Every complex number has a mirror twin across the real axis — and multiplying the two together performs a small miracle. The complex conjugate of a + bi is a − bi: flip the sign of the imaginary part and nothing else. The miracle is the product: (a + bi)(a − bi) = a² + b², a plain real number — the imaginary parts cancel completely. That trick is how you divide complex numbers: to compute (3 + 2i)/(1 − i), multiply top and bottom by the conjugate (1 + i) and the denominator becomes the real number 2.

Electrical engineers use conjugates constantly because AC circuit impedance is complex, and the power delivered depends on conjugate relationships — a junior EE student checking homework will reach for this page weekly. The conjugate also gives the modulus for free: |z|² = z × z̄. Enter the real and imaginary parts and the calculator returns the conjugate plus the real-valued product as a check.

With the pre-filled defaults — Variable A 10, Variable B 5 — the Complex Conjugate Calculator returns Result 50.0000. Change any input and those results update instantly, so you can see exactly how each number you enter moves the answer.