Properties of Conjugate

Four fundamental identities. z·z̄ = a²+b² = |z|² is the most used — it converts division into multiplication. Also: conjugate distributes over addition and multiplication: z₁+z₂ bar = z̄₁+z̄₂, and z₁z₂ bar = z̄₁·z̄₂.

Let $z = a + ib$ and $\bar{z} = a - ib$. All four identities follow by direct computation from this definition.

Identity 1 — Double Conjugate

$$\bar{\bar{z}} = \overline{(a - ib)} = a + ib = z$$

Taking the conjugate twice returns the original number. The conjugate is an involution.

Identity 2 — Sum

$$z + \bar{z} = (a + ib) + (a - ib) = 2a = 2\operatorname{Re}(z)$$

This is the most-used form for extracting the real part of a complex number. Rearranged: $\operatorname{Re}(z) = \dfrac{z + \bar{z}}{2}$.

Identity 3 — Difference

$$z - \bar{z} = (a + ib) - (a - ib) = 2ib = 2i\operatorname{Im}(z)$$

Rearranged: $\operatorname{Im}(z) = \dfrac{z - \bar{z}}{2i}$. Note the denominator is $2i$, not $2$.

Identity 4 — Product

$$z\bar{z} = (a + ib)(a - ib) = a^2 + b^2 = |z|^2$$

This is the most important identity in the chapter. It converts the product of two conjugates into a real number — which is why multiplying numerator and denominator by the conjugate clears the denominator in division.

Distributive Properties

Over addition:

$$\overline{z_1 + z_2} = \overline{(a_1 + a_2) + i(b_1 + b_2)} = (a_1 + a_2) - i(b_1 + b_2) = \bar{z}_1 + \bar{z}_2$$

Over multiplication:

$$\overline{z_1 z_2} = \overline{(a_1 a_2 - b_1 b_2) + i(a_1 b_2 + a_2 b_1)} = (a_1 a_2 - b_1 b_2) - i(a_1 b_2 + a_2 b_1)$$ $$\bar{z}_1 \bar{z}_2 = (a_1 - ib_1)(a_2 - ib_2) = (a_1 a_2 - b_1 b_2) - i(a_1 b_2 + a_2 b_1)$$

Both expressions are identical, so $\overline{z_1 z_2} = \bar{z}_1 \bar{z}_2$.

By induction this extends to any finite product: $\overline{z_1 z_2 \cdots z_n} = \bar{z}_1 \bar{z}_2 \cdots \bar{z}_n$, and in particular $\overline{z^n} = \bar{z}^n$.