### Video Transcript

If π is equal to eight plus two π, what is π plus π star?

In this question, weβve been given a complex number. Itβs π equals eight plus two π. Now, weβre asked to find the value of π plus π star. So, letβs recall what we actually mean by π star. π star is the conjugate of the complex number π . So, letβs recall what we actually mean by the complex conjugate of a number.

Letβs say we have a complex number π§ thatβs of the form π plus ππ. Now, π and π here are real constants. The real part of π§ is π, whereas its imaginary part is the coefficient of π. So, itβs π. When we find the complex conjugate of this number β letβs denote that π§ star β we simply change the sign of the imaginary part. So, for a complex number of the form π plus ππ, its conjugate is π minus ππ.

Well, the real part of our complex number is eight, and its imaginary part is two. So, π star, its complex conjugate, is eight minus two π. All weβve done is change the sign of the imaginary part. Now, weβre looking to find the value of π plus π star. So, thatβs eight plus two π plus eight minus two π. Now, of course, two π minus two π is zero. So, weβre left with eight plus eight, which is 16.

And so, given the complex number π equals π plus two π, π plus π star β thatβs the sum of this complex number and its conjugate β 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 π§ equals π plus ππ, π§ plus π§ star will be two times π.