What is the conjugate of the complex numbers two minus seven 𝑖?
We begin by recalling what we actually mean by the complex number 𝑧. This is of the form 𝑎 plus 𝑏𝑖, where 𝑎 and 𝑏 are real constants. We say that 𝑎 is the real part of 𝑧 whereas 𝑏, the coefficient of 𝑖, is its imaginary part. We then define the complex conjugate of 𝑧 to be equal to 𝑧 star. And we find this by changing the sign of the imaginary part. And so, for a complex number 𝑎 plus 𝑏𝑖, its conjugate is 𝑎 minus 𝑏𝑖.
Let’s look at our complex number then. Well, it’s two minus seven 𝑖. The real part of 𝑧 is two. And its imaginary part is negative seven. We said that, to find the conjugate of a complex number, we change the sign of the imaginary part.
So the complex conjugate of 𝑧, 𝑧 star, will have an imaginary part of positive seven. Its real part remains as two. So we find 𝑧 star is equal to two plus seven 𝑖. And so, we found the conjugate of the complex number two minus seven 𝑖. It’s two plus seven 𝑖.