Complex Conjugate Calculator

Find the complex conjugate of a + bi, compute the product z · z̄, and rationalize complex fractions with step-by-step solutions.

Enter Complex Number

Division / Rationalization (optional)

Enter a second complex number to compute z1 / z2 using conjugate rationalization.

Result

Complex Conjugate (z̄)
3 - 4i
Original z3 + 4i
Conjugate z̄3 - 4i
Product z · z̄25
|z| (Modulus)5
z + z̄6
z - z̄8i
z1 / z2--

Step-by-Step Solution

Conjugate of a + bi = a - bi

Understanding Complex Conjugates

The complex conjugate of a complex number z = a + bi is z̄ = a - bi. It is obtained by simply changing the sign of the imaginary part. Geometrically, the conjugate is the reflection of the complex number across the real axis in the complex plane.

Complex conjugates play a crucial role in simplifying complex arithmetic, particularly in division and rationalization. They are also fundamental in many areas of mathematics, physics, and engineering.

Key Properties of Conjugates

Conjugate Definition

Negate the imaginary part to find the conjugate.

conj(a + bi) = a - bi

Product with Conjugate

The product is always a non-negative real number.

z · z̄ = a² + b² = |z|²

Sum with Conjugate

The sum is always twice the real part.

z + z̄ = 2a = 2 Re(z)

Difference with Conjugate

The difference is always twice the imaginary part times i.

z - z̄ = 2bi = 2i Im(z)

Rationalization

Multiply numerator and denominator by the denominator's conjugate.

z/w = (z · w̄) / (w · w̄)

Double Conjugate

The conjugate of the conjugate returns the original number.

conj(conj(z)) = z

Rationalization of Complex Fractions

To divide one complex number by another, multiply both the numerator and denominator by the conjugate of the denominator. This eliminates the imaginary part from the denominator:

(a + bi) / (c + di) = [(a + bi)(c - di)] / [(c + di)(c - di)] = [(a + bi)(c - di)] / (c² + d²)

The denominator becomes a real number c² + d², making the fraction easy to separate into real and imaginary parts.

Step-by-Step Division Process

  1. Write the division as a fraction: (a + bi) / (c + di).
  2. Find the conjugate of the denominator: c - di.
  3. Multiply both top and bottom by the conjugate.
  4. Expand the numerator using FOIL.
  5. The denominator simplifies to c² + d².
  6. Separate into real and imaginary parts.

Properties of Complex Conjugates

  • Conjugate of a sum: conj(z + w) = conj(z) + conj(w)
  • Conjugate of a product: conj(z · w) = conj(z) · conj(w)
  • Conjugate of a quotient: conj(z / w) = conj(z) / conj(w)
  • Conjugate of a power: conj(zn) = conj(z)n
  • Real number test: z is real if and only if z = conj(z)
  • Pure imaginary test: z is pure imaginary if and only if z = -conj(z)

Applications

Complex conjugates are essential in electrical engineering for analyzing AC circuits (impedance calculations), in signal processing for computing power spectral density, in quantum mechanics where observables must be Hermitian (self-adjoint), and in control theory for analyzing system stability. They also appear in polynomial factoring, where complex roots of real polynomials always come in conjugate pairs.

The Conjugate Root Theorem

If a polynomial has real coefficients and a + bi is a root, then a - bi is also a root. This means complex roots of real polynomials always come in conjugate pairs, which is fundamental to polynomial factorization and equation solving.

Geometric Interpretation

In the complex plane (Argand diagram), the conjugate of z = a + bi is its mirror image across the real axis. Both z and z̄ have the same modulus (distance from the origin) but opposite arguments (angles). This reflection property is useful in many geometric proofs and constructions involving complex numbers.