# What is the complex conjugate of number 12+5i?

Jul 9, 2015

The complex conjugate of $\left(12 + 5 i\right)$ is $\left(12 - 5 i\right)$

#### Explanation:

The complex conjugate is the value you need to multiply a given complex number by

• to eradicate the imaginary component
• using the concept of difference of squares:
$\textcolor{w h i t e}{\text{XXXX}}$$\left({a}^{2} - {b}^{2}\right) = \left(a + b\right) \left(a - b\right)$
$\textcolor{w h i t e}{\text{XXXX}}$or, for complex numbers:
$\textcolor{w h i t e}{\text{XXXX}}$$\left({a}^{2} + {b}^{2}\right) = \left({a}^{2} - {\left(b i\right)}^{2}\right) = \left(a + b i\right) \left(a - b i\right)$

That is
$\textcolor{w h i t e}{\text{XXXX}}$$\left(a + b i\right)$ and $\left(a - b i\right)$ are complex conjugates of each other.