₁ + z₂ = 4 + 2i |
| Case 2 | 3 + 4i | 1 - 2i | 3 | 4 | z₁ × z₂ = 11 - 2i |
| Case 3 | 3 + 4i | 1 - 2i | 3 | 4 | |z₁| = 5 and arg(z₁) ≈ 53.13° |
Formula Used
- Addition: (a + bi) + (c + di) = (a + c) + (b + d)i
- Subtraction: (a + bi) - (c + di) = (a - c) + (b - d)i
- Multiplication: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
- Division: (a + bi)/(c + di) = [(ac + bd) + (bc - ad)i] / (c² + d²)
- Modulus: |z| = √(a² + b²)
- Argument: arg(z) = atan2(b, a)
- Conjugate: z̄ = a - bi
- Power by De Moivre: zⁿ = rⁿ[cos(nθ) + i sin(nθ)]
- Nth Roots: zk = r1/n[cos((θ + 2πk)/n) + i sin((θ + 2πk)/n)]
How to Use This Calculator
- Enter the real and imaginary parts for z₁ and z₂.
- Select the desired operation mode or choose comprehensive analysis.
- Set the decimal precision for displayed results.
- Enter a power exponent if you want z₁ raised to a value.
- Enter a root order if you want all nth roots of z₁.
- Click Solve Complex Variables to generate results.
- Review rectangular, polar, trigonometric, and exponential forms.
- Use the CSV or PDF buttons to export the report.
- Inspect the Plotly Argand graph to compare points visually.
FAQs
1. What does this solver calculate?
It evaluates two complex numbers and returns rectangular, polar, trigonometric, and exponential forms. It also performs arithmetic, powers, roots, conjugates, reciprocals, modulus calculations, and argument calculations.
2. Why are results shown in several forms?
Different forms support different tasks. Rectangular form is convenient for addition and subtraction. Polar and trigonometric forms simplify multiplication, division, powers, and roots.
3. What happens when I divide by zero?
If z₂ equals zero, the division result becomes undefined. The calculator shows a note instead of a numeric output because complex division requires a nonzero denominator.
4. Can this solver handle negative exponents?
Yes. Negative integer exponents are supported through reciprocals of positive powers. If z₁ equals zero, negative powers become undefined.
5. Why are there multiple roots for one number?
An nth root in the complex plane usually has n distinct solutions. These roots are evenly spaced around a circle on the Argand diagram.
6. What is the argument of a complex number?
The argument is the angle formed by the point and the positive real axis. This solver reports it in both radians and degrees for easier interpretation.
7. Why does the graph help?
The graph places complex results on the Argand plane. This helps you compare direction, distance from the origin, and relationships between operations quickly.
8. What is included in the export options?
The CSV export stores computed values and formulas in spreadsheet-friendly rows. The PDF export captures the visible result report for printing, sharing, or study records.