Simplify two plus four 𝑖 over
To work out how to divide two
plus four 𝑖 by 𝑖, we recall the definition of 𝑖. It’s a solution to the equation
𝑥 squared equals negative one. And we say that 𝑖 squared is
equal to negative one. Or often 𝑖 is equal to the
square root of negative one.
If we consider this fraction as
two plus four 𝑖 divided by the square root of negative one, we can see that, to
simplify, we’d need to perform the same process as rationalizing the denominator
when we’re dealing with any other radical. We multiply both the numerator
and the denominator of our fraction by the square root of negative one.
In fact, we know that the
square root of negative one is 𝑖. So we’re going to be
multiplying both the numerator and the denominator of this fraction by 𝑖. And we’re allowed to do that
because multiplying by 𝑖 over 𝑖 is the same as multiplying by one. Essentially, we’re creating an
Let’s apply the distributive
property for 𝑖 multiplied by two plus four 𝑖. 𝑖 multiplied by two is two 𝑖,
and 𝑖 multiplied by four 𝑖 is four 𝑖 squared. Now of course 𝑖 squared is
equal to negative one. So our expression becomes two
𝑖 plus four multiplied by negative one, which is negative four plus two 𝑖.
On the denominator, we have 𝑖
multiplied by 𝑖, which is of course 𝑖 squared, which is negative one. So we can rewrite two plus four
𝑖 over 𝑖 as negative four plus two 𝑖 over negative one. And now we’re simply dividing
by a real number. And to divide a complex number
by a real number, we divide the real part and then separately divide the
imaginary part. Negative four divided by
negative one is four, and two 𝑖 divided by negative one is negative two 𝑖. And we fully simplify two plus
four 𝑖 over 𝑖. It’s four minus two 𝑖.