# Absolute Value – Definition, Function, Symbol, Examples, Facts, FAQs

## What Is the Absolute Value of a Number?

The absolute value of a number tells us how far a given number is from the origin on a number line.

The absolute value only talks about distance, and not direction. We know that distance is a non-negative quantity. Thus, the absolute value of a number can never be negative.

So, what does absolute value mean? Observe the diagram. Here, we can see that

7 is 7 units to the right of 0.

Thus, the absolute value of 7 is 7.

-7 is 7 units to the left of 0.

Thus, the absolute value of -7 is 7.

## Absolute Value: Definition

Absolute value of a number can be defined as the distance of the number from the origin on a number line.

## Absolute Value: Symbol

Absolute value of a number is represented by writing the number between two vertical bars. Note that the vertical bars are not to be confused with parentheses or brackets

The absolute value of x is represented by |x|, and we read it as “absolute value of x.” It is also read as “modulus of x.” Sometimes, the absolute value of a number n is also written using the notion “abs(n).”

In the previous example, we can write

$|7| = | \;–\; 7| = 7$

## How to Find the Absolute Value of a Number

The absolute value of a positive number is the number itself. The absolute value of a negative number is written as the numeric value ignoring the negative sign. The absolute value of 0 is 0.

## Absolute Value of 0

The distance of the point zero from zero is 0 itself. Thus, the absolute value of 0 is 0.

## Absolute Value Properties

Non negativity: Absolute value of a number is always non-negative.

$| n | \ge 0$

Symmetry: The absolute value of a number equals the absolute value of its additive inverse.

$| \;-\; x | = | x |$

Positive definiteness: Absolute value of a number is 0 only if the number is 0.

$| n | = 0$ if and only if $n = 0$

On the same lines, we can say that $| x \;–\; y | = 0 ↔ x = y$

Multiplicativity: The absolute value of the product of two numbers is the product of the absolute values of the individual numbers.

$|x\times y| = | \times | \times | y |$

Preservation of division (equivalent to multiplicativity):

$|\frac{x}{y}| = |\frac{x}{y}| …y \neq 0$

Sub-additivity: The absolute value of the sum of two numbers is less than or equal to the addition of the absolute values of the individual numbers.

$|x + y| \le | x | + | y |$

Equivalent to sub-additivity: $| x \;–\; y | \ge | | x | \;–\; | y | |$

Triangle inequality (equivalent to subadditivity): $| x \;–\; y | \le | x \;–\; z | + | z \;–\; x |$

Idempotence: The absolute value of an absolute value of a number is simply the absolute value.

$| | x | | = | x |$

## Absolute Value Inequalities

Let’s discuss two important absolute value inequalities.

1. $| x | \le a ⇔ \;−\; a \le x \le a$

This is often referred to as AND inequality.

Example: $|x \;−\; 5| \le 7 ⇔ \;−\;7 \le x \;−\; 5 \le 7$

$⇔ \;−\; 2 \le x \le 12$

1. $| x | \ge a ⇔ x \le \;−\; a\; or\; x \ge a$

This is often referred to as OR inequality.

Example: $| x | \ge 3 ⇔ x \le \;−\;3$ or $x \ge 3$

## Absolute Value of Real Numbers

The absolute value of a positive real number is the number itself. The absolute value of a negative real number is the number without a negative sign. The absolute value of 0 is 0.

Thus, for a real number x, the absolute value is given by:

$| x | = x$ if $x \ge 0$

$| x | = \;–\; x$ if $x \lt 0$

## Absolute Value on a Number Line

As mentioned earlier, the absolute value of a number tells us the distance of a number from zero on a number line.

It is easy to understand the absolute value of integers (or any number) using the number line.

Example: The absolute value of 8 is 8.

$| 8 | = 8$.

Note that the absolute value of $\;–\; 8$ is also 8.

$| \;–\; 8 | = 8$.

## Absolute Value of a Number Examples

• $| 1 | = 1$
• $| \;-\;1| = 1$
• $| 3.5 | = 3.5$
• $| 0 | = 0$
• $| 7.9 | = 7.9$
• $| 5 \;–\; 3 | = | 2 | = 2$
• $| \;–\; 1 + 6 | = | 5 | = 5$
• $| \;–\; 6 \times 2| = | \;–\; 12 | = 12$

## Absolute Value Function

The absolute value function can be given by f(x) = |x| such that:

• $| x | = + x$  for $x \gt 0$
• $| x | = \;–\; x$  for  $x \lt 0$

## Applications of Absolute Value

• To find the distance between two points.

The absolute value of the difference of two real numbers (absolute difference) is the distance between them.

