Complex Conjugates

Complex Analysis · Axiom Academy

The "mirror image" of a complex number — a reflection across the real axis — and the one identity that makes complex division work. 1. The Conjugate Is a Reflection The complex conjugate of z = a + bi is obtained by changing the sign of the imaginary part. It is written (with a bar) or z^ * (with a star). Same real part a , opposite imaginary part 2. Examples — Flip the Imaginary Sign To conjugate, leave the real part alone and reverse the sign of the i term. The animation isolates that move: the imaginary component +bi swings down to -bi while the real component a never budges. — there is no imaginary part to flip, so the conjugate equals itself. — the real part is 0 , so flipping the imaginary part negates the whole number. 3. The Product Collapses to a Real Number Multiply z by its conjugate and the imaginary parts cancel exactly. Watch z and in the plane: as the product forms, their imaginary parts annihilate and the result lands on the real axis at a^2 + b^2 — which is precisely |z|^2 . — the conjugate of a product is the product of conjugates. — conjugation distributes over addition. — reflecting twice returns the original (reflect down, then back up). — a reflection preserves distance from the origin. 4. Adding and Subtracting a Number and Its Conjugate Combining z with its reflection has a clean result. Add them tip-to-tail and the imaginary parts cancel, landing on the real axis at 2a . Subtract them and the real parts cancel, landing on the imaginary axis at 2bi .

This is the written version of the interactive lesson above. See the full Complex Analysis course.