Why is $\frac{1}{\frac{1}{X}}=X$?

Question

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?

The reciprocal is by definition the inverse of the fraction --- and you calculate the inverse of the fraction by inverting the fraction. — Gerry Myerson, Jun 16 '13 at 03:53
The reciprocal of $a/b$ is defined to be the unique $x$ such that $a/b\cdot x=1$, denoted by $(a/b)^{-1}$ or $1/(a/b)$. As it happens, $x=b/a$ is a formula for it, which can be easily verified, but there is a difference between a definition and an obvious formula. — anon, Jun 16 '13 at 03:58
@user82306 I always ask my students, how many half pizzas in a pizza? The answer, $1/(1/2)=2$. — James S. Cook, Jun 16 '13 at 04:10

Alexander Gruber · Answer 1 · 2020-04-19T02:56:44.873

183

$1/x$ 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 $x$.

edited Apr 19 '20 at 02:56

answered Jun 16 '13 at 03:55

Alexander Gruber

26,963

25

+1 for explaining why rather than just demonstrating how by algebraic manipulation. – 200_success Jun 16 '13 at 08:18
14

For this to be true, the inverse needs to be unique. And it is in the context of $\mathbb R^\times$, the group of non-zero real numbers under multiplication. – Ayman Hourieh Jun 16 '13 at 12:42
11

(+1) By the magic of democracy, the super-voted answer is indeed the one to heed. – John Bentin Jun 16 '13 at 15:35

amWhy · Accepted Answer · 2013-06-16T19:36:50.120

36

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:

$$\frac 1{\Large \frac 1X}\cdot \frac XX = \frac X1 = X$$ $$ $$

edited Jun 16 '13 at 19:36

answered Jun 16 '13 at 03:55

amWhy

209,954

score 32 · Answer 3 · edited Apr 13 '17 at 12:20

32

$$y=\frac1{\frac 1 x} $$ $$y'(x)=\left(\frac1{\frac 1 x}\right)' = -\left(\frac 1 {\left(\frac 1x\right)^2}\right)\left({\frac 1 x}\right)' = \frac 1 {\left(\frac 1x\right)^2} \cdot {\frac 1 {x^2}} = \frac {y^2(x)}{x^2}$$

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 $\frac {1} {y} = \frac {1}{x} \Rightarrow x=y$.

edited Apr 13 '17 at 12:20

Community

1

answered Jun 16 '13 at 08:40

gukoff

1,500

66

I'm sure OP will find this very helpful. – Gerry Myerson Jun 16 '13 at 11:35
20

It seems people here love to make mathematical Rube-Goldberg machine. – tia Jun 16 '13 at 12:50
2

I tried to explain it in more detail – gukoff Jun 16 '13 at 19:58
4

Some of us (especially with cool names) still do appreciate. :-) – Harold Cavendish Jun 16 '13 at 21:45
13

I think that understanding the question via this method is actually a conceptual step backwards. – Ryan Reich Aug 16 '13 at 07:04
4

Is this a joke? – user2345215 Dec 22 '14 at 20:48
6

I am pretty sure that in one of the derivatives evaluations you have already used that $\frac1{1/x} = x$, so your argument have a good chance of being cyclical. – SBF Jun 13 '16 at 13:23

score 6 · Answer 4 · answered Jun 16 '13 at 03:55

6

Does this work for you? Start with $${1\over1/x}$$ Multiply top and bottom by $x$: $${1\over1/x}{x\over x}={x\over(1/x)x}={x\over1}=x$$

answered Jun 16 '13 at 03:55

Gerry Myerson

179,216

john · Answer 5 · 2013-06-16T11:30:16.847

6

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!

edited Jun 16 '13 at 11:30

answered Jun 16 '13 at 03:58

john

5,633

score 5 · Answer 6 · edited Aug 27 '13 at 14:44

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.

score 4 · Answer 7 · answered Aug 16 '13 at 06:30

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.)

score 2 · Answer 8 · answered Jun 16 '13 at 07:44

2

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, $\frac{1}{\frac{1}{x}} = x$

answered Jun 16 '13 at 07:44

Naffi

295

3

Wouldn't this work better as a direct proof? – Alexander Gruber Jun 21 '13 at 20:06
Probably, but to some people 'indirect' way is more understandable than 'direct' way. – Naffi Jun 22 '13 at 06:39

score 1 · Answer 9 · edited Apr 13 '17 at 12:21

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!

score 1 · Answer 10 · answered Jun 26 '13 at 06:56

1

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 $x\cdot\frac{1}{x}=\frac{x}{1}\cdot\frac{1}{x}=\frac{x\cdot 1}{1\cdot x}=\frac{x}{x}=1$

answered Jun 26 '13 at 06:56

MathTeacher

1,559

Here's another way I look at it: say $x=2$. $1\div\frac{1}{2}$ asks ``How many one halves fit inside one?" Well, two one halves fit inside one. – MathTeacher Dec 05 '13 at 07:48

score 1 · Answer 11 · edited Jan 16 '14 at 07:29

1

Note that: $1/x=x^{-1}$ and also that: $(x^a)^b=x^{ab}$

Therefore $1/(1/x)=(x^{-1})^{-1}=x^1=x$

edited Jan 16 '14 at 07:29

user91500

5,606

answered Jan 16 '14 at 07:03

Matt

31

$(x^a)^b=x^{ab}$ is not always true but $1/1/x = x $ is always true $x\ne =0$. – Feb 22 '17 at 06:39

score -1 · Answer 12 · edited Jun 21 '13 at 12:51

-1

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...

edited Jun 21 '13 at 12:51

Américo Tavares

38,602

answered Jun 16 '13 at 10:27

Aurelius

2,801

Why is $\frac{1}{\frac{1}{X}}=X$?

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