What is the modulus of 𝑧 given that 𝑧 is equal to two minus eight 𝑖?
In this question, we’re given a complex number 𝑧 and we’re asked to find the value of the modulus of 𝑧. So to answer this question, we’re first going to need to recall exactly what we mean by the modulus of a complex number 𝑧. We recall that when we say the modulus of a complex number 𝑧, we mean the distance that 𝑧 is from the origin in an Argand diagram.
Therefore, one way of finding the modulus of 𝑧 will be to plot the point 𝑧 on an Argand diagram and then find its distance from the origin. So let’s do this now. Remember, in an Argand diagram, our horizontal position represents the real part of our number and the vertical position represents the imaginary part of our number.
We want to plot the point 𝑧 onto our Argand diagram. We can see the real part of this complex number is two. So on our Argand diagram, its horizontal coordinate is going to be equal to two. And we can also see the imaginary part of our complex number, that’s the coefficient of 𝑖, is negative eight. So the vertical coordinate on our Argand diagram of the point 𝑧 is going to be negative eight. So we can use this to plot the point 𝑧 on our Argand diagram.
Now, remember, the question wants us to find the modulus of 𝑧. And that’s the distance 𝑧 is from the origin in our Argand diagram. And we can do this by forming the following right-angled triangle. We can see the modulus of 𝑧 will be the hypotenuse of this triangle. We can see the height of this right-angled triangle is the modulus of the 𝑦-coordinate of 𝑧, which is eight. And the width of this right-angled triangle is the modulus of the 𝑥-coordinate. It’s equal to two.
Therefore, by using the Pythagorean theorem, we can find the length of the hypotenuse. It’s equal to the square root of two squared plus eight squared. And of course, in this case, the hypotenuse is our modulus of 𝑧. And we can evaluate this. First, two squared is equal to four and eight squared is equal to 64.
Now, we could add these together. However, we can also notice that both of these share a factor of four. And this is useful because we know four is a perfect square. So we’ll take this factor out. This gives us the square root of four times one plus 16. Now, we’ll use our laws of exponents to take the square root of each of these separately. This gives us the square root of four multiplied by the square root of one plus 16. And of course the square root of four is two and one plus 16 is equal to 17, giving us a final answer of two times the square root of 17.
However, there is a much easier method we could’ve used to answer this question. We can find a formula to help us find the modulus of a complex number given in the following form. 𝜔 is equal to 𝑎 plus 𝑏𝑖, where 𝑎 and 𝑏 are real numbers. We’ll find this formula in exactly the same way we did before. We’ll want to start by plotting 𝜔 onto our Argand diagram. To do that, we need to find the real part and the imaginary part of 𝜔. That’s 𝑎 and 𝑏, respectively. Then we can just plot the point 𝜔 onto our Argand diagram. And the modulus of 𝜔 will be its distance from the origin.
And we can find this distance in the same way we did before. We form the following right-angled triangle. And the modulus of 𝜔 will be the hypotenuse of this right-angled triangle. The height of this right-angled triangle is the absolute value of 𝑏, and the width of this triangle is the absolute value of 𝑎.
Therefore, by using the Pythagorean theorem, we can find a formula for the modulus of 𝜔. It’s equal to the square root of 𝑎 squared plus 𝑏 squared. And this is because the modulus of 𝑎 all squared is equal to 𝑎 squared and the modulus of 𝑏 all squared is equal to 𝑏 squared. So, in fact, an easier way of finding the modulus of 𝑧 given to us in the question is just to apply this formula. In our case, the real part of 𝑧 is equal to two and the imaginary part of 𝑧 is equal to negative eight.
So by our formula, the modulus of 𝑧 is the square root of two squared plus negative eight all squared. And then if we follow the exact same steps we did before, we would also see that this is equal to two times the square root of 17. Therefore, in this question, we were able to show two different methods of finding the modulus of two minus eight 𝑖. In both cases, we saw this was equal to two root 17.