Question Video: Finding the Argument of Powers of Complex Numbers in Algebraic Form Mathematics • Third Year of Secondary School

Video Transcript

Consider the complex number 𝑧 equals seven plus seven 𝑖. 1) Find the argument of 𝑧. 2) Hence, find the argument of 𝑧 to the power of four.

Here, we have a complex number whose real and imaginary parts are positive. This means we would plot this complex number in the first quadrant on the Argand diagram. And we can therefore find the argument by using the formula arctan of 𝑏 divided by 𝑎, where 𝑏 is the imaginary part and 𝑎 is the real part. In our case, that’s the arctan of seven divided by seven. And that’s 𝜋 by four radians.

So, how do we find the argument of 𝑧 to the power of four? Well, what we’re not going to do is evaluate the complex number 𝑧 to the power of four. Instead, we’re going to recall the fact that the argument of the product of two complex numbers is equal to the sum of their arguments. We’re going to extend this and say, well, if we have 𝑧 times 𝑧 times 𝑧 times 𝑧, that’s going to be equal to the argument of 𝑧 plus the argument of 𝑧 plus the argument of 𝑧 plus the argument of 𝑧. But actually, that’s equal to four lots of the argument of 𝑧. And in our example, that’s equal to four lots of 𝜋 by four, which is simply 𝜋 radians. And we can generalize this idea and say that the argument of 𝑧 to the power of 𝑛 is equal to 𝑛 multiplied by the argument of 𝑧.