Video Transcript
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.
