In this explainer, we will learn how to evaluate numerical expressions involving absolute value.
We know that a number line goes in both the positive and the negative directions. Positive numbers are on the right-hand side of zero, and negative numbers are on the left-hand side of zero. Zero is neither positive nor negative.
The sign of a number indicates on which side of zero the number is located, and what is written after the sign gives its distance from zero. This distance from zero is called the absolute value of the number.
Definition: Absolute Value
The absolute value of a number is its distance from zero on a number line.
The absolute value of a number is written using two vertical bars around the number. So, is the absolute value of 3 and is the absolute value of . Since 3 is a distance of 3 to zero, its absolute value is 3; and is a distance of 5 to zero, so its absolute value is 5.
Let us look at a first question to check our understanding.
Example 1: Finding the Absolute Value of a Negative Number
Remember that the vertical bars around a number means that we are considering here its absolute value. The absolute value of a number is its distance from zero on a number line. is a distance of 51 from zero; therefore,
Example 2: Understanding What Numbers Have the Same Absolute Value
Olivia says that two numbers can have the same absolute value, while Jennifer claims that one number can have two absolute values. Who is right?
Remember the definition of the absolute value of a number: it is its distance from zero. A number cannot be located at two different distances from zero, so Jennifer is wrong. However, for any distance from zero greater than zero, there will be two numbers on the number line that are this distance from zero. These two numbers are opposites. So, Olivia is right in saying that two numbers can have the same absolute value. Zero, which has no opposite, is the only number to have an absolute value of zero.
We have seen something very important with the previous example: a number and its opposite have the same absolute value because they are the same distance from zero. And since a distance is always a nonnegative number, we can conclude that the absolute value of a positive number is the number itself, while the absolute value of a negative number is equal to its opposite.
We are going to look now at questions to deepen our understanding of the meaning of absolute value.
Example 3: Understanding That the Absolute Value Represents the Magnitude
On Wednesday, Victoria deposited $27 in her account. On Thursday, she withdrew $30. If you represent both of these situations by integers, which has the higher absolute value?
If we wanted to represent these two operations with integers, we would represent depositing $27 by 27 and withdrawing $30 by . Between these two integers, the one with the higher absolute value is .
Hence, our answer is “withdrawing $30.”
We see that when we represent operations on a bank account with negative and positive numbers, the sign of the number indicates the direction of the operation while its absolute value gives the amount involved in the operation.
Example 4: Understanding That the Absolute Value of a Change Represents Its Magnitude
William and Daniel both weigh 78 kilograms. William would like to gain weight, while Daniel wants to lose weight. They decide to go on a diet. After one month, William weighs 81 kg and Daniel 73 kg.
- What are William’s and Daniel’s changes in weight?
- William: +3 kg, Daniel: kg
- William: +5 kg, Daniel: kg
- William: kg, Daniel: +3 kg
- William: kg, Daniel: +5 kg
- William: +3 kg, Daniel: +5 kg
- Who had the larger change in weight?
- It was the same for both.
William and Daniel both start their diet weighing 78 kg. After one month, William weighs 81 kg, which is 3 kg more than 78 kg. Daniel weighs 73 kg, which is 5 kg less than 78 kg. Therefore, their changes in weight are +3 kg for William and kg for Daniel.
William has gained 3 kg and Daniel has lost 5 kg. Therefore, Daniel has lost more than William has gained. Our answer is that Daniel had the larger change in weight.
The sign of the change in weight indicates whether weight has been gained (positive sign) or lost (negative sign), while the absolute value of the change indicates by how many kilograms the weight has changed.
Example 5: Interpreting the Absolute Value of Elevation as the Distance from Sea Level
The table below shows the elevation with respect to sea level of some locations in the world.
Which of these locations is farthest from sea level?
- Taipei, Taiwan
- Amsterdam, the Netherlands
- Havana, Cuba
- Algiers, Algeria
- Lake Frome, Australia
In the table, elevation is represented by negative and positive numbers, where the sign of the number indicates whether the location is above (positive sign) or below (negative sign) sea level and the absolute value gives the distance (here in meters) from sea level.
We want to find here the farthest location from sea level, that is, the location whose elevation has the highest absolute value. This is Lake Frome in Australia, which is 6 meters from sea level.
- A number line goes in both the positive and the negative directions. Positive numbers are on the right-hand side of zero, and negative numbers are on the left-hand side of zero. Zero is neither positive nor negative.
- The absolute value of a number is its distance from zero on a number line. It is written using two vertical bars around the number, for instance, and .
- A number and its opposite have the same absolute value because they are the same distance from zero.
- A distance is always a nonnegative number, so the absolute value of a positive number is the number itself, while the absolute value of a negative number is equal to its opposite.