Basic division and multiplication We know that 2 x 2 /2 can be solved by removing '2' from the denominator and numerator, we can't do the same if the operation was addition, These "rules" have been established based on the understanding of these operations and how these interact with each other.
Up until now, I've just been accepting it as a rule, but this causes a lot of discomforts while doing math.
I generally get good grades in math, but that's only by following these rules and patterns but there is no genuine understanding as to why we can remove "2" from the numerator and denominator in 2 x 2 /2 but we cannot in 2 + 2/2.
Secondly, 2 x 2 / 2 can be solved by removing "2" from the numerator and the denominator and it gives the same result as 4/2, how?.
Is it an axiom?
I hope to gain an intuition for the above concepts, so I can be free from my long-held guilt.
Thanks.


$\dfrac {2 \times 2}{ 2 }$ can be solved by removing $2$ from the numerator and the denominator and it gives the same result as $\dfrac 4 2$, how? Is it an axiom?

No, it is not an axiom. But, as for every rule for operating with numbers, it is obviously justified by axioms.
Regarding the rationals (i.e. fractions) they are an ordered field.
Regarding division, the axioms state that :

If $a \ne 0$, then the equation $a \cdot x = b$ has a unique solution : $x = \dfrac b a = b \cdot a^{−1}$.

Thus, $a^{−1}$ is the multiplicative inverse of $a$.
We have :

$\dfrac {a \cdot a} a = (a \cdot a) \cdot a^{−1}$.

By Associativity of product we have that :

$(a \cdot a) \cdot a^{−1} = a \cdot (a \cdot a^{−1})$.

But $(a \cdot a^{−1})=1$ and again by axiom :

$a \cdot 1=a$.

Putting all together, we get :

$\dfrac {a \cdot a} a = a$.


At an "intuitive level" it is simply a metter of multipling and dividing quantities.
We multiply two by two the get four and then we divide four by two, getting two.
In real life applications of numerical oeprations we do not "cancel" anything.
