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# Absolute Value of Integers

## Identify absolute value as the distance an integer is from zero.

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Absolute Value of Integers

Credit: Sergey Norin
Source: https://www.flickr.com/photos/5nap/15487502334

Victor's grandparents live in Moscow, Russia. They told him that in the winters the high temperatures average around \begin{align*}-10^\circ C\end{align*} while in the summers the high temperatures average around \begin{align*}23^\circ C\end{align*}. In science class Victor learned that in Celsius, water freezes at \begin{align*}0^\circ C\end{align*}. How could Victor use absolute value to determine which temperature is closer to the freezing point of water, \begin{align*}0^\circ C\end{align*}?

In this concept, you will learn how to find the absolute value of an integer and you will learn how to find the opposite of an integer.

### Guidance

The absolute value of a number is its distance from zero on the number line. The symbol for absolute value is \begin{align*}| \ |\end{align*} .

Let's look at an example.

\begin{align*}|-3|\end{align*}

This is read as “the absolute value of -3”. To figure out the absolute value of -3, think about how far the number -3 is from zero. -3 is 3 units from zero on the number line. This means the absolute value of -3 is 3.

The answer is \begin{align*}|-3| = 3\end{align*}.

Let's look at another example.

\begin{align*}|3|\end{align*}

This is read as “the absolute value of 3”. 3 is 3 units from zero on the number line. This means the absolute value of 3 is 3.

The answer is \begin{align*}|3| = 3\end{align*}.

Notice that both \begin{align*}|-3|\end{align*} and \begin{align*}|3|\end{align*} are equal to 3. This is because both -3 and 3 are 3 units from zero on the number line.

Keep in mind that the absolute value of a number will always be a positive value. This is because all numbers will always be a positive number of units away from zero on the number line.

In general, any positive integer and its opposite will have the same absolute value. To find the opposite of an integer, just change its sign either from positive to negative or from negative to positive.

Let's look at an example.

Find the opposite of -16.

-16 is a negative integer. To find its opposite, change the negative sign to a positive sign.

The answer is that the opposite of -16 is +16 or 16.

Let's look at another example.

Find the opposite of 900.

900 is a positive integer. It's the same as +900. To find its opposite, change the positive sign to a negative sign.

The answer is that the opposite of 900 is -900.

### Guided Practice

Find the absolute value.

\begin{align*}|-234|\end{align*}

First, read the expression. This expression is read as “the absolute value of -234”.

Next, you should think about how far the integer -234 is from zero on the number line. -234 is 234 units from zero on the number line. This means that the absolute value of -234 is 234.

Remember that in general, the absolute value of a number will always be positive.

### Examples

#### Example 1

Find the absolute value.

\begin{align*}|22|\end{align*}

First, read the expression. This expression is read as “the absolute value of 22”.

Next, you should think about how far the integer 22 is from zero on the number line. 22 is 22 units from zero on the number line. This means that the absolute value of 22 is 22.

Notice that our sign didn't change. The absolute value of a number will always be positive.

#### Example 2

Find the absolute value.

\begin{align*}|-222|\end{align*}

First, read the expression. This expression is read as “the absolute value of -222”.

Next, you should think about how far the integer -222 is from zero on the number line. -222 is 222 units from zero on the number line. This means that the absolute value of -222 is 222.

Remember that the absolute value of a number will always be positive.

#### Example 3

Find the opposite of -18.

First, notice that -18 is a negative integer. To find its opposite, change the negative sign to a positive sign.

Credit: OliBac
Source: https://www.flickr.com/photos/olibac/2983779842/

Remember Victor who has grandparents in Moscow? The winter temperature there is around \begin{align*}-10^\circ C\end{align*} while the summer temperature is around \begin{align*}23^\circ C\end{align*}. Victor wondered which of these temperatures is closer to the freezing point of water, \begin{align*}0^\circ C\end{align*}.

First, consider the winter temperature of \begin{align*}-10^\circ C\end{align*}.

\begin{align*}|-10| = 10\end{align*}.

This means that \begin{align*}-10°C\end{align*} is 10 degrees away from the freezing point of water.

Next, consider the summer temperature of \begin{align*}23°C\end{align*}.

\begin{align*}|23| = 23\end{align*}

This means that \begin{align*}23^\circ C\end{align*} is 23 degrees away from the freezing point of water.

The answer is that because \begin{align*}|-10|\end{align*} is less than \begin{align*}|23|\end{align*}, the winter temperature is closer to the freezing point of water.

### Explore More

Write the opposite of each integer.

1. 20

2. -7

3. 22

4. -34

5. 0

6. -9

7. 14

8. 25

Find the absolute value of each integer.

9. \begin{align*}|13|\end{align*}

10. \begin{align*}|-11|\end{align*}

11. \begin{align*}|-5|\end{align*}

12. \begin{align*}|17|\end{align*}

13. \begin{align*}|-9|\end{align*}

### Vocabulary Language: English

fraction

fraction

A fraction is a part of a whole. A fraction is written mathematically as one value on top of another, separated by a fraction bar. It is also called a rational number.
opposite

opposite

The opposite of a number $x$ is $-x$. A number and its opposite always sum to zero.
Whole Numbers

Whole Numbers

The whole numbers are all positive counting numbers and zero. The whole numbers are 0, 1, 2, 3, ...