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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?
$1/x$ 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 $x$.
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:
$$\frac 1{\Large \frac 1X}\cdot \frac XX = \frac X1 = X$$ $$ $$
$$y=\frac1{\frac 1 x} $$ $$y'(x)=\left(\frac1{\frac 1 x}\right)' = -\left(\frac 1 {\left(\frac 1x\right)^2}\right)\left({\frac 1 x}\right)' = \frac 1 {\left(\frac 1x\right)^2} \cdot {\frac 1 {x^2}} = \frac {y^2(x)}{x^2}$$
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 $\frac {1} {y} = \frac {1}{x} \Rightarrow x=y$.
Does this work for you? Start with $${1\over1/x}$$ Multiply top and bottom by $x$: $${1\over1/x}{x\over x}={x\over(1/x)x}={x\over1}=x$$
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!
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.
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.)
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, $\frac{1}{\frac{1}{x}} = x$
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 $x\cdot\frac{1}{x}=\frac{x}{1}\cdot\frac{1}{x}=\frac{x\cdot 1}{1\cdot x}=\frac{x}{x}=1$
Note that: $1/x=x^{-1}$ and also that: $(x^a)^b=x^{ab}$
Therefore $1/(1/x)=(x^{-1})^{-1}=x^1=x$
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!
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...
