Complex Division Input
Enter both numbers in rectangular or polar form. The calculator converts formats, divides them, and displays rectangular and polar results.
Example Data Table
| Dividend | Divisor | Quotient | Modulus | Argument |
|---|---|---|---|---|
| 8 + 6i | 3 - 2i | 0.9231 + 2.6154i | 2.7735 | 70.56° |
| 5 - 4i | 1 + 2i | -0.6 - 2.8i | 2.8636 | -102.09° |
| 12∠40° | 4∠10° | 3∠30° | 3 | 30° |
Formula Used
For rectangular inputs, let z₁ = a + bi and z₂ = c + di. The division formula is:
z₁ / z₂ = [(ac + bd) / (c² + d²)] + [(bc - ad) / (c² + d²)]i
The divisor is rationalized with its conjugate, c - di. The modulus is √(x² + y²), and the argument is atan2(y, x).
For polar inputs, divide magnitudes and subtract angles: (r₁∠θ₁) / (r₂∠θ₂) = (r₁ / r₂)∠(θ₁ - θ₂).
How to Use This Calculator
- Select rectangular or polar entry for the dividend and divisor.
- Enter the real and imaginary parts, or magnitude and angle.
- Choose degrees or radians for all polar angles.
- Set the output precision and decide whether to use scientific notation.
- Press Calculate Division to show the result above the form.
- Review the rectangular form, polar form, denominator, and validation product.
- Use the export buttons to save a CSV summary or PDF report.
FAQs
1. Why multiply by the conjugate?
Multiplying by the conjugate removes the imaginary term from the denominator. That turns the division into a standard real denominator, which makes the real and imaginary parts easier to calculate accurately.
2. Can I mix rectangular and polar inputs?
Yes. Each number has its own input mode. The calculator converts both values internally, then performs the division and reports the answer in rectangular and polar forms.
3. What happens if the divisor equals zero?
Division by zero is undefined. The calculator checks the divisor magnitude and shows an error if both rectangular parts are zero or the polar magnitude is zero.
4. Why does the validation product matter?
The validation product multiplies the quotient by the divisor. If rounding is small, the product should closely match the original dividend, helping you confirm the result.
5. When should I use degrees or radians?
Use the unit that matches your source values. Engineering and classroom exercises often use degrees, while calculus and programming workflows commonly use radians.
6. Does polar division always simplify the process?
Polar form is usually faster when magnitudes and angles are already known. Rectangular form is often better when values are given as real and imaginary parts.
7. Why can tiny rounding differences appear?
Rounded displays hide some digits. The internal calculations use floating-point arithmetic, so minor differences may appear when you compare the validation product with the original dividend.
8. Can I use this for homework checking?
Yes. The step list, denominator breakdown, and polar summary make it useful for checking manual work, revising formulas, and spotting sign mistakes in complex-number division.