Explainer: Comparing and Ordering Integers

In this explainer, we will learn how to compare and order positive and negative integers represented either in a mathematical model or in a real-life situation.

Definition: Integers

The integers are all of the whole numbers, 0,1,2,3,4,5,6,,andsoon as well as their additive inverses (or opposites), which are the negative numbers, 1,2,3,4,5,6,.andsoon

You already know how to locate integers on a number line. Remember that integers are the numbers that can be written without any decimal points. Each positive integer is on the right of zero, and we can think about them as having a distance from zero. For example, 3 is a distance of 3 from zero in the positive direction. Negative numbers appear on the left of zero. The opposite of 3, which is negative 3, or 3, is a distance of 3 from zero in the negative direction.

You should also be comfortable comparing whole numbers (the positive integers together with zero). On a number line showing the whole numbers, numbers get smaller as you move from right to left and larger as you move from left to right.

This is also true when we extend the number line to include negative integers. The numbers still get smaller as you move to the left and bigger as you move to the right.

So, if we wanted to compare the numbers 8, 3, 3, and 8, we could draw them all on a number line. We know how to find 3 and 8 so, to find their opposites (or additive inverses), we have to find the numbers that are a distance of 3 and 8 from zero in the opposite direction.

Then, we know that the smallest numbers are on the left and the largest numbers are on the right. So, 8 is the smallest of the four numbers, 8 is the largest of the numbers, and we can write the following comparison statements between each pair of numbers. 8<33<33<88>33>33>88<33<88>33<88<88>8

Now, let’s look at an example of comparing two negative numbers.

Example 1: Comparing Negative Integers on a Number Line

The table shows the average temperature of two cities in winter. Compare the two temperatures using < or >.

CityTemperature (F)
A5
B2

Answer

To compare 5 and 2, we can plot the numbers on a number line.

Both numbers are negative, so they will be on the left side of zero. The number 2 will be the same distance from zero as 2, but in the opposite direction. Similarly, 5 will be 5 units to the left of zero.

Now, since we know that numbers get larger as we move from left to right on a number line, we know that 5 is smaller than 2. Hence, 5<2.

Next, we will see how to compare a positive number and a negative number.

Example 2: Comparing Positive and Negative Integers

Which of the following is true?

  1. 136<97
  2. 136=97
  3. 136>97

Answer

Here we have to compare 136 and 97.

To do this, think about where the numbers would be located on a number line.

We know that 97 is less than 136 and that these positive numbers are located on the right of zero on a number line. To locate 136, we have to look at the negative numbers, which are on the left of zero. The number 136 is located the same distance away from zero as 136 but in the negative direction (to the left of zero).

Since we know that numbers get larger as we move from left to right on a number line, we know that 136 is smaller than 97. Hence, 136<97.

Finally, we will use what we know to order a set of integers in ascending (least to greatest) or descending (greatest to least) order.

Example 3: Ordering Integers Using a Number Line

The table shows the players in a card game and their corresponding scores. Order the scores descendingly.

PlayerScore
1+16
211
3+2
48
52
619
7+6
8+18

Answer

We need to order the scores from greatest to least. We can do this by plotting the scores on a number line.

To plot the scores, consider their distance from zero, and remember that positive numbers are on the right of zero and negative numbers are on the left. So, 2 and +2 will be the same distance from zero but in opposite directions.

Once you have found all the numbers on a number line, you can use that numbers decrease as you move from right to left. Hence, in descending order, the scores are 18,16,8,6,2,2,11,19.

Example 4: Comparing Amounts by Representing Them as Integers

Last week, Matthew deposited $385 in his bank account, spent $95 on lunch, and loaned $70 to a friend. Express each transaction as an integer and then arrange them in ascending order.

Answer

First, we need to represent each situation with an integer. Remember that positive integers represent gains, or deposits, and negative integers represent losses, or withdrawals.

So, a deposit of $385 represents an increase in the amount of money in his account. We can represent this gain with a positive number: depositingorjust$136+136,136.

Spending $95 represents a decrease in the amount of money in his account. We can represent this loss with a negative number: spending$9595.

Loaning $70 to a friend also represents a loss from his account, so we represent it with a negative number: loaning$7070.

Next, we have to arrange 136, 95, and 70 in ascending order.

Plot the numbers on a number line, remembering that negative numbers appear on the left of zero and are the same distance away from zero as their additive inverse (or opposite). So, 95 and 95 are the same distance from zero in opposite directions. This means that 70 is closest to zero and +136 is furthest from zero.

Since we know that numbers get larger as we move from left to right on a number line, we know that 95 is the smallest of the three numbers and 136 is the largest. Hence, the order is 95,70,136.

We can summarize the steps needed to compare integers as follows.

How to Compare and Order Integers with a Number Line

To compare integers, plot them on a number line remembering the following points:

  • Positive numbers appear to the right of zero, and negative numbers appear to the left.
  • A positive number (e.g., 2) is the same distance from zero as its additive inverse or opposite (e.g., 2).
  • When viewing a number line, numbers increase as you move from left to right.

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