Question Video: Finding the Principal Argument of a Complex Number Mathematics

Video Transcript

Given a complex number 𝑧, where the principal argument of 𝑧 is 𝜃 equals 11𝜋 over 12, determine the principal argument of 10𝑧.

Remember, the argument of a complex number describes the angle that the line segment joining the complex number to the origin makes with the positive real axis. Since this is measured in a counterclockwise direction, an argument of 11𝜋 over 12 radians will correspond to a complex number plotted in the second quadrant. So what do we mean when we talk about the principle argument of a complex number? The principal argument of a complex number is restricted to the open-closed interval negative 𝜋 to 𝜋 radians. So given any argument, specifically one that’s greater than 𝜋 or less than negative 𝜋, we can add or subtract multiples of two 𝜋 to ensure that our argument is within the range for the principal argument.

Now we’re told that the principle argument of 𝑧 is 11𝜋 over 12. But what does that mean for the principal argument of 10𝑧? Let’s begin by looking at an algebraic method, and then we’ll consider a geometric representation. Let’s imagine we’re given the complex number in exponential form: 𝑧 equals 𝑟𝑒 to the 𝑖𝜃. Now here 𝜃 is the argument and 𝑟 is the magnitude or modulus of the complex number. This simply represents the distance that the point lies on the complex plane from the origin. What happens when we multiply this complex number by 10? We multiply the entire expression 𝑟𝑒 to the power of 𝑖𝜃 by 10. So 10𝑧 is 10𝑟𝑒 to the power of 𝑖𝜃.

We notice the value of 𝜃 remains unchanged by multiplying the entire expression by 10. So if the argument of the complex number 𝑧 is some value 𝜃, then multiplying the entire expression by a constant does not change the argument. So the argument of 10𝑧 is 11𝜋 over 12. Now this makes a lot of sense if we think about this geometrically. Imagine we were to plot the complex number 10𝑧 on the complex plane. This essentially is a dilation on the complex plane by a scale factor of 10. This does not affect the argument and it just affects the magnitude or modulus of our complex number. Either way, we showed that the argument of 10𝑧 is 11𝜋 over 12.