Calculator Form
The overall page uses a single-column flow. The calculator fields below use three columns on large screens, two on medium screens, and one on mobile.
Formula Used
For a complex number z = a + bi, the multiplicative inverse exists only when z ≠ 0.
Main identity: 1 / (a + bi) = (a - bi) / (a² + b²)
Why it works: Multiply numerator and denominator by the conjugate, a - bi.
Denominator: (a + bi)(a - bi) = a² + b²
Polar link: If z = r∠θ, then z-1 = (1/r)∠(-θ)
When both a and b are zero, the denominator becomes zero. That makes the inverse undefined.
How to Use This Calculator
- Enter the real part a.
- Enter the imaginary part b.
- Choose your preferred decimal precision.
- Select degrees or radians for the argument display.
- Optionally set a custom graph span.
- Choose how many sample points the graph should use.
- Click Calculate Inverse.
- Review the result, steps, graph, and verification output.
Example Data Table
| Complex number z | Conjugate | |z|² | Multiplicative inverse z-1 |
|---|---|---|---|
| 2 + 3i | 2 - 3i | 13 | 0.153846 - 0.230769i |
| 1 - 1i | 1 + 1i | 2 | 0.5 + 0.5i |
| -4 + 2i | -4 - 2i | 20 | -0.2 - 0.1i |
| 3 + 0i | 3 + 0i | 9 | 0.333333 + 0i |
FAQs
1) What is a multiplicative inverse of a complex number?
It is the number that gives 1 when multiplied by the original complex number. For z, the inverse is written as z-1 or 1/z.
2) When does the inverse not exist?
It does not exist when z = 0 + 0i. In that case, a² + b² = 0, so the formula would divide by zero.
3) Why does the formula use the conjugate?
The conjugate removes the imaginary part from the denominator. That turns the denominator into the real value a² + b².
4) Is the inverse related to modulus and argument?
Yes. The inverse modulus becomes 1/|z|, and the inverse argument becomes the negative of the original argument.
5) Can this calculator handle negative values?
Yes. Both the real and imaginary inputs can be positive, negative, or decimal values. The graph and formulas update automatically.
6) Why is the verification result useful?
It confirms the computation. Multiplying z by z-1 should produce approximately 1 + 0i, allowing for rounding.
7) What does the graph show?
It plots the original complex number, its inverse, the unit circle, and inverse locations for scaled versions of the same input direction.
8) Can I save the result for reports or homework?
Yes. Use the CSV button for data tables and the PDF button for a neat summary that can be shared or stored.