Why is 1 over x/y equal to y/x?

Question

Why is 1 over x/y equal to y/x?

Answer 1

It is $$\frac{1}{1}:\frac{x}{y}=\frac{1}{1}\times \frac{y}{x}=\frac{y}{x}$$

Answer 2

What you want to show is that the multiplicative inverse of $x/y$ is $y/x$. Since (multiplicative) inverses are unique, this follows at once from $$\frac{x}{y} \frac{y}{x} = \frac{xy}{yx} =1.$$

Answer 3

All we need to prove that the multiplicative inverse of $$\frac{1}{a} \text{ is } a$$. i.e. to prove that $$\frac{1}{1/a}=a$$ This is very simpme once you observe the multiplication $$a\times\frac{1}{a}=1$$.

Answer 4

Because by definition, if $ab=1,$ and $b\ne 0,$ then we have that $a=\frac1b.$ Now since $\frac yx\cdot \frac xy=1,$ it follows that $$\frac{1}{x/y}=\frac yx.$$

Answer 5

Well, the division is a derived operation:

$a : b = a\cdot b^{-1}$ where $b\ne 0$.

I.e., division by $b$ means multiplication with the inverse of $b$, where $b\cdot b^{-1}=1$.

Here as lulu said,

$(\frac{x}{y})^{-1} = \frac{y}{x}$.

So

$\frac{1}{\frac{x}{y}} = 1: \frac{x}{y} = 1\cdot (\frac{x}{y})^{-1} = 1\cdot \frac{y}{x} =\frac{y}{x}$.