Why is $\frac{1}{\frac{1}{X}}=X$? Can someone help me understand in basic terms why $$\frac{1}{\frac{1}{X}} = X$$
And my book says that "to simplify the reciprocal of a fraction, invert the fraction"...I don't get this because isn't  reciprocal by definition the invert of the fraction? 

$$y=\frac1{\frac 1 x} $$
$$y'(x)=\left(\frac1{\frac 1 x}\right)' = -\left(\frac 1 {\left(\frac 1x\right)^2}\right)\left({\frac 1 x}\right)' = 
\frac 1 {\left(\frac 1x\right)^2} \cdot {\frac 1 {x^2}} = \frac {y^2(x)}{x^2}$$
So we have that $$x^2dy = y^2dx\\
\int \frac{dy}{y^2} = \int \frac{dx}{x^2}\\
-\frac{1}{y} = -\frac 1 x + C\\$$
Let's take a look at $y(1)$. $\frac 1 1 = 1$, this is already explained in a more common problem here: 
Why is $n$ divided by $n$ equal to $1$?
So $y(1)=\frac{1}{\frac{1}{1}} =\frac 1 1 = 1$.
Note that I lost one possible solution, $y(x)=0$, by dividing by $y$. But since $y(1)=1$, it isn't really the solution.
Again: $y(1)=1$, so $~-\frac 1 1 = -\frac 1 1 + C ~~\Rightarrow~~ C=0$. Then $\frac {1} {y} = \frac {1}{x} \Rightarrow x=y$.
