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.