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$\require{cancel}$ 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?

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    $\begingroup$ Say the average kid has 3 litres of blood in their body. If you give two kids a bar of 12 grams of chocolate, you will have 2 grams of chocolate per litre of blood. If you now take one kid, you will have 4 grams of chocolate per litre of blood. If you take one step forward and take half a kid, you will have 8 grams of chocolate per litre. And so on. $\endgroup$ – Pedro Tamaroff Jan 3 '15 at 2:02
  • $\begingroup$ @PedroTamaroff I love the example :) $\endgroup$ – Johanna Jan 3 '15 at 2:18
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    $\begingroup$ @PedroTamaroff That is a nice description of integer/ fraction division, but there are several complex examples of division between fractions where you're explanation isn't clear enough to provide a description of what's happening, like the example I gave above $\endgroup$ – 21Brunoh Jan 4 '15 at 23:40
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enter image description here I'm sorry I can't do better pictures.

Defining multiplication of fractions to mean lots of makes sense in practical situations. Multiplication in other contexts can be defined differently e.g. the dot product of vectors. Another thought: Suppose you only had integers and multiplication. You might want to solve $$3a=1$$ Either accept there is no solution or introduce $\frac{1}{3}$ to do the job. In this sense fractions adhere to the definition of multiplication you already had.

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    $\begingroup$ Fractions can be made more vigorous by defining equivalence classes for ordered pairs. After all 6/9 of 4/3 had better give the same answer. $\endgroup$ – Karl Jan 2 '15 at 19:50
  • $\begingroup$ Nice picture, appreciated the answer. But my question is even more elementary. Why "$\times$" means "of"? Although your answer is visually easy to understand what did guarantee us in the first place that taking $2/3$ of $4/3$ is the same as doing the multiplication between this two fractions? $\endgroup$ – 21Brunoh Jan 2 '15 at 23:53
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    $\begingroup$ You could start by rotating the picture... $\endgroup$ – Pedro Tamaroff Jan 3 '15 at 1:57
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as for multiplying by the reciprocal, the function of dividing fractions is too long to understand for most. however, multiplying is a much easier concept, so they use the equivalent in multiplication of what would be completed dividing. if you divide a number by 2, it cuts the number in half. the way to do this with multiplication is to the number by .5 or 1/2. same concept, if you divide 1/3 by 2, the equivalent multiplication problem is 1/3 times 1/2.

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