Complex Conjugate Calculator

What is the complex conjugate?

The complex conjugate of a complex number z = a + bi is the number z̄ = a − bi. You form it by keeping the real part unchanged and flipping the sign of the imaginary part.

The term “conjugate” comes from the Latin conjugare, meaning “to join together.” Complex conjugates are paired in a specific way: they have the same real part and imaginary parts that are equal in magnitude but opposite in sign. This pairing has deep mathematical consequences — polynomials with real coefficients always have complex roots in conjugate pairs, and the product of a number with its conjugate is always a non-negative real number.

The complex conjugate is denoted several ways depending on the textbook: z̄ (z with an overline), z* (z with an asterisk, common in physics and engineering), or conj(z). All three mean the same thing.

Quick examples:

• Conjugate of 3 + 4i is 3 − 4i
• Conjugate of −2 − 7i is −2 + 7i
• Conjugate of 5 (real number) is 5 (the conjugate of a real number is itself)
• Conjugate of 6i (pure imaginary) is −6i

A number is equal to its own conjugate if and only if it is real — that is, when its imaginary part b = 0.

The conjugate formula

For z = a + bi:

The conjugate of a conjugate returns the original number: z̄̄ = z. This makes the conjugate operation an involution — applying it twice is the same as doing nothing.

Worked example: Find the conjugate of z = −3 + 5i.

Here a = −3, b = 5. So z̄ = −3 − 5i.

Check: the real part stays −3, the imaginary part changes from +5 to −5. ✓

Worked example: Find the conjugate of z = 4.

Here a = 4, b = 0. So z̄ = 4 − 0i = 4. A real number is its own conjugate. ✓

The modulus |z|

The modulus (also called the absolute value or magnitude) of z = a + bi is the distance from the origin to the point (a, b) in the complex plane:

The modulus is always a non-negative real number. It equals zero only when z = 0.

The key identity: The product of z and its conjugate equals the square of the modulus:

This identity is one of the most useful in complex arithmetic. It explains why multiplying by a conjugate eliminates imaginary parts — the cross terms +abi and −abi cancel, leaving only the sum of squares.

Note that 3 + 4i has modulus exactly 5 — this is a complex analog of the 3-4-5 Pythagorean triple.

Argument and polar form

The argument of z = a + bi is the angle θ that the line from the origin to (a, b) makes with the positive real axis, measured counter-clockwise:

The argument is typically expressed in radians (range −π to π) or in degrees (range −180° to 180°). Using atan2 (the two-argument arctangent) correctly handles all four quadrants.

The polar form uses the modulus and argument to represent z as:

This is often abbreviated z = re^(iθ) using Euler’s formula e^(iθ) = cos θ + i sin θ.

The conjugate in polar form is simply z̄ = r(cos(−θ) + i sin(−θ)) = re^(−iθ) — same modulus, negated argument. This geometric interpretation makes sense: reflecting the point (a, b) across the real axis gives (a, −b), which is z̄. The reflection preserves distance from the origin, so the modulus stays the same.

Example: For z = 3 + 4i:

• r = |z| = 5
• θ = atan2(4, 3) ≈ 0.9273 rad ≈ 53.13°
• Polar form: 5(cos 0.9273 + i sin 0.9273)
• Conjugate z̄ = 3 − 4i has argument ≈ −0.9273 rad ≈ −53.13°

Key properties of conjugates

The conjugate interacts cleanly with all the standard complex arithmetic operations. These properties are used constantly in calculations:

The distributive properties (conjugate of a sum = sum of conjugates; conjugate of a product = product of conjugates) make the conjugate map a field automorphism of the complex numbers — it preserves the algebraic structure. For polynomials with real coefficients, this means that if z is a root, so is z̄.

The real part extraction trick: Since z + z̄ = 2a, you can isolate the real and imaginary parts:

The reciprocal and division of complex numbers

Dividing by a complex number uses the conjugate to eliminate the imaginary part from the denominator. The key technique is multiplying numerator and denominator by the conjugate of the denominator:

More generally, to compute w / z for two complex numbers:

Example: Compute (2 + 3i) / (1 − 2i).

