Given that 𝑧 equals eight plus four 𝑖, find the modulus of 𝑧.
Well, to enable us to find the modulus of our complex number, what we need to do is actually consider a rule. And the rule is that for a complex number in the form 𝑧 equals 𝑎 plus 𝑏𝑖, its modulus is found by the equation: the modulus of the complex number equals the square root of 𝑎 squared plus 𝑏 squared.
And just to help us remember what the form 𝑧 equals 𝑎 plus 𝑏𝑖 actually means, well, it actually means where we have the complex number split into different parts. So we have the Re 𝑧 is 𝑎 and that’s our real part. And then, we have 𝑏 and 𝑏 is Im 𝑧, which is the imaginary part of our complex number. Okay, great, so now we have this formula for finding the modulus. We can do that and solve the problem.
Okay, given that our complex number is equal to eight plus four 𝑖, then eight is actually going to be our 𝑎. And our positive four — that’s because we’ve got a plus sign in front of our four — is going to be our 𝑏. Okay, so we can now substitute these into our equation for the modulus. So we’re gonna get the modulus of 𝑧 is equal to the square root of and then eight squared because that’s our 𝑎 and then plus four squared because four is our 𝑏.
Okay, great, so now let’s calculate this. And this gives us that the modulus of 𝑧 is equal to the square root of 64 plus 16. So therefore, the modulus of 𝑧 is equal to root 80. Okay, so is this the final answer? Well, actually, if we get to this point and we’ve actually got a surd, what we always say is trying to simplify the surd where we can. And we’re actually gonna use this surd rule to actually help us simplify our surd root 80 because we know that root 𝑎𝑏 is equal to root 𝑎 multiplied by root 𝑏. And what we want to do is actually find the highest square number. That’s one of the factors of 80.
So therefore, we’re going to get the modulus of 𝑧 is equal to root 16 root five. That’s because 16 multiplied by five is 80 sounds like our 𝑎𝑏. But also 16 is actually the highest square number that goes into 80. So therefore, we can say that the modulus of 𝑧 is going to be equal to four root five. And that’s because the square root of 16 is four. So it gives us four root five.
And we actually found that by using the formula that the modulus of a complex number is equal to square root of 𝑎 squared plus 𝑏 squared, where the complex number is in the form 𝑧 is equal to 𝑎 plus 𝑏𝑖.