Note : from, if I dare say, a semantic point of view, " multiplicative inverse of a number $N$ " is clearly defined as " a number $M$ such that $N\times M = M\times N = 1$" ( the multiplicative identity). My question deals with the syntactic aspect of the inverse, I mean , with the syntactic manipulations and transformations in which a multiplicative inverse can be involved.
- At first sight, it looks as though there were two definitions of " multiplicative inverse" :
(1) multplicative inverse of a single number $n$ , that is, $\frac 1n$
(2) multiplicative inverse of a quotient of two numbers $\frac nm$ , that is $\frac mn$.
Which definition should be considered as prior?
Should one consider that the most fundamental one is the second, and that, in the first definition, number $n$ has to be understood implicitly as $\frac n1$ : in that case , multiplicative inverse means in general : " the reverse fraction $\frac mn$ ", and $\frac 1n$ has to be considered as a particular case.
Or is it the other way round? In that case , the primitive definition of inverse is $\frac 1n$ and $\frac mn$ is a particular case.
If I am correct, it is the second answer that is the standard one.
But it brings the question : how to derive : " inverse of $\frac nm = \frac mn$" from " inverse of $n = \frac 1n$"?
Recognize that $$\frac{m}{n}=m\cdot\frac1n$$ and the property that the multiplicative inverse of a product is the product of the multiplicative inverses of each factor.
The multiplicative inverse of $m$ is $\dfrac1m$ and the multiplicative inverse of $\dfrac1n$ is $n$. Therefore, the multiplicative inverse of $\dfrac{m}{n}$ is $$\frac1m\cdot n=\frac{n}{m}$$ as desired.