Find the conjugate of a+bi, compute the product z·z̄ = a²+b², and use conjugates to rationalize complex denominators.
z = a + bi → z̄ = a − bi
The conjugate of z = a + bi is z̄ = a − bi. Geometrically, it is the reflection of z across the real axis in the complex plane (the Argand diagram).
Key identities that always hold:
1/(a+bi) = (a−bi) / (a²+b²)
To divide by a complex number, multiply numerator and denominator by the conjugate. This turns the denominator into a real number:
1/(a+bi) · (a−bi)/(a−bi) = (a−bi)/(a²+b²)
In general:
(c+di)/(a+bi) = (c+di)(a−bi) / (a²+b²)
One-on-one Algebra 2 tutoring covers conjugates, modulus, polar form, and dividing complex numbers — with clear explanations that stick, not just steps to memorize.