Multiply numerator and denominator by the conjugate of 1 − 2i, which is 1 + 2i:

• Numerator: (2 + 3i)(1 + 2i) = 2 + 4i + 3i + 6i² = 2 + 7i − 6 = −4 + 7i
• Denominator: (1 − 2i)(1 + 2i) = 1 + 4 = 5

So (2 + 3i) / (1 − 2i) = (−4 + 7i) / 5 = −0.8 + 1.4i

The conjugate is what makes this possible: multiplying denominator by its conjugate gives a real number (the modulus squared), and dividing a complex number by a real number is trivial — just divide real and imaginary parts separately.

Second example: Compute 1 / (2 − 3i).

Multiply numerator and denominator by the conjugate 2 + 3i:

• Numerator: 1 · (2 + 3i) = 2 + 3i
• Denominator: (2 − 3i)(2 + 3i) = 4 + 9 = 13

So 1 / (2 − 3i) = (2 + 3i) / 13 ≈ 0.154 + 0.231i

The reciprocal 1/z = (a − bi) / (a² + b²) is undefined only when z = 0 (modulus = 0), since there is no number you can multiply by zero to get one.

Complex conjugate reference table

The complex plane: geometry of the conjugate

In the complex plane (also called the Argand diagram), a complex number z = a + bi is plotted as the point (a, b): the real part on the horizontal axis, the imaginary part on the vertical axis.

The conjugate z̄ = a − bi is the reflection of z across the real axis. This geometric picture explains all the conjugate’s algebraic properties at a glance:

• Reflection preserves horizontal distance (real part unchanged) but flips vertical position (imaginary part negated).
• Reflection preserves the distance from the origin, so |z̄| = |z|.
• Reflection negates the angle, so arg(z̄) = −arg(z).
• Two points that are reflections of each other are equidistant from every point on the mirror line — which is why z + z̄ is always real (it lies on the real axis).

Multiplying by the conjugate rotates the angle to zero. When you multiply z = r·e^(iθ) by z̄ = r·e^(−iθ), the product has angle θ + (−θ) = 0 and modulus r². A complex number at angle 0 with any modulus is a positive real number — which is why z · z̄ is always a non-negative real.

This geometric view also makes the conjugate root theorem intuitive: a polynomial with real coefficients has real-valued outputs for real-valued inputs, which means its graph is symmetric about the real axis when viewed in the complex plane. If z = a + bi is a root (a point where the polynomial hits zero), its mirror image a − bi must also be a root.

Real-world uses of the complex conjugate

Electrical engineering: AC circuit analysis

In AC circuits, voltages and currents are represented as complex numbers (phasors). The impedance Z = R + jX, where j is the engineering symbol for i. To find the power delivered to a circuit, engineers compute the complex power S = V · I*, where I* is the complex conjugate of the current phasor.

The real part of S is the active power (watts), the imaginary part is the reactive power (VAR). The conjugate is essential because it is what makes the real and reactive parts separate cleanly.

Maximum power transfer from a source to a load occurs when the load impedance is the complex conjugate of the source impedance — this is the conjugate matching condition, fundamental to antenna and RF circuit design.

Quantum mechanics: inner products and expectation values

In quantum mechanics, quantum states are represented by complex-valued wave functions ψ(x). The probability of finding a particle at position x is |ψ(x)|² = ψ(x) · ψ*(x) — the wave function times its complex conjugate. This guarantees the probability is always real and non-negative.

Expectation values of observables are computed as integrals involving both ψ and ψ*. Without the conjugate, these integrals could be complex-valued, which would be physically meaningless for an observable quantity.

Signal processing: Fourier analysis

The Fourier transform of a real-valued signal has a specific symmetry: F(−f) = F(f)*, the value at negative frequency is the complex conjugate of the value at positive frequency. This Hermitian symmetry means that for a real signal, the negative-frequency components carry no new information — you only need to store the positive-frequency half.

Correlation between two signals is computed using conjugate multiplication in the frequency domain. The inverse Fourier transform of F₁(f) · F₂*(f) gives the cross-correlation of the two signals — a fundamental operation in radar, sonar, and communications.

Polynomial algebra: conjugate root theorem

For any polynomial with real coefficients, complex roots always come in conjugate pairs. If 2 + 3i is a root of a real polynomial, then 2 − 3i must also be a root. This is the complex conjugate root theorem, a consequence of the fact that the conjugate map is a field automorphism.

In practice, this means a degree-n real polynomial always has an even number of non-real roots. If you find one complex root numerically, you get the second one for free — it’s just the conjugate.