The value which refers to the distance of a number from the origin of a number line is called **absolute values**. It is represented as |a|, which defines the magnitude of any integer ‘a’. The absolute value of any integer, whether positive or negative, will be the real numbers, regardless of which sign it has. It is represented by two vertical lines |a|, which are known as the modulus of a.

For example: 5 is the absolute value for both 5 and -5.

**|-5| = +5 and |+ 5| = +5**

It helps in connecting the absolute value of complex numbers and the magnitudes of the vector.

We can define the absolute values like the following

**{ a if a ≥ 0 }**

**|a| = { -a if a < 0 }**

Note: There is no absolute value for 0 because the absolute value changes the sign of the numbers into positive and zero has no sign.

## Definition

The absolute value of a number or integer is the actual distance of the integer from zero, in a number line. Therefore, the absolute value is always a positive value and not a negative number.

If the number is positive then it will result in a positive number only. And if the number is negative, then the modulus of this number will also be a positive number. It is denoted as |n|, where n is an integer. Let us see some examples here.

- |-1| = 1
- |-14| = 14
- |1| = 1
- |0| = 0
- |7| = 7
- |7-2| = |5| = 5
- |2+3| = |5| = 5

## Absolute Value Symbol

The distance of any number from the origin on the number line is the absolute value of that number. It also shows the polarity of the number whether it is positive or negative. It can be negative ever a s it shows the distance and the distance can’t be negative. So, it is always positive.

‘| |’ – Absolute value symbols, ‘| | pipes’ are used to represent the absolute values.
For example: ‘|a|’, where “a” is the number whose absolute value has to be determined. |

**Also, read:**

## Absolute Value Properties

If x and y are real numbers and then the absolute values are satisfying the following properties,

- Non-negativity
- Positive-definiteness
- Multiplicativeness
- Subadditivity
- Symmetry
- Identity of indiscernible (equivalent to positive-definiteness)
- Triangle inequality (equivalent to subadditivity)
- Preservation of division (equivalent to multiplicativeness)
- Equivalent to subadditivity

1. **Non-negativity :**

| x | ≥ 0

2. **Positive-definiteness:**

| x | = 0 ↔ a = 0

3. **Multiplicativeness:**

| x × y| = |x| × |y|

4. **Subadditivity:**

| x + y| ≤ | x | + | y |

5. **Symmetry :**

| – x | = | x |

6. **Identity of indiscernible (equivalent to positive definiteness) :**

| x – y | = 0 ↔ a = b

7.** Triangle inequality (equivalent to subadditivity):**

| x – y | ≤ | x – z | + | z – x |

8. **Preservation of division (equivalent to multiplicativeness):**

| x / y| = | x | / | y |

9.** Equivalent to subadditivity:**

| x – y | ≥ | | x | – | y | |

## Absolute Value of a Real Number

If a real number x, the absolute value will satisfy the following conditions.

| x | = x, if x ≥ 0

| x | = – x, if x < 0

Let’s look at the absolute value of 2 in the number line given below. Here, |2| is the distance of 2 from 0(zero). So, both +2 and -2 is the distance of 2 from the origin. But it would be taken as 2 because distance is never measured in negative.

## Absolute Value of Complex Number

Complex numbers consist of real numbers and imaginary numbers. Hence, unlike integers, it is difficult to find the absolute value for them. Suppose, x+iy is the given complex number.

z = x+iy

The absolute value of z will be;

|z|= √[Re(z)^{2}+Im(z)^{2}]

|z| = √(x^{2}+y^{2})

Where x and y are the real numbers.

## Problems and Solutions

Let’s understand this topic better with the help of examples.

**Question 1: **Arrange the given numbers in ascending order.

-|-14|, |12|, |7|, |-91|, |-5|, |-8|, |-65|, |6|

**Solution:**

Initially, find the absolute values of the given numbers

-14, 12, 7, 91, 5, 8, 65, 6

Now, arrange the numbers in ascending order (smallest to the largest number)

-14, 5, 6, 7, 8, 12, 65, 91

**Question 2: **Arrange the given numbers in ascending order.

-|-24|, |21|, 17|, |-109|, |-15|, |-19|, |-75|, |16|

**Solution:**

Find the absolute values of the given numbers

-24, 21, 17, 109, 15, 19, 75, 16

Now, arrange the numbers in ascending order (smallest to the largest number)

-24, 15, 16, 17,19, 21, 75, 109

## Absolute Value Number Line/Graph

The graph of absolute values is called the absolute value graph. As we know the absolute value of any real number is positive, so the absolute value of any number or function graph will lie on the positive side only.

**Example:** Graph the absolute value of the number -9.

**Solution: **Absolute value of |-9| is +9.

So the graph of the number -9 will look like following

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