1.07 Comparing and ordering integers
We have now seen that an integer is a whole number or its opposite:
$...,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,...$...,−6,−5,−4,−3,−2,−1,0,1,2,3,4,5,6,...
Notice that zero ($0$0) is an integer but it is neither positive or negative.
Comparing the sizes of integers
In what we have seen so far the integers are increasing from left to right on the number line. This means that when we compare two integers, the integer further to the right is always greater and the integer further to the left will always be lesser.
The number line increases from left to right.
Inequality symbols can be used to show the relative ordering of two integers on the number line.
The symbol $<$< represents the phrase is less than. For example, $-3$−3 is less than $4$4 can be represented by $-3<4$−3<4.
The symbol $>$> represents the phrase is greater than. For example, $4$4 is greater than $-3$−3 can be represented by $4>-3$4>−3.
We can use a number line to clearly see the relationship between different integers.
- Since the point at $-4$−4 is to the left of $0$0, we know that $-4$−4 is less than $0$0, so $-4<0$−4<0.
- Since the point at $0$0 is to the left of $3$3, we know that $0$0 is less than $3$3, so $0<3$0<3.
- Since the point at $8$8 is to the right of $3$3, we know that $8$8 is greater than $3$3, $8>3$8>3.
Ordering integers
We can arrange these four integers in ascending order by writing them left to right in order from the least integer to the greatest integer. We can use the $<$< symbol to arrange the integers like so, $-4<0<3<8$−4<0<3<8. Here are the integers written in ascending order:
$-4,0,3,8$−4,0,3,8
Now using the $>$> symbol, we can arrange these same integers in descending order, written left to right from greatest to least. Rearranging $-4<0$−4<0 to $0>-4$0>−4 and $0<3$0<3 to $3>0$3>0, we can arrange the integers like so, $8>3>0>-4$8>3>0>−4. Here are the integers written in descending order:
$8,3,0,-4$8,3,0,−4
Notice that the descending order of the integers is the reverse of the ascending order.
We've seen before that the further an integer is to the right on a number line, the larger the integer is.
Worked examples
Question 1
Arrange $0,7,-1$0,7,−1, and $6$6 in ascending order.
Think:
Let's remember the number line:
$-1$−1 is the smallest number as it is furthest to the left on the number line. $7$7 is furthest to the right, so it is the biggest.
Do: The numbers in ascending order are $-1,0,6,7$−1,0,6,7.
question 2
Arrange $-15,-2,-7$−15,−2,−7 and $-5$−5 in descending order.
Think: $-2$−2 is furthest to the right, so it is the biggest number. $-15$−15 is the smallest number as it is furthest to the left.
Do: $-2,-5,-7,-15$−2,−5,−7,−15
Practice questions
Question 3
Which is the largest number marked on the number line?
A horizontal number line with integers marked at five-unit intervals, ranging from -10 on the left to 15 on the right. Each labeled point is denoted by a longer vertical tick mark. Shorter vertical tick marks, signifying one-unit intervals, are placed between the labeled points. Additionally, three diamond-shaped markers are positioned at the specific points of -9, 7, and 13 on the number line.
$7$7
A$-9$−9
B$13$13
C
Question 4
Arrange the following numbers in ascending order:
$-11,15,-19,28$−11,15,−19,28
$\editable{}$, $\editable{}$, $\editable{}$, $\editable{}$
Question 5
Consider the numbers $-3$−3 and $-9$−9.
Graph $-3$−3 and $-9$−9 on the number line.
Insert either $<$< or $>$> to make a true statement.
$-3\editable{}-9$−3−9