What is the argument of the complex number four 𝑖?
In this question, we’re asked to find the argument of the complex number four 𝑖. To answer this question, we’re first going to need to recall what we mean by the argument of a complex number. We recall the argument of a complex number 𝑧 written arg of 𝑧 is the angle that 𝑧 makes with the positive real axis on an Argand diagram. And usually, we give the argument of a complex number in radians. And we also know in radians there’s lots of different values which represent the same angle. For example, zero, two 𝜋, and four 𝜋 all represent the same angle. And to get around this problem, usually when we find the argument of a complex number, we try to give our answer between negative 𝜋 and 𝜋, where we include 𝜋. And this value is usually called the principal argument of 𝑧.
Now there’s a few different ways of finding the argument of a complex number. For example, you might know a few formulas to find it. However, we should always sketch a picture before we try and find the argument of a complex number. So we’ll start with our Argand diagram. Remember, the horizontal axis is the real part of our complex number, and the vertical axis is the imaginary part of our complex number.
So to plot four 𝑖 onto our Argand diagram, we need to find the real part of four 𝑖 and the imaginary part of four 𝑖. The imaginary part of four 𝑖 will be the coefficient of 𝑖, which in this case is four. And the real part of four 𝑖 is equal to zero because there’s no added constant. So this tells us, on an Argand diagram, the complex number four 𝑖 will be represented by the point zero, four. And we can input this onto our Argand diagram.
Next, we’re going to want to sketch the argument of four 𝑖 onto our Argand diagram. And to do this, it can be helpful to add the ray from the origin to our complex number four 𝑖. Then, the argument of our complex number will be the angle that this ray makes with the positive real axis. Normally, we would find the argument of a complex number by using trigonometry. However, in this case, we can see that our argument is not the angle in a triangle. Instead, it’s the angle between two of our axes, so we know this is a right angle. And we know that a right angle is represented by 𝜋 by two. And since this angle is counterclockwise from our positive real axis, this means it will be positive 𝜋 by two. So we were able to show the argument of four 𝑖 is 𝜋 by two.
However, there is something we can notice about this example. We can ask the question, what is the argument of 𝑏𝑖 if our value of 𝑏 is positive? Using the exact same reasoning we did for four 𝑖, we can show that the argument of 𝑏𝑖 would also be 𝜋 by two. This means we’ve shown a useful result. If our value of 𝑏 is positive, then the argument of 𝑏𝑖 will always be equal to 𝜋 by two. We could have also used this result to answer our question.
Therefore, we were able to show the argument of the complex number four 𝑖 is 𝜋 by two.