Calculator Inputs
Example data table
| Example | z₁ | z₂ | Quotient | Magnitude Ratio | Phase Difference |
|---|---|---|---|---|---|
| Worked example | 4 + 2i | 1 - 1i | 1 + 3i | 3.1623 | 71.5651° |
| Check case | 3 + 4i | 1 + 2i | 2.2 - 0.4i | 2.2361 | -10.3048° |
| Equal imaginary signs | 6 + 8i | 2 + 2i | 3.5 + 0.5i | 3.5355 | 8.1301° |
Formula used
If z₁ = a + bi and z₂ = c + di, then
z₁ / z₂ = (a + bi) / (c + di)
z₁ / z₂ = ((a + bi)(c - di)) / ((c + di)(c - di))
z₁ / z₂ = ((ac + bd) + (bc - ad)i) / (c² + d²)
Real part = (ac + bd) / (c² + d²)
Imaginary part = (bc - ad) / (c² + d²)
|z₁ / z₂| = |z₁| / |z₂|
|z| = √(real² + imaginary²)
Arg(z₁ / z₂) = Arg(z₁) - Arg(z₂)
Arg(z) = atan2(imaginary, real)
How to use this calculator
- Enter the real and imaginary parts for z₁.
- Enter the real and imaginary parts for z₂.
- Choose decimal precision for the displayed values.
- Select degrees or radians for angle reporting.
- Choose signed or positive angle normalization.
- Set your preferred output style: rectangular, polar, or both.
- Click Calculate Ratio to display the result above the form.
- Review the summary cards, worked steps, result table, and graph.
- Use the export buttons to download the current result as CSV or PDF.
Frequently asked questions
1) What is a complex ratio?
A complex ratio is the division of one complex number by another. It produces a new complex number with its own real part, imaginary part, magnitude, and angle.
2) Why must the denominator be nonzero?
Division by 0 + 0i is undefined because the denominator modulus becomes zero. That makes the conjugate formula impossible to evaluate safely.
3) Why is the conjugate used in division?
The conjugate removes the imaginary term from the denominator. This converts the division into a clean rectangular form with separate real and imaginary components.
4) What does magnitude ratio mean?
Magnitude ratio compares the lengths of z₁ and z₂ on the complex plane. It equals the modulus of the quotient and shows the relative scaling between both values.
5) What is phase difference?
Phase difference is the angular separation between z₁ and z₂. It is found by subtracting the argument of the denominator from the argument of the numerator.
6) Should I use degrees or radians?
Use degrees for quick interpretation and reporting. Use radians for higher-level mathematics, calculus, signal analysis, and many engineering formulas.
7) Can the calculator verify the result?
Yes. The summary table includes a verification row showing that multiplying the quotient by z₂ reconstructs z₁ within the selected rounding precision.
8) What does the graph show?
The plot places z₁, z₂, and the quotient as points and vectors on the Argand plane. This helps you compare direction, sign, and relative size visually.