Complex Conjugate Calculator

For any complex number z = a + bi, compute its conjugate z̄ = a − bi, modulus |z|, argument θ, and polar representation. Interactive complex plane visualization shows symmetry across the real axis.

Examples:
3+4i
1−2i
−2+5i
7i (pure imag)
−4 (real)
2−3i
Client-side computation — No data leaves your browser. Instant and private.

Deep Dive into Complex Conjugates

The complex conjugate of a complex number \( z = a + bi \) is defined as \( \overline{z} = a - bi \). Geometrically, it reflects the point across the real axis. Conjugates are fundamental in electrical engineering (impedance), quantum mechanics (probability amplitudes), signal processing (Hermitian symmetry), and polynomial root finding (non-real roots come in conjugate pairs).

Key identities: \( \overline{z_1 + z_2} = \overline{z_1} + \overline{z_2} \), \( \overline{z_1 \cdot z_2} = \overline{z_1} \cdot \overline{z_2} \), \( z \cdot \overline{z} = |z|^2 \), and \( \overline{\overline{z}} = z \).
Conjugate Root Theorem (real coefficients): If a polynomial has real coefficients, any non‑real complex root must occur in conjugate pairs.
Example: \( x^2 - 2x + 5 = 0 \) has discriminant \( -16 \), roots \( 1 + 2i \) and \( 1 - 2i \), which are conjugates. The calculator instantly verifies that \( \overline{1+2i} = 1-2i \), showing the symmetry.

How the Tool Works: Step-by-Step Mathematics

Given \( z = a + bi \):

  • Conjugate: \( \overline{z} = a - bi \) – simply flips the sign of the imaginary part.
  • Modulus (absolute value): \( |z| = \sqrt{a^2 + b^2} \), representing the distance from origin.
  • Argument (angle): \( \theta = \text{atan2}(b, a) \) in radians, converted to degrees. The principal argument lies typically in \( (-\pi, \pi] \).
  • Polar form: \( z = r(\cos\theta + i\sin\theta) = r e^{i\theta} \) (Euler's formula).

The interactive plot visualizes both \( z \) and \( \bar{z} \) along with a line segment representing modulus. Symmetry across the real axis is immediate.

Properties & Applications Across Disciplines

Property / Application Description
Polynomial Roots (Real Coefficients) If coefficients are real, non-real roots occur in conjugate pairs — essential for solving quadratics with negative discriminant. The Conjugate Root Theorem is a direct consequence of \( \overline{P(z)} = P(\overline{z}) \).
AC Circuit Theory Impedance \( Z = R + iX \); its conjugate ensures power calculations and resonance conditions.
Quantum Mechanics The inner product uses complex conjugates to ensure probability amplitudes are real.
Signal Processing Hermitian symmetry in Fourier transforms yields real-valued signals.
Division of Complex Numbers \( \frac{z_1}{z_2} = \frac{z_1 \overline{z_2}}{|z_2|^2} \) rationalizes denominator.
Engineering Insight: Conjugate Matching

In RF engineering, maximum power transfer occurs when the load impedance is the complex conjugate of the source impedance (\( Z_L = \overline{Z_S} \)). Our calculator helps visualize why conjugates cancel reactance. For a source \( 50 + j30 \, \Omega \), the optimal load is \( 50 - j30 \, \Omega \). The conjugate operation directly yields the required imaginary sign reversal.

Frequently Asked Questions

It reflects the complex number across the real axis. The point (a, b) maps to (a, -b). The modulus remains unchanged while the argument changes sign.

Because \( (a+bi)(a-bi) = a^2 + b^2 \) which is a non-negative real number, equal to |z|².

Absolutely. For a real number like −4, the conjugate is itself (−4), and the graph shows the point on the real axis overlapped.

We use the standard JS Math.atan2(y, x) which gives double-precision accuracy, typically 15 decimal digits. Suitable for all educational and practical purposes.

Euler's formula states \( e^{i\theta} = \cos\theta + i\sin\theta \), so any complex number can be written as \( r e^{i\theta} \). This representation is used extensively in control theory and quantum physics.

Trusted mathematical reference — This calculator implements validated formulas derived from analytic geometry and complex analysis. Reviewed by the GetZenQuery tech team, referencing standard texts (Churchill & Brown, "Complex Variables and Applications", MathWorld). Last updated April,2026

References: MathWorld: Complex Conjugate; Needham, T. "Visual Complex Analysis" (1997); Wikipedia: Complex conjugate.