Question Video: Relationship between the Complex Conjugate and the Modulus Mathematics

Video Transcript

Consider the complex number 𝑧 equals negative four plus 𝑖 root five. Calculate the modulus of 𝑧. Calculate the modulus of 𝑧 star. And determine 𝑧 times 𝑧 star.

So here we have a complex number. It’s 𝑧 equals negative four plus 𝑖 root five. Now, the general form of a complex number 𝑧 is 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. And then, these two parts denote the modulus of 𝑧. That’s the square root of the sum of the squares of the real and imaginary parts.

So, in our question, the real part of 𝑧 is negative four whereas the imaginary part is the square root of five. And we can therefore say that the modulus of this is the square root of negative four squared plus the square root of five squared. Negative four squared is 16 whereas the square root of five squared is simply five. Then, 16 plus five is 21, so we find the modulus of 𝑧 to be equal to the square root of 21.

Now, the next part of this question is fairly similar, but we’re being asked to calculate the modulus of 𝑧 star. Now, 𝑧 star is called the conjugate of 𝑧. And we find this by changing the sign of the imaginary part of our complex number. So, if 𝑧 is equal to 𝑎 plus 𝑏𝑖, 𝑧 star is 𝑎 minus 𝑏𝑖. And so for our complex number 𝑧, its conjugate is negative four minus 𝑖 root five. This time, its real part is negative four, but its imaginary part is negative root five. So the modulus of 𝑧 star, the modulus of the conjugate, is the square root of negative four squared plus negative root five squared. But, of course, negative root five squared is still five. So we get an answer again of the square root of 21.

The third and final part of this question asks us to determine the product of 𝑧 and 𝑧 star. That’s the product of negative four plus 𝑖 root five and negative four minus 𝑖 root five. Let’s use the FOIL method to distribute these parentheses. Negative four times negative four is positive 16. Then negative four times negative 𝑖 root five is four 𝑖 root five. 𝑖 root five times negative four is negative four 𝑖 root five. And, finally, 𝑖 root five times negative 𝑖 root five is negative 𝑖 squared root five squared.

We can see that four 𝑖 root five minus four 𝑖 root five is zero. And we should also recall that 𝑖 squared is equal to negative one. So negative 𝑖 squared times the square root of five squared becomes positive five. And 𝑧 times 𝑧 star is 16 plus five, which is equal to 21. And so we’ve answered all three parts of this question.

But what information can we generally quote? Firstly, the modulus of a complex number and its conjugate will always be equal. And secondly, the product of a complex number and its conjugate will always be equal to the modulus of that complex number squared.