Put the number 𝑧 equals five root
two over two minus five root six over two 𝑖 in exponential form.
This complex number is currently in
algebraic form. It has a real part of five root two
over two and an imaginary part of negative five root six over two. Remember a complex number in
exponential form is 𝑟𝑒 to the 𝑖𝜃, where 𝑟 is the modulus and 𝜃 is the argument
in radians. The modulus is fairly
straightforward to calculate. For a complex number of the form 𝑎
plus 𝑏𝑖, its modulus is the square root of the sum of the square of 𝑎 and 𝑏.
In this case, it’s the square root
of five root two over two all squared plus negative five root six over two all
squared. Five root two over two all squared
is 25 over two. And negative five root six over two
all squared is 75 over two. The sum of 25 over two and 75 over
two is 100 over two, which is simply 50. So the modulus of 𝑧 is the square
root of 50, which we can simplify to five root two. But what about the argument?
If we put this complex number on
the Argand plane, it’s represented by the point whose Cartesian coordinates are five
root two over two and negative five root six over two. This means it lies in the fourth
quadrant. We can find the argument for
complex numbers that lie in the first and fourth quadrant by using the formula
arctan of 𝑏 divided by 𝑎 or arctan of the imaginary part divided by the real
In this example, that’s arctan of
negative five root six over two divided by five root two over two, which is negative
𝜋 by three. So the argument for our complex
number is negative 𝜋 by three. We calculated the modulus of 𝑧 to
𝑏 five root two and its argument to be negative 𝜋 by three. So in exponential form, we can say
it’s five root two 𝑒 to the negative 𝜋 by three 𝑖. And at this point, it’s worth
recalling that the argument is periodic with a period of two 𝜋. So we can add or subtract multiples
of two 𝜋 to our argument.
If we add two 𝜋 to negative 𝜋 by
three, we get five root two 𝑒 to the five 𝜋 over three 𝑖.