If 𝑠 is equal to eight plus two 𝑖, what is 𝑠 plus 𝑠 star?
In this question, we’ve been given a complex number. It’s 𝑠 equals eight plus two 𝑖. Now, we’re asked to find the value of 𝑠 plus 𝑠 star. So, let’s recall what we actually mean by 𝑠 star. 𝑠 star is the conjugate of the complex number 𝑠. So, let’s recall what we actually mean by the complex conjugate of a number.
Let’s say we have a complex number 𝑧 that’s of the form 𝑎 plus 𝑏𝑖. Now, 𝑎 and 𝑏 here are real constants. The real part of 𝑧 is 𝑎, whereas its imaginary part is the coefficient of 𝑖. So, it’s 𝑏. When we find the complex conjugate of this number — let’s denote that 𝑧 star — we simply change the sign of the imaginary part. So, for a complex number of the form 𝑎 plus 𝑏𝑖, its conjugate is 𝑎 minus 𝑏𝑖.
Well, the real part of our complex number is eight, and its imaginary part is two. So, 𝑠 star, its complex conjugate, is eight minus two 𝑖. All we’ve done is change the sign of the imaginary part. Now, we’re looking to find the value of 𝑠 plus 𝑠 star. So, that’s eight plus two 𝑖 plus eight minus two 𝑖. Now, of course, two 𝑖 minus two 𝑖 is zero. So, we’re left with eight plus eight, which is 16.
And so, given the complex number 𝑠 equals 𝑎 plus two 𝑖, 𝑠 plus 𝑠 star — that’s the sum of this complex number and its conjugate — is 16. And actually, this is a result that we can generalize. The sum of a complex number and its conjugate will always be two times the real part of that complex number. So, in this case, it was two times eight. And in the general case of the complex number 𝑧 equals 𝑎 plus 𝑏𝑖, 𝑧 plus 𝑧 star will be two times 𝑎.