Calculator Inputs
Example Data Table
| Input Form | Original Number | Conjugate | |z| | z × conjugate(z) |
|---|---|---|---|---|
| Rectangular | 3 + 4i | 3 - 4i | 5 | 25 |
| Rectangular | -2 + 7i | -2 - 7i | 7.2801 | 53 |
| Polar | 6[cos(30°) + i sin(30°)] | 6[cos(-30°) + i sin(-30°)] | 6 | 36 |
| Polar | 10[cos(1.2) + i sin(1.2)] | 10[cos(-1.2) + i sin(-1.2)] | 10 | 100 |
Formula Used
Rectangular form
If z = a + ib, then the complex conjugate is z̄ = a - ib.
The modulus is |z| = √(a² + b²).
The product with its conjugate is z × z̄ = a² + b² = |z|².
The reciprocal is 1 / z = (a - ib) / (a² + b²), provided z ≠ 0.
Polar form
If z = r[cos(θ) + i sin(θ)], then z̄ = r[cos(-θ) + i sin(-θ)].
The conjugate keeps the same modulus r.
Only the angle changes sign, which mirrors the point across the real axis.
How to Use This Calculator
Choose either rectangular or polar input mode first. Enter the required values, set your angle unit, choose the imaginary symbol, then select the desired decimal precision. Press the calculate button to display the conjugate immediately above the form, along with modulus, argument, reciprocal, product, and graph.
Use rectangular form when your number looks like a + bi. Use polar form when you already know the modulus and angle. The export buttons save the current result table as CSV or PDF for notes, assignments, reports, or revision sheets.
Frequently Asked Questions
1. What is a complex conjugate?
A complex conjugate is formed by changing only the sign of the imaginary part. For z = a + bi, the conjugate is a - bi. The real part stays the same, while the point reflects across the real axis.
2. Why is the conjugate useful?
It simplifies division, helps find reciprocals, removes imaginary terms from denominators, and is used in algebra, signal processing, physics, and engineering. It also gives a direct geometric reflection on the complex plane.
3. Does the modulus change after conjugation?
No. A number and its conjugate always have the same modulus. They are the same distance from the origin because conjugation only flips the sign of the imaginary coordinate.
4. What happens to the argument?
The argument changes sign, except at the origin where the argument is undefined. If z has angle θ, then its conjugate has angle -θ, after angle normalization.
5. Can I enter numbers in polar form?
Yes. This calculator accepts polar inputs using modulus and angle. It converts them to rectangular form, computes the conjugate, and also shows the conjugate in polar form with the negated angle.
6. Why is z × conjugate(z) always real?
Because the imaginary terms cancel. For z = a + bi, multiplying by a - bi gives a² + b². That result is always real and nonnegative, and it equals the modulus squared.
7. When is the reciprocal undefined?
The reciprocal is undefined only when z = 0. That is because division by zero is not allowed. Every nonzero complex number has a valid reciprocal computed from its conjugate and modulus squared.
8. Can I use i or j notation?
Yes. You can choose either i or j as the imaginary symbol. This is useful for mathematics courses using i and engineering contexts where j is often preferred.