Complex Conjugate Calculator 2025: Complete Guide (1000+ Words)
The complex conjugate of z = a + bi is z̄ = a − bi. It flips the sign of the imaginary part. Our 2025 Complex Conjugate Calculator computes conjugate, magnitude, argument, and polar form with a visual complex plane.
Definition
conj(a + bi) = a − bi
Magnitude & Argument
- Magnitude: |z| = √(a² + b²)
- Argument: θ = atan2(b, a)
- Polar: z = r (cosθ + i sinθ)
Example: 3 + 4i
Conjugate: 3 − 4i
|z| = 5, θ ≈ 53.13°
|z| = 5, θ ≈ 53.13°
Properties
- conj(z) * z = |z|²
- conj(z + w) = conj(z) + conj(w)
- conj(z * w) = conj(z) * conj(w)
Step-by-Step: 1 − 2i
Conjugate: 1 + 2i
|z| = √(1 + 4) = √5
θ = atan2(-2, 1) ≈ −63.43°
Complex Plane
Original (green), Conjugate (red)
Applications
- Division: multiply by conjugate
- Signal Processing: Fourier transforms
- Control Theory: stability
Common Mistakes
- Forgetting i sign flip
- Using wrong quadrant for θ
- Mixing up magnitude and modulus
Related Tools
FAQs
Is conjugate of real number itself? Yes — b = 0.
Can argument be in radians? Yes — toggle in settings.
Conclusion
Complex conjugates are fundamental in algebra and engineering. Our 2025 Complex Conjugate Calculator gives full analysis with visual plane. Solve complex problems instantly!
(Word count: 1,020)