Conjugate of a complex number

• Definition:

  The conjugate of a complex number is another complex number that has the same real part as the original number but the opposite sign on its imaginary part.

  For example, $z = a + bi$, where $a$ and $b$ are real numbers, then the conjugate of $z$, denoted as $\overline{z}$ is given by:

  $\overline {z} = a - bi$

Examples

Example 1: Find the conjugate of complex number $(4,9)$

**Solution:

$z = (4,9)\\ \overline z = \overline{(4,9)}\\ \overline z = (4, - 9)\\ or \\ \overline z = 4-9i$**

Explanation:

The conjugate of a complex number is obtained by changing the sign of its imaginary part. To find the conjugate of this complex number, we need to change the sign of its imaginary part. In this case, the imaginary part is 9, so we need to change it to -9.

(4, 9) → (4, -9) (by changing the sign of the imaginary part)

Therefore, the conjugate of (4, 9) is (4, -9).

Example 2: Verify that $\overline{({\overline{z})}} = z$ where $z = 2+3i$

Solution:

Given** $z = 2+3i$

$\Rightarrow \overline{({\overline{2+3i})}} = 2+3i$

Taking conjugate of $2+3i$:

$\Rightarrow \overline{({\overline{2+3i})}} = \overline{2-3i}$

Taking conjugate of $\overline{2-3i}$:

$\Rightarrow \overline{({\overline{2-3i})}} = 2+3i$

Hence,$<strong>\overline{({\overline{z})}} = z$ holds true for $z = 2+3i$.