Definition: The conjugate of the complex number a + bi is a – bi, the same number with the sign of its imaginary part flipped; their product is always a real number.
What is a conjugate?
For a complex number a + bi, the conjugate is a – bi. Geometrically it is the mirror image across the horizontal axis of the complex plane. Conjugates always show up in pairs when a polynomial with real coefficients has a non-real root.
Why is it useful?
Multiplying a number by its conjugate removes the imaginary part: (a + bi)(a – bi) = a2 + b2. That makes it the standard tool for dividing complex numbers, because you can clear i from the denominator.
Worked example: divide 6 + 2i by 1 + 3i. Multiply top and bottom by 1 – 3i.
Top: (6 + 2i)(1 – 3i) = 6 – 18i + 2i – 6i2 = 12 – 16i. Bottom: 1 + 9 = 10.
The quotient is 1.2 – 1.6i.
What is a common mistake?
Students sometimes flip both signs and write -a – bi, which is the negative, not the conjugate. Only the sign in front of the i term changes.