Fractions on a number line

 

Fractions between integers - really?

The number line contains all of the integers - the positive integers such as $1,2,3,4,...$1,2,3,4,... etc and the negative integers $-1,-2,-3,-4,...$−1,−2,−3,−4,... and the special integer $0$0 (although all numbers are special).

But there are other numbers on the number line besides integers.

There are numbers between the integers called fractions. The word fraction comes from the old Latin word fractus meaning "broken".

We can break the gap between integers into equal parts.

For example, we can break the gap between the integer $2$2 and the integer $3$3 into three equal parts, known as thirds, as shown here:

  

The number $X$X would therefore be $2\frac{1}{3}$213​ and the number $Y$Y would be $3\frac{2}{3}$323​.

 

How small can the pieces be?

We could break the gaps into halves ($2$2 equal pieces) or quarters ($4$4 equal pieces) or even twelfths ($12$12 equal pieces). In fact you could break the gap between consecutive integers into any number of equal pieces. If you had the patience, you could break the gap into a million equal pieces if you like!

Here is the gap between the integer $0$0 and the integer $1$1 broken up into $12$12 pieces. You may want to count them to be sure! Four of the new numbers have been labelled $A$A, $B$B, $C$C and $D$D.

The number $B$B for example is the fraction $\frac{6}{12}$612​ because it is six pieces along from the integer $0$0. You should be able to see that it is half-way between $0$0 and $1$1, and so the number could be equivalently be called $\frac{1}{2}$12​. Can you imagine two longer equal pieces? One going from $0$0 to $B$B and the other from $B$B to $1$1?

The number $C$C is also interesting. It is the number $\frac{9}{12}$912​ or equivalently $\frac{3}{4}$34​. Can you see why the fractions are equivalent?

We could also say that the distance between the number $A$A and $B$B is $\frac{5}{12}$512​ , and between $B$B and $C$C is $\frac{3}{12}$312​ or $\frac{1}{4}$14​. The distance between $A$A and $D$D is $\frac{10}{12}$1012​ or, using sixths, $\frac{5}{6}$56​. 

A number line has many advantages, and one of these is to allow us to visualise equivalent fractions.  

 

Weird numbers?

Mathematicians once thought they could fill up the interval between $0$0 and $1$1 completely by breaking it up into smaller and smaller pieces (maybe a million equal pieces or a billion equal pieces or a trillion equal pieces or even more than that!).

Unfortunately for them, they eventually discovered very strange numbers on the line that could never be named as fractions like the ones above, no matter how many pieces they used. At first they made no sense at all, but in the end there were too many of them to ignore.  But that of course is an entirely different story to tell.  

Worked Examples

Question 1

Question 2

Question 3

Question 4