Question

How do I find the complex conjugate of $10+6i?$

Answer 1

Hint: The complex conjugate or simply conjugate of a complex number $z=x+iy$ is defined as the complex number $\overline{z}=x-iy.$ We will find the imaginary part of the complex number and then change the sign of the imaginary part.

Complete step by step answer:
Let us consider the given complex number $10+6i.$
We are asked to find the complex conjugate of this number.
There are two parts in a complex number: the real part which contains only the real numbers and the imaginary part which contains the imaginary number $i.$
Suppose that we are given with a complex number $z=x+iy$ where $x$ and $y$ are real numbers. In this complex number, the real part that contains only the real number is $x$ and the imaginary part that contains the imaginary number $i$ is $iy.$
 The complex conjugate of this number is defined as the complex number $\overline{z}=x-iy.$
If we compare the complex number and its conjugate which is also a complex number, there is only one difference between these two numbers. And that is the sign of the imaginary part. In the conjugate complex number, the sign of the imaginary part is changed.
So, what we have to do is to change the sign of the imaginary part in the given complex number.
In the given complex number $10+6i,$ the real part is $10$ and the imaginary part is $6i.$
We will change the sign of the imaginary part to get the conjugate.
Hence the complex conjugate of the given number $10+6i$ is the complex number $10-6i.$

Note:
The modulus or absolute value of a complex number and that of the complex conjugate are the same. Suppose that $x+iy$ the complex number and $x-iy$ is its conjugate. Then the modulus is $\sqrt{{{x}^{2}}+{{y}^{2}}}=\sqrt{{{x}^{2}}+{{\left( -y \right)}^{2}}}.$