$\require{cancel}$ I'm having some issues regarding division so I will start by asking how this concept was developed throughout the ages: What was the first civilization to introduce the idea of division? What possibly motivated them to define division that way and how related was to our present notion/definition of division?
There are several rules I take for granted and when doing calculations I don't know what's really going on and why it actually works. For instance, when dividing fractions, why do we invert and multiply?
$\frac{\frac{1}{3}}{\frac{2}{3}} = \frac{1}3 \cdot \frac{3}2$
Some answers I found are like this one: "You must use the multiplicative inverse to cancel the operation and obtain the final result". Which is a shortcut for: $\frac{\frac{1}{3}}{\frac{2}{3}} = \frac{\frac{1}{3}\cdot \frac{3}{2}}{\frac{2}{3}\cdot \frac{3}{2}} = \frac{\frac{1}{3}\cdot \frac{3}{2}}{\cancel{\frac{2}{3}}\cdot \cancel{\frac{3}{2}}} = \frac{1}{3}\cdot \frac{3}{2} $
But this is still non intuitive for me, I cannot visually understand why this always work. This also rises another problem, the rules used for multiplying fractions.
$\frac{2}3 \cdot \frac{4}3 = \frac{2\cdot4}{3\cdot3} = \frac{8}{9}$
I do this somehow mechanically and I don't have the grasp of what is really happening through this steps. So my question is why this rules always work, and more important, how through intuition or visualization I can be sure that this results are indeed true?