The Complex Conjugate

When an engineer needs the strength of a radio or audio signal, or a physicist the probability hidden in a quantum state, they take a complex number and multiply it by its mirror image to squeeze out a real, positive answer. That mirror image is the complex conjugate.

Take any complex number z = a + bi and flip the sign of its imaginary part. The result is the complex conjugate, written \bar{z}:

\overline{a + bi} = a - bi.

Geometrically, conjugation is a reflection across the real axis: the point (a, b) mirrors to (a, -b). This one tiny operation is the workhorse of complex arithmetic — it is what lets us divide complex numbers, and what turns z\bar{z} into a plain real number.

The properties that make it useful

Every rule below follows from the definition by ordinary algebra. The two most important — conjugate of a sum and conjugate of a product — say that conjugation passes straight through addition and multiplication, which is exactly why it behaves so well.

For complex numbers z = a + bi and w:

conjugate of a sum is the sum of conjugates: \overline{z + w} = \bar{z} + \bar{w};
conjugate of a product is the product of conjugates: \overline{z\,w} = \bar{z}\,\bar{w};
z\bar{z} = |z|^2 = a^2 + b^2 — always real and \ge 0;
z + \bar{z} = 2a = 2\operatorname{Re}(z);
z - \bar{z} = 2bi = 2i\operatorname{Im}(z);
conjugating twice returns the original: \overline{\bar{z}} = z;
z is real \iff z = \bar{z}.

To see the key one, multiply z by \bar{z} using the multiplication rule with c = a and d = -b: the cross terms cancel, i^2 = -1 flips the last sign, and

z\bar{z} = (a + bi)(a - bi) = a^2 - b^2 i^2 = a^2 + b^2.

Dividing complex numbers

Division is the reason the conjugate earns its keep. A quotient \dfrac{a + bi}{c + di} has an i stuck in the denominator, and we cannot leave it there. The trick: multiply top and bottom by the conjugate of the denominator, which clears the i below.

Step 1 — multiply by the conjugate of the denominator. Multiplying by \dfrac{c - di}{c - di} = 1 changes nothing but the form:

\frac{a + bi}{c + di} = \frac{(a + bi)(c - di)}{(c + di)(c - di)}.

Step 2 — the denominator becomes real. By the z\bar{z} = c^2 + d^2 rule, the bottom is now a plain real number:

(c + di)(c - di) = c^2 + d^2.

Step 3 — expand the numerator and split into parts. Multiply out the top and collect real and imaginary terms:

\frac{a + bi}{c + di} = \frac{(ac + bd) + (bc - ad)i}{c^2 + d^2}.

That is a clean complex number again — real part over c^2 + d^2, imaginary part over c^2 + d^2.

A worked example

Compute \dfrac{3 + 2i}{1 + 4i}.

Step 1 — multiply top and bottom by 1 - 4i (the conjugate of the denominator):

\frac{3 + 2i}{1 + 4i} = \frac{(3 + 2i)(1 - 4i)}{(1 + 4i)(1 - 4i)}.

Step 2 — the denominator. Here c^2 + d^2 = 1^2 + 4^2 = 17:

(1 + 4i)(1 - 4i) = 1 + 16 = 17.

Step 3 — the numerator. Expand and use i^2 = -1:

(3 + 2i)(1 - 4i) = 3 - 12i + 2i - 8i^2 = 3 - 10i + 8 = 11 - 10i.

Step 4 — assemble. Put numerator over denominator:

\frac{3 + 2i}{1 + 4i} = \frac{11 - 10i}{17} = \frac{11}{17} - \frac{10}{17}i.

The most useful special case is dividing 1 by z. Multiply top and bottom by \bar{z}:

\frac{1}{z} = \frac{\bar{z}}{z\bar{z}} = \frac{\bar{z}}{|z|^2} = \frac{a - bi}{a^2 + b^2}.

So the reciprocal of z is its conjugate scaled down by |z|^2. For example \dfrac{1}{3 + 4i} = \dfrac{3 - 4i}{25} = \dfrac{3}{25} - \dfrac{4}{25}i, since |3 + 4i|^2 = 25.

It's tempting to conjugate only the first piece of an expression and call it done. But conjugation must flip the sign of every imaginary part in the whole expression — nothing gets left untouched, no matter how deeply it's buried.

For example, to conjugate \dfrac{2 + 3i}{5 - i} + 4i, a common slip is to write \dfrac{2 - 3i}{5 - i} + 4i — flipping only the numerator's leading term while leaving the denominator and the trailing 4i untouched. The correct conjugate flips every i in sight, at every level of the expression:

\overline{\frac{2 + 3i}{5 - i} + 4i} = \frac{2 - 3i}{5 + i} - 4i.

A safe rule of thumb: conjugation distributes over every +, -, \times and \div in the expression. Work from the outside in and conjugate each piece as you go — this matters most when the complex number you care about is nested deep inside a larger formula.

See the reflection

Slide a and b to move z = a + bi. Its conjugate \bar{z} = a - bi is the mirror image across the real axis — same real part, opposite imaginary part. The readout shows z\bar{z} = a^2 + b^2 recomputed live, always a non-negative real number.