Determine whether each statement is true or false. If false, expl...

Question

Determine whether each statement is true or false. If false, explain why. The product of a complex number and its conjugate is always a real number.

Accepted Answer

Hey, everyone here we are asked to identify whether the given statement is true or not. And if it is false, we are asked to write the reason we are given the multiplication of the complex number with its conjugate always gives a real number. Here we have two answer choice options, answer choice, a true or answer choice B false. So here to solve this problem, we need to come up with a scenario that matches our given statement. So first, we need to select some complex number. In this case, we can just have, let's say two plus I. And now we need to identify its conjugate. So the conjugate of two plus I is just two minus I. And now we need to multiply these two expressions. So we have the quantity of two plus I times the quantity of two minus I. And now multiplying these expressions together, we can do this in multiple ways by either foiling out the terms. Or we can also recall the difference of squares property which states that the quantity of A plus B times the quantity of A minus B is equivalent to A squared minus B squared. Now comparing this formula to our expression, we have so far, we can rewrite the product as two squared minus I squared. And so now we just want to simplify our result here. And we first need to recall that I squared is equivalent to negative one. So now simplifying first, we have two squared which gives us four, then we have minus I squared, which we know is negative one. So we have four minus negative one. Now simplifying further, we actually have four plus one and adding these terms four plus one gives us five. So here we see our results after multiplying the con complex number with its conjugate is a whole number here of five. So we did get a real number by multiplying our complex number with its conjugate. Therefore, our answer here is true that the multiplication of a complex number with its conjugate always gives a real number. So we are left with answer choice. A where our given statement is true. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

Determine whether each statement is true or false. If false, explain why. The product of a complex number and its conjugate is always a real number.

Key Concepts

Complex Numbers

Conjugate of a Complex Number

Product of a Complex Number and Its Conjugate