Video Transcript
What does the argument of a complex number represent? Is it (A) its imaginary coordinate in the complex plane? (B) Its real coordinate in the complex plane. Is it (C) the angle it makes with the positive real axis? Or (D) the angle it makes with the positive imaginary axis. Finally, is it (E) its distance from the origin in the complex plane.
Let’s begin by reminding ourselves the different ways that we can represent a complex number. There’s algebraic form. 𝑧 equals 𝑎 plus 𝑏𝑖. In this case, 𝑎 and 𝑏 must be real numbers. When a number is written in this form, we say that 𝑎 is the real part of the complex number, whilst 𝑏 is the imaginary part. And if we were to plot this point on the complex plane, 𝑎 would be the real coordinate and 𝑏 would be the imaginary coordinate.
So far, we haven’t seen anything that describes the argument. So the next type that we’re interested in is the polar form of a complex number. And we also have the exponential form of a complex number. The polar form is 𝑧 equals 𝑟 cos 𝜃 plus 𝑖 sin 𝜃, whereas the exponential form is 𝑟𝑒 to the power of 𝑖𝜃. So what do the values 𝑟 and 𝜃 represent? Well, in both forms, 𝑟 is the modulus of the complex number. And it’s found by finding the square root of the sum of the squares of the real and imaginary parts of the algebraic form. This means, essentially, when we plot the point 𝑎, 𝑏 on the complex plane, 𝑟 tells us the distance away from the origin it lies.
Finally, 𝜃 is the argument. Now, this is the bit we’re interested in. Let’s imagine that both the real and imaginary part of our complex number are positive. We draw a line that connects this point to the origin and then drop in a right-angle triangle. 𝜃 is then the angle that the line 𝑟 makes with the positive real axis. And in fact, it doesn’t matter if 𝑎 and 𝑏 are not real. 𝜃 is still the angle it makes with the positive real axis.
So let’s compare this to our options. We see that (A), the imaginary coordinate in the complex plane, is given by the value of 𝑏. The real coordinate in the complex plane is given by the value of 𝑎 in the algebraic form. 𝜃 the argument is the angle it makes with the positive real axis. We haven’t defined the angle it makes with the positive imaginary axis, although we could calculate it. And 𝑟 is its distance from the origin in the complex plane. So the answer is (C). The argument is the angle it makes with the positive real axis.
