# What is conjugate of 11 + 17i?

Jul 16, 2015

$11 - 17 i$

#### Explanation:

The complex conjugate of an arbitrary complex number $z = a + b i$ (with $a , b \setminus \in \mathbb{R}$) is $\overline{z} = a - b i$.

Geometrically, in the complex plane, $\overline{z} = a - b i$ is a reflection of the point $z = a + b i$ across the horizontal (real) axis.

Algebraically, it has the properties that the product $z \cdot \overline{z}$ is real and, moreover, it equals the square of the modulus ("length") $| z {|}^{2}$ of the complex number $z = a + b i$. Just do the algebra to see this:

$z \cdot \overline{z} = \left(a + b i\right) \cdot \left(a - b i\right) = {a}^{2} - a b i + a b i - {b}^{2} \cdot {i}^{2} = {a}^{2} + {b}^{2}$ since ${i}^{2} = - 1$. Also note that ${a}^{2} + {b}^{2} = {\left(\sqrt{{a}^{2} + {b}^{2}}\right)}^{2} = | z {|}^{2}$.

This is useful for dividing complex numbers. Given a division problem, written as a fraction, such as $\frac{2 + 7 i}{3 + 8 i}$, the trick to determining what this equals in the form $a + b i$ is to multiply the top (numerator) and bottom (denominator) of this fraction by the complex conjugate of the bottom:

$\frac{2 + 7 i}{3 + 8 i} \cdot \frac{3 - 8 i}{3 - 8 i} = \frac{6 - 16 i + 21 i - 56 {i}^{2}}{9 + 64} = \frac{62}{73} + \frac{5}{73} i$.

If you think about this trick for the abstract expression $\frac{1}{z}$, it implies that you can write it as $\frac{1}{z} = \frac{1}{z} \cdot \frac{\overline{z}}{\overline{z}} = \frac{\overline{z}}{|} z {|}^{2} = \frac{1}{|} z | \cdot \frac{\overline{z}}{|} z |$.

This has a nice geometric meaning. It implies that the "inversion mapping" (function) that sends $z$ to its multiplicative inverse $\frac{1}{z}$ (if $z \ne 0$) can be thought of geometrically as first reflecting $z$ across the horizontal (real) axis to get $\overline{z}$, then "normalizing" $\overline{z}$ to $\frac{\overline{z}}{|} z |$ to get a point 1 unit from the origin on the same ray from the origin through $\overline{z}$. Finally, multiply this number by the scale factor $\frac{1}{|} z |$ to get the number $\frac{1}{z} = \frac{1}{|} z | \cdot \frac{\overline{z}}{|} z |$ on the same ray through the origin as $\overline{z}$, but with a modulus of $\frac{1}{|} z |$.

Here's a picture to illustrate this process with $z = \frac{3}{2} + 2 i$, $\overline{z} = \frac{3}{2} - 2 i$, $\frac{\overline{z}}{|} z | = \frac{\frac{3}{2} - 2 i}{\sqrt{\frac{9}{4} + 4}} = \frac{\frac{3}{2} - 2 i}{\sqrt{\frac{25}{4}}} = \left(\frac{3}{2} - 2 i\right) \cdot \frac{2}{5} = \frac{3}{5} - \frac{4}{5} i$, and $\frac{1}{z} = \frac{\overline{z}}{|} z {|}^{2} = \frac{\frac{3}{2} - 2 i}{\frac{25}{4}} = \frac{6}{25} - \frac{8}{25} i$ 