Question

What is the conjugate of $8 + 4i$ ?

Answer 1

Complex number consists of both a real part and an imaginary part. To find the conjugate, just change the sign of an imaginary part of the complex number. Check for the complex number properly .

Complete step-by-step solution:
The given term is $8 + 4i$.
In this complex number, we have both a real part and an imaginary part.
Here the real part is $8$.
And the imaginary part is $4i$.
Whenever a conjugate is asked to find, we should change the sign of the imaginary part.
That is in the above question conjugate of 8 + 4i will be changed as 8 – 4i .
Since in the given problem, the imaginary part of the given complex number has positive sign.
In its conjugate it should have negative sign.
$+ 4i$ will become $- 4i$.
So the required answer will be $8 - 4i$.
Additional information :
Whenever conjugate is asked , keep in mind that the complex and its conjugate number when multiplied we should get a real number.
Here the complex number is 8 + 4i and its conjugate is $8 - 4i$.
Product of it is ( $\left( {{\text{ }}a + b} \right){\text{ }}\left( {a - b} \right){\text{ }} = {\text{ }}{a^2}-{\text{ }}{b^2})$ )
$\left( {{\text{ }}{i^2} = {\text{ }} - 1{\text{ }}} \right)$
Therefore,
Finally on solving the above equations, we got $80$ which is a real number..

Note: Complex number is a combination of real number and an imaginary number. The term without i is called real part and the term with i is called an imaginary part. Sometimes only imaginary parts will be given but still we can express as a complex number by taking 0 as the real number. Some students make the mistake by considering the conjugate with reciprocal.