Find the complex conjugate of any complex number by reflecting it across the real axis. Perfect for simplifying complex expressions, solving polynomial equations, and electrical engineering calculations.
Last updated: 2026-05-24T22:58:31.818Z | By ForgeCalc Engineering
The real component of your complex number
The imaginary component (coefficient of i)
Original Complex Number
3 + 4i
Complex Conjugate
3 - 4i
Enter real and imaginary components; the conjugate flips the sign of the imaginary component.
The complex conjugate of a complex number is found by changing the sign of its imaginary part. If z = a + bi, then its conjugate z̄ = a - bi. Geometrically, this represents a reflection across the real axis on the complex plane.
When a complex number is multiplied by its conjugate, the result is always a real number equal to the square of its magnitude: z · z̄ = (a + bi)(a - bi) = a² + b².
Original Complex Number
3 + 4i
Conjugate
3 - 4i
Product: 25
Original Complex Number
5 - 2i
Conjugate
5 + 2i
Product: 29
Original Complex Number
-1 + i
Conjugate
-1 - i
Product: 2
Original Complex Number
2i
Conjugate
-2i
Product: 4
(z₁ + z₂)̄ = z̄₁ + z̄₂
The conjugate of a sum equals the sum of conjugates
(z₁ · z₂)̄ = z̄₁ · z̄₂
The conjugate of a product equals the product of conjugates
(z₁ / z₂)̄ = z̄₁ / z̄₂
The conjugate of a quotient equals the quotient of conjugates
(z̄)̄ = z
The conjugate of a conjugate returns the original number
The result is always a real number equal to the square of its magnitude: (a + bi)(a - bi) = a² + b². For example, (3 + 4i)(3 - 4i) = 9 + 16 = 25.
Yes! Real numbers have no imaginary part (b = 0), so the conjugate of a real number is itself. For example, the conjugate of 5 is 5.
To divide complex numbers, multiply both numerator and denominator by the conjugate of the denominator. This eliminates the imaginary part from the denominator.
They're essential in electrical engineering for AC circuits, signal processing, and control systems. Conjugates help simplify calculations involving impedance and frequency responses.
On the complex plane (Argand diagram), the conjugate of a point is its reflection across the real (horizontal) axis. The real part stays the same, the imaginary part's sign flips.
A complex number and its conjugate have the same magnitude (distance from origin). If z has magnitude r, then its conjugate z̄ also has magnitude r.
Complex Conjugate Formula
z = a + bi → z̄ = a - bi
Multiplication by Conjugate
z · z̄ = (a + bi)(a - bi) = a² + b² = |z|²
Division Using Conjugates
z₁ / z₂ = (z₁ · z̄₂) / (z₂ · z̄₂) = (z₁ · z̄₂) / |z₂|²
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