# Thread: Imaginary number proof problem

1. ## Imaginary number proof problem

I've just started imaginary numbers.
I'm trying to derive what the inverse of j is.
Where $\displaystyle j =\sqrt{-1}$

So I did the following:

$\displaystyle \frac{1}{j}=\frac{z}{1}$

Multiplied j by both sides.

$\displaystyle jz=1$

squared both sides

$\displaystyle -z^2 =1$

and by solving I get that z=j

which is obviously wrong since $\displaystyle j^2 \neq 1$

Any insights on what I'm over looking?
I derived it another why, but I want to know why this approach isn't working.
Thanks!

2. Originally Posted by elieh
Multiplied j by both sides $\displaystyle jz=1$ squared both sides $\displaystyle -z^2 =1$ and by solving I get that z=j
You should get $\displaystyle z=\pm j$ . Both are solutions of $\displaystyle -z^2 =1$ but only $\displaystyle z=-j$ is solution of $\displaystyle jz=1$. Why?. Because if we square both sides of an equation, can appear solutions that are not in the original one.

Another example: the only solution of $\displaystyle x=2$ is $\displaystyle 2$ and the solutions of $\displaystyle x^2=4$ are $\displaystyle \pm 2$

3. Alright, Thank you!