Proving that fractions are the same as division

[logicgate]

TL;DR

I'm trying to prove that fractions are the same as division using the axiom of multiplicative inverse.

So using the multiplicative inverse axiom we have :
1) x . x^-1 = 1
2) x . (1/x) = 1
I have no idea why do mathematicians define the multiplicative inverse of a number x to be the "fraction" 1/x.
But I know for sure that multiplying any number a for example by the multiplicative inverse of x is the same as "a divided by x"

 

[PeroK]

TL;DR Summary: I'm trying to prove that fractions are the same as division using the axiom of multiplicative inverse.

So using the multiplicative inverse axiom we have :
1) x . x^-1 = 1
2) x . (1/x) = 1
I have no idea why do mathematicians define the multiplicative inverse of a number x to be the "fraction" 1/x.
But I know for sure that multiplying any number a for example by the multiplicative inverse of x is the same as "a divided by x"

If we start with the integers, and assume we can add and multiple them, we can ask:

Given two integers $a$ and $b$, is there a number $c$ such that $a \times c = b$?

Sometimes there is a suitable integer and sometimes there isn't. If we want that equation to be solvable for all non-zero $a$, then we have to introduce some new numbers that aren't integers. So, by definition, we say:

$c \equiv \frac b a$ is the number such that $a \times c = b$.

And that effectively defines the rational numbers, as quotients of integers.

Then we have a special case where $b = 1$, hence $c = \frac 1 a$ and $c \times a = 1$. In this case, we call $c$ the multiplicative inverse of $a$ and we also write $c = a^{-1}$.

That leads more generally to the equation: $\frac b a = ba^{-1}$.

 

[WWGD]

This may be of interest:

https://en.m.wikipedia.org/wiki/Field_of_fractions

Every integral domain, i.e., a ring R in which a.b=0 implies a=0 or b=0, may be embedded in a field, called the Field of Fractions, through a specific process , described in the link.In your case, the Integral Domain of the Integers, can be embeded in the field of Rationals. In this case, if r is an element of said integral domain, the element (1/r) is used to denote it's multiplicative inverse in the associated field of fractions. Notice other examples like that of the Integral Domain $\{ a+ib: a,b \in \mathbb Z \}$, has $\{c+id: c,d \in \mathbb Q \}$ as its field of fractions.
Maybe @fresh_42 can elaborate.