Question Video: Finding the Modulus of Imaginary Numbers

Video Transcript

What is the modulus of the complex number two 𝑖?

In this question, we’re asked to calculate the modulus of the complex number two 𝑖. And to do this, we’re going to need to recall what we mean by the modulus of a complex number. We recall the modulus of a complex number is its distance from the origin in an Argand diagram. So one way of answering this question is to plot the complex number two 𝑖 onto an Argand diagram. And remember, in an Argand diagram, our horizontal axis represents the real component of the complex number and the vertical axis represents the imaginary part of the complex number.

So when we want to plot the complex number two 𝑖 onto this diagram, this has no real component. It only has an imaginary component. And the imaginary component is equal to two. So its vertical coordinate is going to be equal to two. And therefore, its distance from the origin is also going to be equal to two. And therefore, because the modulus of a complex number is its distance from the origin, we’ve shown the modulus of two 𝑦 is equal to two.

However, it is worth pointing out this is not the only way we could’ve answered this question. We can see the complex number we’re given in the question, two 𝑖, is given in a special form. It’s given in algebraic form. And the algebraic form of a complex number is the form 𝑎 plus 𝑏𝑖, where our values of 𝑎 and 𝑏 are real numbers.

And when we’re asked to find the modulus of a complex number given in algebraic form, we can just use the formula. The modulus of 𝑎 plus 𝑏𝑖 is going to be equal to the square root of 𝑎 squared plus 𝑏 squared. And in fact, we can use our Argand diagram to see why this is true. We can plot the point 𝑎 plus 𝑏𝑖 onto our Argand diagram. Its real component is going to be equal to 𝑎, and its imaginary component is going to be equal to 𝑏.

This means that we can plot the point 𝑎 plus 𝑏𝑖 onto our Argand diagram. And this distance from the origin or modulus is going to be the hypotenuse of a right-angled triangle. And we can find the two shorter lengths of this right-angled triangle. It’s 𝑎 and 𝑏. So by the Pythagorean theorem, the hypotenuse of this right-angled triangle or the modulus of 𝑎 plus 𝑏𝑖 will be equal to the square root of 𝑎 squared plus 𝑏 squared.

Then, all we would need to do is use two 𝑖 in our formula. And of course, for two 𝑖, our value of 𝑎, the real part of two 𝑖, is equal to zero and our value of 𝑏, the imaginary part of two 𝑖, is equal to two. Substituting these values in, we get the modulus of two 𝑖 is equal to the square root of zero squared plus two squared, which of course we can calculate is equal to two.

Therefore, we were able to see two different ways of finding the modulus of the complex number two 𝑖. In both cases, we saw the modulus of two 𝑖 is equal to two.