Question

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

Answer 1

We first explain the concept of conjugate number of complex numbers $a+ib$. The general form helps in identifying the coefficients. We use that to find the conjugate of $8+4i$.

Complete step by step solution:
The general form of a complex number is $a+ib$. Here $a,b$ are scalar coefficients.
The conjugate of that complex number is $a-ib$.
The concept of conjugate is where the multiplication of the complex numbers gives the square value of the modulus value of those complex numbers.
We can see that the modulus value of $a+ib$ and $a-ib$ is equal to $\sqrt{{{a}^{2}}+{{b}^{2}}}$.
Now we multiply the complex numbers $a+ib$ and $a-ib$.
We get $\left( a+ib \right)\left( a-ib \right)={{a}^{2}}+{{b}^{2}}$.
Now we find the conjugate of $8+4i$. Equating with the general form of $a+ib$, we get $a=8,b=4$.
The conjugate becomes $8-4i$.
**Therefore, the conjugate of $8+4i$ is $8-4i$.

Note:
We can verify the modulus value for $8+4i$ and $8-4i$.
$\left| 8+4i \right|=\left| 8-4i \right|=\sqrt{{{8}^{2}}+{{4}^{2}}}=\sqrt{80}$.
The multiplication gives $\left( 8+4i \right)\left( 8-4i \right)={{8}^{2}}+{{4}^{2}}=80$.