# Question #fea46

Sep 1, 2016

$8 \div 2$ is the same as $\frac{8}{2} = 4$

A fraction is a quicker and easier way of writing a division, especially in the algebra which lies ahead.

#### Explanation:

Writing a division as a fraction is a more useful way of showing the division. They mean exactly the same thing!

$2 \div 3 = \frac{2}{3} \text{ } 20 \div 4 = \frac{20}{4} = 5$

$5 \div 10 = \frac{1}{2}$ It can also be shown as $\frac{5}{10} = \frac{1}{2}$

As you work more and more with Algebra you will find that you use the $\div$ sign less and less, and divisions are shown as fractions.

Sep 5, 2016

#### Explanation:

$\textcolor{b r o w n}{\text{The big question is: does it actually mean divide?}}$

$\textcolor{g r e e n}{\text{I can only leave you with an open question}}$

A fraction is something that has been so much a part of my life for so long that I have not really given much thought to this question.

Consider what a fraction is:

$\left(\text{count")/("size indicator of what you are counting") ->("numerator")/("denominator}\right)$

The size indicator is how many of what you are counting it takes to make a whole.

The line's prime function is to separate the two numbers. Its very existence and agreed format also tells us that we to consider the relationship between the two numbers in a particular way. That we are counting parts of a whole and declaring the size of what we are counting.

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
The problem comes from the tendency and convention to disregard something. Let me explain:

It is all to do with ratio and I wish to demonstrate this using a right triangle
.

If you are familiar with a little trig consider the value (ratio) of $\tan \left(\theta\right)$ full name of Tangent.

$\tan \left(\theta\right) = \left(\text{Length of AC = 4")/("Length of BC=3}\right) = \frac{4}{3}$

Suppose we were to change the 3 into 1 we have:

$\tan \left(\theta\right) = \frac{4 \div 3}{3 \div 3} = \frac{1.33 \overline{3}}{1}$

People do not bother to write the 1 so you end up with just $\tan \left(\theta\right) = 1.33 \overline{3}$

This is actually saying we have $1.33 \overline{3}$ of AC for 1 of BC

However there is a tendency for people to say divide the 3 into 4 and you get $1.33 \overline{3}$ which is true but is it what $\tan \left(\theta\right)$ is actually declaring. $\tan \left(\theta\right)$ is saying that for 1 along you get $1.33 \overline{3}$ up.

Looking at a fraction, say $\frac{1}{2}$. Divide 2 into 1 and you get 0.5. However, like in the ratio, what is missed out is that it really is $\frac{0.5}{1}$ because we have $\frac{1 \div 2}{2 \div 2} = \frac{0.5}{1}$

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{g r e e n}{\text{The open question!}}$

$\textcolor{g r e e n}{\text{The problem is that numerically they behave in the same way. So does it mean divide or not?}}$