36.6: Absolute Value of Numbers

Lesson

Let's explore distances from zero more closely.

Exercise \(\PageIndex{1}\): Number Talk: Closer to Zero

For each pair of expressions, decide mentally which one has a value that is closer to 0.

\(\frac{9}{11}\) or \(\frac{15}{11}\)

\(\frac{1}{5}\) or \(\frac{1}{9}\)

\(1.25\) or \(\frac{5}{4}\)

\(0.01\) or \(0.001\)

Exercise \(\PageIndex{2}\): Jumping Flea

Move the bug to a starting point, choose a jump distance, and press the jump button. You may need to zoom in or out if your bug jumps off the screen.

1. A bug is jumping around on a number line.
   1. If the bug starts at 1 and jumps 4 units to the right, where does it end up? How far away from 0 is this?
   2. If the bug starts at 1 and jumps 4 units to the left, where does it end up? How far away from 0 is this?
   3. If the bug starts at 0 and jumps 3 units away, where might it land?
   4. If the bug jumps 7 units and lands at 0, where could it have started?
   5. The absolute value of a number is the distance it is from 0. The bug is currently to the left of 0 and the absolute value of its location is 4. Where on the number line is it?
   6. If the bug is to the left of 0 and the absolute value of its location is 5, where on the number line is it?
   7. If the bug is to the right of 0 and the absolute value of its location is 2.5, where on the number line is it?
2. We use the notation \(|-2|\) to say “the absolute value of -2,” which means “the distance of -2 from 0 on the number line.”
   1. What does \(|-7|\) mean and what is its value?
   2. What does \(|-1.8|\) mean and what is its value?

Exercise \(\PageIndex{3}\): Absolute Elevation and Temperature

1. A part of the city of New Orleans is 6 feet below sea level. We can use “-6 feet” to describe its elevation, and “\(|-6|\) feet” to describe its vertical distance from sea level. In the context of elevation, what would each of the following numbers describe?
   1. \(25\) feet
   2. \(|25|\) feet
   3. \(-8\) feet
   4. \(|-8|\) feet
2. The elevation of a city is different from sea level by 10 feet. Name the two elevations that the city could have.
3. We write “\(-5^{\circ}\text{C}\)” to describe a temperature that is \(5\) degrees Celsius below freezing point and “\(6^{\circ}\text{C}\)” for a temperature that is \(5\) degrees above freezing. In this context, what do each of the following numbers describe?
   1. \(1^{\circ}\text{C}\)
   2. \(-4^{\circ}\text{C}\)
   3. \(|12|^{\circ}\text{C}\)
   4. \(|-7|^{\circ}\text{C}\)
4. 1. Which temperature is colder: \(-6^{\circ}\text{C}\) or \(3^{\circ}\text{C}\)?
   2. Which temperature is closer to freezing temperature: \(-6^{\circ}\text{C}\) or \(3^{\circ}\text{C}\)?
   3. Which temperature has a smaller absolute value? Explain how you know.
5. Are you ready for more?

Summary

We compare numbers by comparing their positions on the number line: the one farther to the right is greater; the one farther to the left is less.

Sometimes we wish to compare which one is closer to or farther from 0. For example, we may want to know how far away the temperature is from the freezing point of \(0^{\circ}\text{C}\), regardless of whether it is above or below freezing.

The absolute value of a number tells us its distance from 0.

The absolute value of -4 is 4, because -4 is 4 units to the left of 0. The absolute value of 4 is also 4, because 4 is 4 units to the right of 0. Opposites always have the same absolute value because they both have the same distance from 0.

Figure \(\PageIndex{1}\)

The distance from 0 to itself is 0, so the absolute value of 0 is 0. Zero is the only number whose distance to 0 is 0. For all other absolute values, there are always two numbers—one positive and one negative—that have that distance from 0.

