The conjugate of a complex number vs an expression?

Question

The Conjugate of the complex number $a+bi$ is $a+b(-i)$. Therefore the conjugate of a real number $a$ is $a$.

But I have seen that the conjugate of the real number $a+b$ is $a-b$.

Shouldn't it be $a+b$?

Answer 1

It depends on context, really. The term "conjugate" has uses all over mathematics. For example, one might consider $1-\sqrt{2}$ the "conjugate" of $1+\sqrt{2}$, since multiplying them gives you a rational number (in this case, an integer). In the context of the picture you sent, I think this is the meaning of "conjugate". When dealing with complex numbers, usually it's appropriate to call $a-bi$ the complex conjugate of $a+bi$. Of course, $\sqrt{x-5}+\sqrt{5}$ is probably not the complex conjugate of $\sqrt{x-5}-\sqrt{5}$.