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 refers to the distance of a point from the zero or origin on the number line, regardless of the direction. The absolute value is always positive.

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.

## What are the applications of absolute?

There are several applications of absolute value in mathematics and other areas. Some of the most common applications are:

### 1. Measuring distance

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

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

### 2. Equations and inequalities with absolute value

Absolute value is used to solve equations and inequalities involving the distance between two values. For example, the equation |x-3| = 5 has the solutions x = -2 and x = 8.

### 3. Module of complex numbers

The absolute value of a complex number is also known as its modulus. It is used to find the distance from the complex number to the origin in the complex plane.

### 4. Data analysis

Absolute value is used in statistics and data analysis to find the difference between two values, such as the difference between the mean and a data point.

### 5. Physics and engineering

Absolute value is used in physics and engineering to find the magnitude of physical quantities such as velocity, acceleration, and force.

### 6. Deviations from the standard

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 actual temperature and the y-axis would represent the deviation of the temperature from the average temperature.

### 7. Bank withdrawals, credit cards and the nature of money

Even if there is a negative balance in any debit or savings account when money is owed to the bank, you can never withdraw a negative amount of money from the bank’s teller windows or ATMs.

The negative sign on your balance only indicates that you have an obligation to repay the bank which, once paid, will only return to zero (if paid exactly) or to a positive amount (if paid or deposited more than the negative balance).

The same can be said for credit cards, where the bank lends you money in advance periodically. This means that no matter how negative or positive your balance is, there is no such thing as a negative amount of money.

In general, the absolute value is a versatile concept that can be applied in many different fields to represent the distance between two values, the magnitude of a number or a physical quantity, etc.

## See also

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