To say “the absolute value of 4,” we write: \(|4|\)

To say that “the absolute value of -8 is 8,” we write: \(|-8|=|8|\)

Glossary Entries

Definition: Absolute Value

The absolute value of a number is its distance from 0 on the number line.

Figure \(\PageIndex{2}\)

The absolute value of -7 is 7, because it is 7 units away from 0. The absolute value of 5 is 5, because it is 5 units away from 0.

Definition: Negative Number

A negative number is a number that is less than zero. On a horizontal number line, negative numbers are usually shown to the left of 0.

Figure \(\PageIndex{3}\)

Definition: Opposite

Two numbers are opposites if they are the same distance from 0 and on different sides of the number line.

For example, 4 is the opposite of -4, and -4 is the opposite of 4. They are both the same distance from 0. One is negative, and the other is positive.

Figure \(\PageIndex{4}\)

Definition: Positive Number

A positive number is a number that is greater than zero. On a horizontal number line, positive numbers are usually shown to the right of 0.

Figure \(\PageIndex{5}\)

Definition: Rational Number

A rational number is a fraction or the opposite of a fraction.

For example, 8 and -8 are rational numbers because they can be written as \(\frac{8}{1}\) and \(-\frac{8}{1}\).

Also, 0.75 and -0.75 are rational numbers because they can be written as \(\frac{75}{100}\) and \(-\frac{75}{100}\).

Definition: Sign

The sign of any number other than 0 is either positive or negative.

For example, the sign of 6 is positive. The sign of -6 is negative. Zero does not have a sign, because it is not positive or negative.

Practice

Exercise \(\PageIndex{4}\)

On the number line, plot and label all numbers with an absolute value of \(\frac{3}{2}\).

Figure \(\PageIndex{6}\)

Exercise \(\PageIndex{5}\)

The temperature at dawn is \(6^{\circ}\text{C}\) away from 0. Select all the temperatures that are possible.

1. \(-12^{\circ}\text{C}\)
2. \(-6^{\circ}\text{C}\)
3. \(0^{\circ}\text{C}\)
4. \(6^{\circ}\text{C}\)
5. \(12^{\circ}\text{C}\)

Exercise \(\PageIndex{6}\)

Put these numbers in order, from least to greatest.

\(|-2.7|\qquad 0\qquad 1.3\qquad |-1|\qquad 2\)

Exercise \(\PageIndex{7}\)

Lin’s family needs to travel 325 miles to reach her grandmother’s house.

1. At 26 miles, what percentage of the trip’s distance have they completed?
2. How far have they traveled when they have completed 72% of the trip’s distance?
3. At 377 miles, what percentage of the trip’s distance have they completed?

(From Unit 5.4.3)

Exercise \(\PageIndex{8}\)

Elena donates some money to charity whenever she earns money as a babysitter. The table shows how much money, \(d\), she donates for different amounts of money, \(m\), that she earns.

Table \(\PageIndex{1}\)

\(d\)	\(4.44\)	\(1.80\)	\(3.12\)	\(3.60\)	\(2.16\)
\(m\)	\(37\)	\(15\)	\(26\)	\(30\)	\(18\)

1. What percent of her income does Elena donate to charity? Explain or show your work.
2. Which quantity, \(m\) or \(d\), would be the better choice for the dependent variable in an equation describing the relationship between \(m\) and \(d\)? Explain your reasoning.
3. Use your choice from the second question to write an equation that relates \(m\) and \(d\).

(From Unit 6.4.1)

Exercise \(\PageIndex{9}\)

How many times larger is the first number in the pair than the second?

1. \(3^{4}\) is _____ times larger than \(3^{3}\).
2. \(5^{3}\) is _____ times larger than \(5^{2}\).
3. \(7^{10}\) is _____ times larger than \(7^{8}\).
4. \(17^{6}\) is _____ times larger than \(17^{4}\).
5. \(5^{10}\) is _____ times larger than \(5^{4}\).

(From Unit 6.3.1)