The absolute value of a number is the distance from 0 to that number on the number line. The absolute value is related to the measure of distances or differences in cases where the direction is not important.

In this article, we will look at a more detailed definition of absolute value, learn about its properties, and learn about some of its real-life applications.

## What is the definition of absolute value?

The absolute value is denoted by two vertical lines that enclose the number or the expression. For example, the absolute value of the number 3 is written |3|. This means that the distance from 0 is 3.

Similarly, the absolute value of negative 3 is written |-3|. This also means that the distance from 0 is 3.

Consider the expression |x|>3. To represent this on the number line, we need all the numbers that have an absolute value greater than 3. This can be graphed by placing an open point on the number line.

Now, let’s consider the expression |x|≤3. This expression includes all absolute values that are equal to or less than 3. We can graph this expression by placing a closed point on the number line.

An easy way to represent absolute value with inequalities is as follows:

- For |
*x*|<3, we can write -3<*x<*-3. - For |x|=3, we can write
*x*=5 or*x*=-5. - For |
*x*+2|>3, we can write 3>*x*+2>-3.

## Properties of absolute value

The absolute value has the following fundamental properties:

1. Non-negativity |x| ≥ 0.

2. Multiplicativity |xy| = |x| |y|.

3. Subadditivity |x+y| ≤ |x|+|y|.

4. Idempotency ||x|| = |x|.

5. Symmetry |-x| = |x|.

6. Identity of discernible |x-y| = 0, ⇔ x=y.

7. Triangle of inequality |x-y| ≤ |x-z| + |z-y|.

8. Division preservation |x/y| = |x|/|y|, if we have y≠0.

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## What are the applications of absolute?

The measurement of distance is one of the most common applications of the absolute value. Distance is the absolute value of the difference in position between two points.

Therefore, given two points A and B, the distance between them is |A-B| which is equivalent to |B-A|. The distance does not depend on the direction. In general, the absolute value is used when the direction is not important.

Another important application is with money transfer. Suppose I have a spreadsheet to record my finances. If someone owes me money, that would show up on my spreadsheet as a positive number.

If I owe someone money, that would appear as a negative number. Regardless of who owes whom, when that debt is paid, the amount of money moved is always positive.

An absolute value function can be used to show how much a value deviates from the norm. For example, the average internal temperature of humans is 37 ° C. The temperature can vary by 0.5 ° C and still be considered normal. As a function, we can have the equation *y* = |*x*-37|.

If we were to graph this function, the *x*-axis would represent the current temperature and the *y*-axis would represent the deviation of the temperature from the average temperature.

## See also

Interested in learning more about absolute value? Take a look at these pages:

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