• To identify and remove outliers in data analysis.
• To set tolerance in quality analysis.
• To determine the degree of variation in the data compared from a certain point.

• The absolute value of x, denoted by | x |, is also known as “modulus” of x.
• $| x | \lt$ a implies that  $\;-\;a \lt x \lt a$
• $| x | \gt$ a implies that $x\lt (\;-\;a)$ or $x \gt a$

## Conclusion

In this article, we learned about the absolute value of a number, its symbol, properties, absolute value function, and we also learned its applications. Let’s move ahead and solve a few examples and MCQs based on the concept of absolute value for practice.

## Solved Examples on Absolute Value

1. What is the absolute value of  $\;–\;18$?

Solution:

The absolute value of a negative number is written as the numeric value, ignoring the negative sign.

The absolute value of $\;-\;18 = |\;-\;18| = 18$

2. Find the value of $-|\frac{\;-\;15}{22}|$.

Solution:

First, focus on the absolute value part.

The absolute value of $\;-\;\frac{15}{22} = |\frac{-15}{22}| = |\frac{15}{22}|$

There’s a minus sign outside the absolute value. It will get tagged along with the answer.

$-|\frac{-15}{22}| = \frac{-15}{22}$

3. Find: $|-9| + |10|$.

Solution:

Here,

|\;-\; 9 = 9|

$|10| = 10$

Thus, $|\;-\;9| + |10| = 9 + 10 = 19$

4. Find the absolute value.

$|5|, |\;-\;11|, |0|, |\;-\;7|, |\;-\;10|$

Solution:

$|5| = 5$

$|\;-\;11| = 11$

$|0| = 0$

$|\;-\;7| = 7$

$|\;-\;10| = 10$

5. Simplify: $|x \;−\; 1| \le 2$

Solution:

$|x \;−\; 1| \le 2$

$\;−\;2 \le x \;−\; 1 \le 2$

$(\;−\;2 + 1) \le x \le (2 + 1)$

$\;−\; 1 \le x \le 3$

## Practice Problems on Absolute Value

1

### The absolute value of $\;–\; 25$ is ___.

$\;–\; 25$
25
0
50
CorrectIncorrect
$|\;–\; 25| = 25$
2

### $|11| \times | \;-\; 12| =$ ___

$\;–\; 132$
0
132
$\;–\; 264$
CorrectIncorrect
$|11| = 11$
$|\;-\;12| = 12$
$|11| \times | \;-\;12| = 11 \times 12 = 132$
3

### Find the value of $|\frac{-3}{5}| \times |\frac{5}{6}|$

$\;– \;\frac{1}{2}$
0
1
$\frac{1}{2}$
CorrectIncorrect
Correct answer is: $\frac{1}{2}$
$|\frac{-3}{5}| \times |\frac{5}{6}| = \frac{3}{5} \times \frac{5}{6} = \frac{1}{2}$
4

### $|x \;-\; y| = 0$ if and only if

$x = \;-\; y$
$x = y$
$x \neq y$
$xy = 0$
CorrectIncorrect
Correct answer is: $x = y$
$|x\;-\;y| = 0$ if and only if $x = y$.
5

### $| 0 | =$

0
1
$\;-\;1$
Not defined
CorrectIncorrect
Absolute value of 0 is 0.
6

### The absolute value of 8 is the distance between ____ on a number line.

8 and -8
8 and 1
8 and 0
-8 and 1
CorrectIncorrect
Correct answer is: 8 and 0
The absolute value of 8 is the distance between 8 and 0 on a number line.

$|\;-\;x| = x$

So, $-\; |\;-\;x| = \;-\; x$

$| x | \le$ a can be written as a compound inequality $\;−\; a \le x \le a$.

$| x | \ge a$ can be simplified as $x \le \;−\; a$ or $x \ge a$.

Example: $| x | \ge 1$ means $x \le \;−\;1$ or $x \ge 1$.

Yes, two different numbers can have the same absolute value. The two numbers $\;–\;10$, and 10 have the same absolute value of 10.
$| \;–\; 10 | = | 10 | = 10$.

The absolute value of 0 is always 0 as the distance of 0 from 0 is 0.

Absolute numbers or absolute values simply refer to numeric values, ignoring the sign of the given numbers.

Absolute value is a distance, and distance can never be negative. Thus, the absolute value is always positive. To be more accurate, the absolute value is a non-negative number. Absolute value of 0 is 0. 0 is neither positive nor negative. It is considered a non-negative as well as a non-positive integer.

We use a bar on either side of a number to mean absolute value. The absolute value bars or the vertical bars | | represent “modulus symbol.” It gives us the distance of the number from 0.

Absolute value is also commonly referred to as numerical value or magnitude.