Understanding the concepts of division and fractions

Question

$\require{cancel}$ 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?

Answer 1

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.

Answer 2

If you are interested in how these rules are derived algebraically, that's how you can do it using a few division laws: Proof

Why do these rules always work? Because all numbers (with a few very specific exceptions) behave the same way and obey the same laws. Why? I don't know. I think that's just how it is, and we simply discovered it. Even laws of arithmetic were not just invented, they were designed to represent how numbers truly behave. If for example $7+1$ would not be the same as $1+7$, there wouldn't be a law which states $a+b=b+a$ for every two numbers. By the way, that's why algebra was created, to make arithmetic and its solutions more general.

How through intuition or visualization I can be sure that these results are indeed true? I am sure that there is a visualization of dividing fractions, maybe by cutting rectangles into parts. Although, intuition is a bit tricky. Math is not always intuitive. I can't imagine many things either. But I believe that a clear definition of fractions and division may help you to understand it better. And maybe develop an intuitive understanding!

Fractions are just an alternative way to write division. Division is an inverse operation of multiplication, defined in this way: $a÷b=c$ if and only if $c×b=a$. It simply means that $a÷b$ is such a number, that if you multiply it by b, you get a. For example, $10÷2$ is such a number, that if we multiply it by $2$, we get $10$. In other words, what number should I add to itself twice in order to get $10$? $5$.

Also, since multiplication is commutative, which just means that $a×b=b×a$, division can also be defined as $a÷b=c$ if and only if $b×c=a$. I just replaced $c×b$ with $b×c$ because it is the same thing, but now it allows division to have an alternative meaning. According to new definition, $10÷2$ is such a number, that if I take $2$ and multiply it by that number, I get $10$. In other words, how many times should I add $2$ to itself to get $10$? $5$ times.

You also asked about division of fractions. When you divide by a fraction, you can instead multiply by a flipped fraction, because it is a reciprocal of the original fraction. https://en.wikipedia.org/wiki/Multiplicative_inverse Dividing by a number is same as multiplying by its reciprocal, that's why it works. However, I prefer not to use reciprocals for such explanations, because if, for example, you work only with natural numbers, not every natural number has a reciprocal. You can explain division of fractions without this concept.

Since fractions are just a way to write division, we can write $\frac{x}{y}÷\frac{z}{w}$ as $(x÷y)÷(z÷w)$. Now, we can open the brackets according to laws of operations, and we are done.

Begin with $(x÷y)÷(z÷w)$. First, let's open the brackets of $(z÷w)$. There is a very useful formula which proves that $a÷(b÷c)=a÷b×c$. I attached a picture with a proof of this formula, but you can also think about it intuitively. Let's say you have $12÷6$ and $12÷(6÷2)$. In the first case you divide $12$ by $6$, in the second case you divide $12$ by a number which is twice less than $6$. Again, you divide by a number which is twice less. $12$ divided by $6$ is $2$, which means that $2$ is such a number that fits in $12$ $6$ times ($12=2×6=2+2+2+2+2+2$). So $12$ divided by a number which is twice less than $6$, is such a number that fits in $12$ twice less times. That means that such number is twice bigger, isn't it? So, $12÷(6÷2)$ is twice bigger than $12÷6$, in math words $12÷(6÷2)=(12÷6)×2$.

$(x÷y)÷(z÷w)=(x÷y)÷z×w$. We can remove the brackets from $x÷y$ because it does not change the order of operations - with or without brackets $x÷y$ will be performed first. We get $x÷y÷z×w$. Look closely at $x÷y÷z$. It can be rewritten as $x÷(y×z)$. Again, there is an algebraic proof for it, but let's explain it intuitively. $12÷2÷3=k$ where $k$ is a number that fits $3$ times in a number that fits twice in $12$. How many times does $k$ fit in $12$? $6$ times, or $3×2=2×3$ times. So, $12÷(3×2)=k=12÷2÷3$.

$x÷y÷z×w=x÷(y×z)×w$. $x$ divided by $(y×z)$ and then multiplied by $w$. According to another law, $a÷b×c=a×c÷b$, we can first multiply by $w$ and after that divide by $(y×z)$. Let's see why. $6×2÷3$ - a number which fits three times in a number twice bigger than $6$ . $6÷3$ - a number which fits three times in $6$. I think it is clear that $6×2÷3$ is twice bigger than $6÷3$, therefore $6×2÷3=6÷3×2$.

$x÷(y×z)×w=x×w÷(y×z)$. Now put the brackets on $x×w$ (it won't change the order of operation, it is done just for visual clarity). We get $(x×w) ÷(y×z)$. Rewrite it as fractions: $\frac{x×w}{y×z}$

We just showed that $\frac{a}{b}÷\frac{c}{d}=\frac{a×d}{b×c}$

Answer 3

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.