Conjugate Calculator — Complex Conjugate of a + bi
Enter the real and imaginary parts of a complex number z = a + bi to instantly find its conjugate z̄ = a − bi, modulus |z|, argument, and the product z·z̄.
- 1
Sum of squares (a² + b²)
3² + 4² = 25 - 2
Modulus |z| = √(a²+b²)
√25 = 5
How does this calculator work?
The complex conjugate of z = a + bi is z̄ = a − bi — flip the imaginary sign. Both share modulus |z| = √(a²+b²). Their product z·z̄ = a²+b² is always real, making conjugates essential for rationalising complex denominators and proving that complex roots come in pairs.
Formula
How this is calculated
The complex conjugate of z = a + bi is z̄ = a − bi: the real part stays the same and the sign of the imaginary part is flipped. Geometrically, z̄ is the reflection of z across the real (horizontal) axis in the complex plane — both vectors have the same length but point symmetrically above and below the axis.
The modulus |z| = √(a² + b²) is the Euclidean distance from the origin to z (and to z̄, since conjugation preserves length). The argument arg(z) = atan2(b, a) is the angle the vector makes with the positive real axis, measured in radians (divide by π/180 for degrees). The argument of z̄ is −arg(z) — a reflection in angle.
Multiplying a complex number by its conjugate always gives a real, non-negative result: z·z̄ = (a+bi)(a−bi) = a²+b² = |z|². This property is used constantly in rationalising complex denominators: dividing 1/(a+bi) = (a−bi)/(a²+b²), which converts the result into real and imaginary parts without any i in the denominator.
Frequently asked questions
The conjugate of z = a + bi is z̄ = a − bi. It matters because z·z̄ = a² + b², a positive real number. This lets you rationalise complex fractions (multiply numerator and denominator by the conjugate), solve polynomial equations (complex roots come in conjugate pairs for real-coefficient polynomials), and compute modulus squared without a square root.
Yes — for any polynomial with real coefficients, if a + bi (b ≠ 0) is a root then a − bi is also a root. This is the Complex Conjugate Root Theorem. It does NOT apply to polynomials with complex coefficients (e.g. x² − 2i = 0 has roots ±√(2i), which are not conjugates of each other).
To compute (c+di)/(a+bi), multiply both numerator and denominator by the conjugate (a−bi): result = (c+di)(a−bi) / (a²+b²). The denominator becomes a real number, and the numerator expands to (ac+bd) + (ad−bc)i, giving a clean real/imaginary split.
TG we-Calculate Editorial Team. (2026). Conjugate Calculator — Complex Conjugate of a + bi [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/conjugate-calculator
TG we-Calculate Editorial Team. "Conjugate Calculator — Complex Conjugate of a + bi." TG we-Calculate. 2026. https://we-calculate.com/calculator/conjugate-calculator.
TG we-Calculate Editorial Team, "Conjugate Calculator — Complex Conjugate of a + bi," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/conjugate-calculator
@misc{wecalculate_conjugate_calculator, title = {Conjugate Calculator — Complex Conjugate of a + bi}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/conjugate-calculator}}, year = {2026}, note = {TG we-Calculate} }
Did this calculator help you?