# Understanding the concepts of division and fractions

$\require{cancel}$ 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?

• 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. – Pedro Tamaroff Jan 3 '15 at 2:02
• @PedroTamaroff I love the example :) – Johanna Jan 3 '15 at 2:18
• @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 – 21Brunoh Jan 4 '15 at 23:40

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.
• 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? – 21Brunoh Jan 2 '15 at 23:53