Ordering and comparing rational numbers

To order / compare rational numbers;

a )You can order rational numbers either by using logic, based on division which is what a rational number means.

b) You can think of a rational number as a fraction and then sort it. The way
we sort fractions is the same way to sort rational numbers.

We recommend that you also look at the topic of ordering in fractions .

Order   \( \displaystyle \frac{8}{5} \) and \( \displaystyle \frac{12}{5} \)

\( \displaystyle \frac{8}{5} \)   “8 divided by 5” , divide 8 by 5 

\( \displaystyle \frac{12}{5} \)  “12 divided by 5” , divide 12 by 5

You can find the solution to the equation either using a calculator or manual and compare the results or you can use logic to find out which one is larger or smaller.

Let’s do this by dividing ;

\( \displaystyle \frac{8}{5}=1,6 \)

\( \displaystyle \frac{12}{5}=2,4 \)

it is clear that \( \displaystyle \frac{12}{5}>\frac{8}{5}\)

Now let’s use logic ;

Let’s say we have an 8 unit staff in our hand. We need to divide this to 5 equal parts.
We also have a 12 unit staff in our other hand, and we must also divide this by 5. Note
that I am dividing both staffs into 5 equal parts, but the lengths of the staffs are different...
Of course the longer the staff, the individual pieces will also be larger.

\( \displaystyle \frac{12}{5}>\frac{8}{5}\)

Compare \( \displaystyle \frac{15}{6}\) and \( \displaystyle \frac{15}{10}\)

We have a 15 unit staff in one hand and I am dividing it into 6 equal parts;
I also have a 15 unit staff in one hand and am dividing this into 10 equal parts - which one is larger?

The more you divide the staffs of equal lengths, the more pieces you will have – getting
smaller each time. There will be less value in each piece. Therefore;

\( \displaystyle \frac{15}{6}>\frac{15}{10}\) 

Let’s find the result using division;

\( \displaystyle \frac{15}{6}=2,5\)  We have divided 15 with 6

\( \displaystyle \frac{15}{10}=1,5\) We have divided 15 with 10.

\( \displaystyle \frac{15}{6}>\frac{15}{10}\) 

Compare \( \displaystyle \frac{3}{4}\) and \( \displaystyle \frac{5}{6}\)

Firstly; neither the number that is divided nor the number that is used for division are
the same, also both rational numbers are between 0 and 1. If we can make the numbers that
are being divided equal, we can find a solution using the same logic from the previous section that you have just learnt

Let us equalise the denominators. Just like expanding in fractions…

 \( \displaystyle \frac{3}{4}.\frac{3}{3}=\frac{9}{12}\) There is no difference between dividing 3 by 4 and 9 by 12; the results are the same.

 \( \displaystyle \frac{5}{6}.\frac{2}{2}=\frac{10}{12}\)  In the same way there is no difference between dividing, 5 by 6 or 10 by 12. The results are the same.

The problem is now as follows;

\( \displaystyle \frac{9}{12}\) and \( \displaystyle \frac{10}{12}\)

Imagine that we have a stick of 9 units and that we are going to divide this into 12
equal parts; we also have a stick of 10 units and will divide it by 12 equal parts.
Which one will be larger? I am dividing both sticks by 12, so however long our stick,
the longer the individual parts.

\( \displaystyle \frac{10}{12}>\frac{9}{12}\)

\( \displaystyle \frac{5}{6}>\frac{3}{4}\)

Compare \( \displaystyle \frac{4}{6}\) and \( \displaystyle \frac{5}{8}\) 

Both numbers are between, 0 and 1 and the divided numbers are not the same as the numbers used for dividing; so let us equalise one or the other.

This time let us equalise the numbers being divided. I equalise through expanding;

\( \displaystyle \frac{4}{6}.\frac{5}{5}=\frac{20}{30}\)

\( \displaystyle \frac{5}{8}.\frac{4}{4}=\frac{20}{32}\)

We have a stick of 20 units and we divide this into 30equal parts, we also have another
stick of 20 units and this time we divide by 32. The more we divide the stick; the pieces
will get smaller; for bigger pieces we need to divide less. So;

\( \displaystyle \frac{20}{30}>\frac{20}{32}\)

So the result is ;

\( \displaystyle \frac{4}{6}>\frac{5}{8}\)

For numbers where it is not clear whether or not it is larger or smaller; for rational numbers where the divided and dividing numbers are different – by equalising you can use logic to find the result.

\( \displaystyle 2\frac{1}{4}\) and \( \displaystyle 1\frac{3}{5}\)

You do not have to pair in anyway; 2 as a whole number with some pieces, is larger than 1 and some pieces.

\( \displaystyle 2\frac{1}{4}\) It is between 2 and 3.

\( \displaystyle 1\frac{3}{5}\) It is between 1 and 2.

\( \displaystyle 2\frac{1}{4}>1\frac{3}{5}\) 

\( \displaystyle 1\frac{3}{5}\) and \( \displaystyle 1\frac{1}{4}\)

Both numbers are made up of 1 whole number and some pieces. To figure out which is larger, we need to compare the fractions.

\( \displaystyle \frac{3}{5}\)  Larger than half

\( \displaystyle \frac{1}{4}\)   Quarter
By equalising the divided or dividing number, you can
find out which rational number is larger than the other.

\( \displaystyle 1\frac{3}{5}>1\frac{1}{4}\)

Comparison in negative rational numbers

The rules for sorting integers are the same for rational numbers.
For example ; When \( \displaystyle 4>3\)
 , with negative integers it is \( \displaystyle -4<-3\) 

In the same way, half negative is smaller than a quarter negative .

\( \displaystyle \frac{1}{2}>\frac{1}{4}\) becomes \( \displaystyle -\frac{1}{2}<-\frac{1}{4}\)

Thus whilst ordering negative rational numbers, we order according to
positive, add the – symbol, and turn the ‘smaller than’ and ‘larger than’
symbols and turn them on their head. If the symbol is smaller than it
is smaller, and smaller becomes larger.

\( \displaystyle -\frac{3}{4}\) and \( \displaystyle -\frac{5}{6}\)

\( \displaystyle \frac{5}{6}>\frac{3}{4}\) Let us sort according to positive rational numbers, we have explained this above.

 \( \displaystyle -\frac{5}{6}<-\frac{3}{4}\)  Now, let us add the - sign, let s turn the ‘smaller than’ and ‘greater than’ symbols on their head.

Both rational numbers are between, 0 and -1 but \( \displaystyle -\frac{5}{6}\) is closer to -1. So it is smaller.

\( \displaystyle -\frac{5}{6}<-\frac{3}{4}\)

\( \displaystyle -2\frac{1}{4}\) and \( \displaystyle -1\frac{3}{5}\)

If you have learnt between which two numbers the rational numbers should be in,
there is then no need to sort according to positive integers.

\( \displaystyle -2\frac{1}{4}\)  Between -2 and -3

 \( \displaystyle -1\frac{3}{5}\)  Between -1 and -2

\( \displaystyle -2\frac{1}{4}< -1\frac{3}{5}\)

Ordering Positive Rational Numbers and
Negative Rational Numbers

\( \displaystyle \frac{1}{2}\) and \( \displaystyle -1\frac{3}{5}\)

Positive numbers, no matter what the  number are always bigger than negative numbers.

So it is  \( \displaystyle \frac{1}{2}> -1\frac{3}{5}\)