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? 

Well, I think this is a matter of what is multiplication and what is division.
First, we denote that
$$\frac{1}{x}=y$$
which means
$$xy=1\qquad(\mbox{assuming $x\ne0$ in fundamental mathematics where there isn't Infinity($\infty$)})$$
Now,
$$\frac{1}{\frac{1}{x}}=\frac{1}{y}$$ by using the first equation. Here, 
by checking the second equation ($xy=1$), it is obvious that
$$\frac{1}{y}=x$$
thus
$$\frac{1}{\frac{1}{x}}=x$$
Q.E.D.
