1.10 Compare and order rational numbers
Introduction
We know that rational numbers include integers, all fractions, terminating and repeating decimals, and percents
\ldots ,-8,-7.4,-7,-6,-5.33387,-4,-2,\,0,\,\dfrac{1}{2},75\%,\,1,\,2,\,3,\,3.5656,\,\ldots
Now let's look at how we can compare and order rational numbers that are in the same form.
Compare rational numbers
Just as we can compare and order integers on the number line, we can compare rational numbers on the number line.
As we move to the right on the number line the numbers get larger.
Likewise, as we move to the left on the number line the numbers get smaller.
For example, -3.12, is to the left of 1.3, so -3.12 is smaller than 1.3. We can write this statement as the following inequalities:
| \displaystyle -3.12 | \displaystyle < | \displaystyle 1.3 | because -3.12 is further to the left on the number line |
| \displaystyle 1.3 | \displaystyle > | \displaystyle -3.12 | because 1.3 is further to the right on the number line |
Examples
Example 1
Which is the largest rational number marked on the number line?
Example 2
Consider the values 1.25 and 0.75.
Plot the pair of numbers on a number line.
State the inequality sign that makes the statement true.
1.25 \,⬚\, 0.75
The symbol < represents the phrase "is less than".
The symbol > represents the phrase "is greater than".
Order rational numbers
We can also use a number line to help us arrange rational numbers in ascending order by plotting them on a number line and then listing them in order from the number that is the farthest to the left to the number that is farthest to the right. Here are the rational numbers 0,\, 3.7,\, -4.5,\, 1.3 plotted on a number line:
Order the numbers from least to greatest:-4.5,\,0,\,1.3,\,3.7
We can use the same number line to arrange the rational numbers in descending order by finding the greatest number, which is the number farthest to the right, and writing them in order to the number that is farthest to the left.
Here are the rational numbers written in descending order:
3.7,\,1.3,\,0,\,-4.5
Examples
Example 3
Arrange the following numbers in ascending order:
2.25, \, 2.75, 1.5, \, -2.5, \,-2.75
Descending rational numbers get smaller as we move to the left on the number line.
Ascending rational numbers get larger as we move to the right on the number line.