The conjugate of sum of two complex numbers is equal to sum of their conjugates.
A complex number can be added to another complex number to find their sum, whereas the sum of them is also a complex number and its conjugate is also a complex number.
The conjugate of sum of two complex numbers is mathematically equal to the sum of conjugates of both complex numbers because the conjugation is distributed over the addition of complex numbers. So, the property is called the distributive property of complex conjugates over addition.
Now, let’s learn how to express the distributive property over addition of complex conjugates in mathematical form. Let $z_1$ and $z_2$ denote two complex numbers and their conjugates are written as $\overline{z_1}$ and $\overline{z_2}$ in mathematics.
The conjugate of sum of complex numbers $z_1$ and $z_2$ is equal to the sum of conjugates of them. It can be written mathematically as follows.
$\overline{z_1+z_2}$ $\,=\,$ $\overline{z_1}+\overline{z_2}$
The above mathematical equation clearly explains how the conjugation is distributed over the addition of complex numbers and it can be used as a formula in mathematics.
Now, let’s have a look at an example to know whether the distributive property over addition of complex numbers is real or not.
$z_1$ $\,=\,$ $2+3i$ and $z_2$ $\,=\,$ $4-5i$
Firstly, let’s add both complex numbers to find sum of both complex numbers.
$\implies$ $z_1+z_2$ $\,=\,$ $2+3i$ $+$ $4-5i$
$\implies$ $z_1+z_2$ $\,=\,$ $6-2i$
The sum of two complex numbers is also a complex number. Now, let’s find its conjugate and it is written as $\overline{z_1+z_2}$.
$\,\,\,\therefore\,\,\,\,\,\,$ $\overline{z_1+z_2}$ $\,=\,$ $6+2i$
Now, find the conjugates of both complex numbers $\overline{z_1}$ and $\overline{z_2}$.
Finally, let’s add both conjugates of complex numbers to find the sum of them.
$\implies$ $\overline{z_1}+\overline{z_2}$ $\,=\,$ $2-3i$ $+$ $4+5i$
$\,\,\,\therefore\,\,\,\,\,\,$ $\overline{z_1}+\overline{z_2}$ $\,=\,$ $6+2i$
Now, let’s compare the result of conjugate of the sum of complex numbers with the result of sum of their conjugates. You will understand that they both are equal mathematically.
$\,\,\,\therefore\,\,\,\,\,\,$ $\overline{z_1+z_2}$ $\,=\,$ $\overline{z_1}+\overline{z_2}$ $\,=\,$ $6+2i$
It is called the distributive property of complex conjugates over addition.
The above explanation helps you to know the distributive property of complex conjugates over addition. Now, let’s begin to learn how to prove the distributive property over addition of complex conjugates mathematically.
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