How is $-16/4i$ equal to $4i$?

Question

I came across a problem: $-16/4i$. Every time I put it into a calculator, it comes out as $4i$, but when I try to solve it is $-4i$, because of the negative one in front of the $16$.

Answer 1

Hint: What is $\frac{1}{i}$ equal to?

Answer 2

One way to do it: $$ \frac{-16}{4i} = \frac{-16}{4i}\frac{-i}{-i} = \frac{16i}{4i(-i)} = \frac{16i}{4} = 4i. $$ The key thing is that $i(-i) = 1$.

Answer 3

I think the problem is that you are entering $-16/4i$ into your calculator, but you ought to be entering $-16/(4i)$. The calculator is (correctly) interpreting your input to mean $-16 \div 4 \times i$, which evaluated from left to right is in fact $-4i$. If you want the calculator to compute $-16 \div (4 \times i)$, you need to include the parentheses.

And, as others have pointed out, dividing by $i$ is the same thing as multiplying by $-i$, because $\frac{1}{i} = -i$.

Answer 4

Notice, $i^2=-1$ hence, $$\frac{-16}{4i}=\frac{-16}{4i}\times\frac{i}{i}=\frac{-16i}{4i^2}=\frac{-16i}{4(-1)}=\frac{-16i}{-4}=\color{red}{4i}$$

Answer 5

Another way of doing it is: $\frac{-16}{4i}$ Then reduce to get $\frac{-4}{i}$. We know that $i^2=-1$, there fore we can make it $\frac{4i^2}{i}$ and by reduceing again we get $4i$.