Video Transcript
What is the principal argument of the complex number 𝑧 equals 𝑎 plus 𝑏𝑖, where 𝑎 and 𝑏 are real, which lies in the second quadrant of the Argand diagram?
Let’s begin by visualizing our complex number on the Argand plane. Suppose both 𝑎 and 𝑏 are positive real numbers. Remember, 𝑎 is the real part of our complex number, and 𝑏 is the imaginary part. So the point 𝑧 plotted on an Argand plane will have coordinates 𝑎, 𝑏.
The argument of 𝑧, let’s call this 𝜃, is the angle that this line segment makes with the positive real axis as shown. Now, in fact, we can form a right triangle and add some dimensions. The included angle is 𝜃 and then the side opposite this is 𝑏 units, while the side adjacent to this angle is 𝑎 units. Since the tangent of an angle is equal to the ratio of the opposite and the adjacent side, we can say that tan 𝜃 here is equal to 𝑏 over 𝑎. Then, we could solve for 𝜃 by taking the inverse tan of both sides or the arctan of both sides. So 𝜃 is equal to the inverse tan of 𝑏 over 𝑎.
So how do we extend this idea to a point which lies in the second quadrant? This will now be a point where 𝑎 is less than zero and 𝑏 is greater than zero. Now, since the magnitudes of 𝑎 and 𝑏 remain unchanged, this new complex number is achieved by reflecting the original complex number across the imaginary axis. Since the angle that this new line segment makes with the negative real axis is equal to the angle that the original complex number makes with the positive real axis, we could say that the argument is equal to 𝜋 minus the inverse tan of 𝑏 over the magnitude of 𝑎.
Now, we’ve chosen the magnitude of 𝑎 because we know, of course, that it’s negative. And we want this to match our earlier value for 𝜃, but in fact we can rewrite this further. We know 𝑎 is negative, so 𝑏 divided by 𝑎 is itself going to be negative. And we know that both tan and arctan are odd functions. So the arctan or inverse tan of negative 𝑥 is equal to negative the inverse tan of 𝑥.
So when 𝑎 is negative — in other words, when our complex number lies in the second quadrant — the argument of 𝑧 is in fact equal to 𝜋 minus negative inverse tan of 𝑏 over 𝑎 or 𝜋 plus the inverse tan of 𝑏 over 𝑎. And so, whilst it’s generally sensible to sketch the complex number on an Argand diagram, we can use the values of 𝑏 and 𝑎 directly from the algebraic form.
