Complex Numbers: Conjugates and Division

The conjugate of a + bi is a - bi. You flip the sign of the imaginary part. Key property: (a + bi)(a - bi) = a^2 + b^2 (always a real number!) This is because: a^2 - abi + abi - b^2i^2 = a^2 + b^2. The conjugate is used to eliminate i from denominators.

Example: Find the conjugate of 3 + 7i and multiply them.

To divide frac{a + bi}{c + di}, multiply top and bottom by the conjugate of the denominator: [formula] The denominator becomes a real number, and you simplify.

Example: Simplify frac{5 + i}{2 - 3i}