Simplify negative five plus five
root three 𝑖 over negative root three minus 𝑖, giving your answer in both
algebraic and trigonometric form.
To divide complex numbers in
rectangular or algebraic form, that’s the general form 𝑧 equals 𝑎 plus 𝑏𝑖, we
first need to find the complex conjugate of the denominator. For a complex number 𝑎 plus 𝑏𝑖,
its complex conjugate is 𝑎 minus 𝑏𝑖. We denote the complex conjugate
with a little bar as shown. Once we have this, we can multiply
both the numerator and the denominator by the complex conjugate. The denominator of our fraction is
negative root three minus 𝑖. So its complex conjugate is
negative root three plus 𝑖. We need to multiply both the
numerator and the denominator of the fraction by negative root three plus 𝑖.
We multiply these just like we
multiply any normal pair of brackets. Negative five multiplied by
negative root three is five root three. Negative five multiplied by 𝑖 is
negative five 𝑖. Five root three 𝑖 multiplied by
negative root three is negative five root three squared multiplied by 𝑖. And five root three 𝑖 multiplied
by 𝑖 is five root three 𝑖 squared.
Remember, 𝑖 squared is simply
negative one. So five root three multiplied by 𝑖
squared is negative five root three. Root three squared is three. So negative five multiplied by root
three squared multiplied by 𝑖 is negative 15𝑖. And this means that the numerator
of this fraction simplifies to negative 20𝑖.
Let’s now multiply the
denominator. Negative root three multiplied by
negative root three is root three squared. Negative root three multiplied by
𝑖 is negative root three 𝑖. Negative 𝑖 multiplied by negative
root three is positive root three 𝑖. And negative 𝑖 multiplied by 𝑖 is
negative 𝑖 squared. And this simplifies to four. So our fraction becomes negative
20𝑖 over four, which is negative five 𝑖.
In rectangular or algebraic form,
the complex number is 𝑧 is equal to negative five 𝑖. Comparing this to the general
algebraic form for a complex number, we can see that 𝑎 must be equal to zero. And 𝑏 must be equal to negative
five for our complex number. When we write a complex number in
trigonometric or polar form, we write it as 𝑧 is equal to 𝑟 multiplied by cos 𝜃
plus 𝑖 sin 𝜃, where 𝑟 is called the modulus of the complex number and 𝜃 is the
So we need to find a way to
represent the real and complex components of our number in terms of 𝑟 and 𝜃. In fact, we do have some formulae
we can use. The modulus is the square root of
𝑎 squared plus 𝑏 squared. This comes from the Pythagorean
theorem. And to find 𝜃, we use tan 𝜃 is
equal to 𝑏 over 𝑎. Let’s substitute what we know about
our complex number into these formulae.
𝑟 is equal to the square root of
zero squared plus negative five squared, which is five. And tan 𝜃 is equal to negative
five divided by zero. Now, this is actually
undefined. However, we know that the tangent
function is undefined at negative 𝜋 over two and at intervals of 𝜋. So 𝜃 must be negative 𝜋 over two
for this to be true.
Notice how we chose a negative 𝜋
over two as opposed to positive 𝜋 over two since the complex component of the
number in rectangular form was negative five. This means that the number would
sit on the axes between the third and fourth quadrant on the complex coordinate
plane. Since we measure this in an
anticlockwise direction from zero to negative 𝜋, this coordinate has to be negative
𝜋 over two.
The trigonometric form of our
complex number is five cos of negative 𝜋 over two plus 𝑖 sin of negative 𝜋 over