Is negative nine the complex conjugate of the number negative nine?
Let’s begin by recalling what we actually mean by the complex conjugate of a number. Let 𝑍 be a complex number of the form 𝑎 plus 𝑏𝑖, where 𝑎 and 𝑏 are real constants. We say that 𝑎 is the real part of our complex number, whereas the imaginary part is 𝑏. It’s the coefficient of 𝑖. We define 𝑍 star, which we sometimes also call 𝑍 bar, as the complex conjugate of the number 𝑍. It’s 𝑎 minus 𝑏𝑖. And we find the complex conjugate by changing the sign of the imaginary part.
So let’s write our number negative nine as a complex number. Negative nine itself is a real number. So if we’re going to write it as a complex number, we’re going to write it as negative nine plus zero 𝑖. Its real part is negative nine, and its imaginary part is zero. We said that the complex conjugate is found by changing the sign of the imaginary part. So the complex conjugate of our complex number is negative nine minus zero 𝑖. But of course, that’s simply negative nine. And so negative nine is indeed the complex conjugate of the number negative nine.