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? 
 A: Does this work for you? Start with $${1\over1/x}$$ Multiply top and bottom by $x$:  $${1\over1/x}{x\over x}={x\over(1/x)x}={x\over1}=x$$
A: Symbolically:
$$ \frac{1}{\frac{1}{X}}=1\div \frac{1}{X}=1\times\frac{X}{1}=X$$
OR
$$ \frac{1}{\frac{1}{X}}= \frac{1}{\frac{1}{X}} \times \frac{X}{X}=\frac{X}{1}=X$$
Intuitively:
We are asking how many 1/X pieces we can fit into a whole. Clearly there must be X of them!
A: 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.
A: Naturally, if we inverse some inverted object, we will have the object itself!
This matter take place for numbers and their operations:
$$-(-x)=x$$
and for any non-zero $x$ $$\frac{1}{\frac{1}{x}}=x$$
(if we know "our limits" this fact is true for all objects in mathematics also for functions $(f^{-1})^{-1}=f$, directions(vectors, matrices, ...) and so on.)
A: Maybe this will help you see why $\;\dfrac 1{\large \frac 1X}= X.\;$ We multiply numerator and denominator by $X$, which we can do because we can multiply any number by $\dfrac XX = 1$ without changing the actual value of the number:
$$\frac 1{\Large \frac 1X}\cdot \frac XX = \frac X1 = X$$
$$ $$
A: $$y=\frac1{\frac 1 x} $$
$$y'(x)=\left(\frac1{\frac 1 x}\right)' = -\left(\frac 1 {\left(\frac 1x\right)^2}\right)\left({\frac 1 x}\right)' = 
\frac 1 {\left(\frac 1x\right)^2} \cdot {\frac 1 {x^2}} = \frac {y^2(x)}{x^2}$$
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 $\frac {1} {y} = \frac {1}{x} \Rightarrow x=y$.
A: Let, $\frac{1}{\frac{1}{x}} = y$, where $y \neq x$. Now,
$\frac{1}{\frac{1}{x}} = y \\\Rightarrow 1 = \frac{y}{x}\\\Rightarrow x = y$
A contradiction! So, $\frac{1}{\frac{1}{x}} = y$, where $y \neq x$ is false. So, $\frac{1}{\frac{1}{x}} = x$
A: $1/x$ is, by definition, the number that you multiply $x$ by to get $1$.
Similarly, $1/\left(1/x\right)$ is the number that you multiply $1/x$ by to get $1$.
But wait a sec: we just learned in the first sentence that that number is $x$.
A: I look at it this way. How do you check division? With multiplication.$\frac{15}{5}=3$ because $3\cdot 5=15$. Similarly, $\frac{1}{\frac{1}{x}}=x$ because $x\cdot\frac{1}{x}=\frac{x}{1}\cdot\frac{1}{x}=\frac{x\cdot 1}{1\cdot x}=\frac{x}{x}=1$
A: Note that: $1/x=x^{-1}$
and also that: $(x^a)^b=x^{ab}$
Therefore $1/(1/x)=(x^{-1})^{-1}=x^1=x$
A: All the other answers use algebra, but I prefer the intuitive explanation based on this comment.
Suppose that you need to divide a large pizza for infants, who cannot safely  eat an entire pizza. Instead, each infant can eat only a slice. Each slice is smaller than the original pizza; so the size of each slice must necessarily be a fraction $< 1$.
In the picture below, each slice is $1/6$ of the original pizza. 

Then $\dfrac{\color{red}{1}}{\color{forestgreen}{\dfrac{1}{6}}} = \color{red}{1}$ pizza divided by slices of size $\color{forestgreen}{\dfrac{1}{6}}$. 
Well, what occurs if you divide $\color{red}{1}$ pizza totally into slices of size $\color{forestgreen}{\dfrac{1}{6}}$?
You end with 6 slices of pizza, after you cut the pizza per the 3 black straight lines as pictured above! 
A: Simply because:
$$
\frac{1}{\frac{1}{X}}=\frac{1}{(\frac{X}{1})^{-1}}=\frac{1}{1}\frac{X}{1}=\frac{1X}{1}=X
$$
And yes, the reciprocal of a fraction, is the inverted fraction...
