Input complex number
Example data table
Use the button above to add your calculated values to this table, then export your dataset as CSV or PDF for documentation or reports.
| # | Real part a | Imaginary part b | Inverse Re(1/z) | Inverse Im(1/z) | |z| | |1/z| | arg(z) (rad) | arg(z) (deg) |
|---|---|---|---|---|---|---|---|---|
| 1 | 2.0000 | 3.0000 | 0.1538 | -0.2308 | 3.6056 | 0.2774 | 0.9828 | 56.3099 |
| 2 | -1.0000 | 4.0000 | -0.0588 | -0.2353 | 4.1231 | 0.2425 | 1.8158 | 103.3048 |
Formula used
For a complex number z = a + bi, the multiplicative inverse is defined as the number w such that z · w = 1.
We compute the inverse using the complex conjugate and the squared magnitude:
- Conjugate of z: conj(z) = a - bi
- Squared magnitude: |z|² = a² + b²
- Inverse: 1/z = conj(z) / |z|² = (a - bi) / (a² + b²)
Breaking the inverse into real and imaginary components:
- Re(1/z) = a / (a² + b²)
- Im(1/z) = -b / (a² + b²)
In polar form, write z = |z|(cos θ + i sin θ). Then 1/z = (1/|z|)(cos(−θ) + i sin(−θ)), so the magnitude inverts and the angle changes sign.
How to use this calculator
- Select Rectangular (a + bi) if you know the real and imaginary parts directly.
- Select Polar (r, θ) if your complex number is given by magnitude and angle.
- Enter either a and b, or r and θ with the correct angle unit.
- Choose the decimal precision for all displayed quantities.
- Click Calculate inverse to compute the multiplicative inverse, its magnitude, and its argument in radians and degrees.
- Review rectangular and polar forms of z and 1/z in the results panel.
- Click Add current result to example table to store the calculation, then use Download CSV or Download PDF to export your data.
This extended setup is useful for switching between rectangular and polar descriptions while checking algebraic work in complex analysis or circuit problems.
Geometric view of the complex reciprocal
On the Argand diagram, taking 1/z reflects the angle of z across the real axis and rescales its distance from the origin. Points near the origin move far away after inversion, while distant points approach the unit circle when |z| is large.
Rectangular and polar consistency checks
Because the calculator shows both rectangular and polar views of z and 1/z, you can confirm algebraic work visually. Matching magnitudes and opposite arguments across forms help students validate manual derivations from complex algebra or phasor manipulation in one consistent environment.
Using multiplicative inverses in algebraic simplification
Complex reciprocals frequently appear when rationalizing denominators, simplifying transfer functions, or isolating variables in equations. Computing 1/z quickly lets you substitute exact rectangular values back into symbolic steps without losing track of conjugates and magnitude relationships along the way.
Connections to modular multiplicative inverses
Although this tool works over the complex plane, it mirrors the idea of multiplicative inverses in modular arithmetic. For integer problems under a modulus, you can switch to the Modular Multiplicative Inverse Calculator to work with discrete congruences instead of continuous complex numbers.
Imaginary‑dominant numbers and stability concerns
When the imaginary part dominates, small perturbations in b can cause noticeable changes in 1/z. The calculator helps you experiment interactively and compare results with the dedicated Imaginary Number Division Calculator when focusing purely on division of imaginary‑heavy values.
Combining reciprocals with roots and powers
Reciprocals, roots, and powers are closely related operations on complex numbers. After finding 1/z, you may want to explore square roots or higher roots for the same input using the Complex Square Root Calculator to deepen intuition about magnitude and multi‑valued behavior.
Frequently Asked Questions (FAQs)
What is the multiplicative inverse of a complex number?
The multiplicative inverse of a complex number z is another complex number w such that z·w = 1. This calculator computes w from rectangular or polar input, showing both forms and useful supporting values like magnitude and argument.
How is the multiplicative inverse different from the conjugate?
The conjugate of z = a + bi is a − bi, while the multiplicative inverse is (a − bi)/(a² + b²). Conjugation reflects across the real axis. Inversion rescales by 1/|z|² and flips the argument sign, giving a distinct geometric transformation.
Does every complex number have a multiplicative inverse?
Every nonzero complex number has a multiplicative inverse, but z = 0 + 0i does not. The formula divides by a² + b², so if both a and b are zero the denominator vanishes and no complex number can satisfy z·w = 1.
Why does the magnitude of the inverse equal 1 divided by |z|?
Write z in polar form as z = |z|(cos θ + i sin θ). Its inverse is 1/z = (1/|z|)(cos(−θ) + i sin(−θ)). The magnitudes multiply to one, so |z|·|1/z| = 1, which immediately implies |1/z| = 1/|z|.
How accurate are the values produced by this calculator?
Calculations use standard floating‑point arithmetic in your browser. For most educational and engineering scenarios the precision slider is sufficient, but when rigorous proofs or safety‑critical work are required, confirm results with symbolic methods or additional reliable computational tools.
Can I work directly in polar form?
Yes. In polar mode you enter magnitude and angle, then choose radians or degrees. The calculator converts to rectangular form internally, applies the same inverse formula, and displays both rectangular and polar descriptions of the original number and its reciprocal.
Where can I find related complex number calculators?
To explore other complex operations, use the Complex Square Root Calculator on this site. It complements this tool by focusing on roots instead of reciprocals and deepens understanding of magnitude, argument, and multi‑valued behavior.