O**absolute value of a number**refers to the distance of a number from the origin of a number line. It's called |a| represented, which defines the size of any integer "a". The absolute value of any integer, whether positive or negative, is thereal numbers, regardless of its sign. It is represented by two vertical lines |a| represented what is known as the modulus of a.

Example: 5 is the absolute value for 5 and -5.

**|-5| = +5 e |+ 5| = +5**

In this article, we will learn what is the absolute value of a number, symbols, examples, absolute value on the number line, absolute value of real numbers, absolute value of complex numbers in detail.

**Index:**

- Definition
- Symbol
- examples
- absolute value function
- Characteristics
- Absolute value of the real number
- Absolute value of a complex number
- Absolute value on the number line
- Solved Problems
- practical questions
- common questions

## What is the absolute value of a number?

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

We can define the absolute values as follows:

**{ a with a ≥ 0 }**

**|and| = { -a with a < 0 }**

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

If the number is positive, this will only result in a positive number. And if the number is negative, then the modulus of that number is also a positive number. It is displayed as |n| denotes where n is an integer.

## absolute value symbol

The absolute value symbol is represented by the modulus symbol "|" |', with the numbers between them. For example, the absolute value of 9 is represented as |9| designated.

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 always be negative as it indicates distance, and distance cannot be negative. So always positive.

For example: "|a|", where "a" is the number whose absolute value is to be determined.

## Absolute value of a number Examples

Let's look at some examples of the absolute value of a number.

- |-1| = 1
- |-14| = 14
- |1| = 1
- |0| = 0
- |7| = 7
- |7-2| = |5| = 5
- |2+3| = |5| = 5
- |-3×5| = |-15| = 15

### Related articles:

- absolute value formula
- whole numbers
- Adding and subtracting whole numbers

## absolute value function

The absolute value function is given by f(x) = |x| for what:

- |x| = +x para x > 0
- |x| = -x para x < 0

## absolute value properties

If x and y are real numbers and the absolute values satisfy the following properties:

Property | Expression |

no negativity | | x | ≥ 0 |

positive certainty | | x | = 0 ↔ a = 0 |

multiplicative | | x × y| = |x| × |y| |

subadditivity | | x + y| ≤ | x | + | y | |

symmetry | |-x| = |x| |

Identity of the indistinguishable (corresponds to positive determinateness) | | x – y | = 0 ↔ a = b |

Triangular inequality (equivalent to subadditivity) | | x – y | ≤ | x – z | + | z – x | |

Conservation of division (equivalent to multiplicative) | | x/y| = | x | / | y | |

Equivalent to subadditivity | | x – y | ≥ | | x | – | y | | |

## Absolute value of a real number

When x is a real number, the absolute value satisfies the following conditions.

| x | = x, so x ≥ 0

| x | = – x, let x < 0

Let's look at the absolute value of 2 in the series of numbers below. Here |2| is the distance of 2 from 0 (zero). So both +2 and -2 are 2 away from the origin. But it would be taken as 2 because distance is never measured negatively.

## Absolute value of the complex number

Complex numbers consist of real and imaginary numbers. Therefore, 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}+j^{2})

Where x and y are the real numbers.

## Absolute value on the number line

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

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

**Solution:**Absolute value of |-9| it's +9.

So the graph for the absolute value of -9 looks like this

### problems and solutions

Let's understand this topic better with examples.

**Question 1:**

Arrange the given numbers in ascending order.

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

**Solution:**

First find the absolute values of the given numbers

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

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

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

**Example 3:**

Find the absolute value of a number -12/5

**Solution:**

Where to find: |-12/5|

|-12/5| = 12/5

So the absolute value of a number is -12/5 12/5

**Example 4:**

Find the absolute value for the following numbers:

(a) |-1/2|

(b) |72|

(c) |3/4|

**Solution:**

The absolute values of the numbers are:

(a) |-1/2| = 1/2

(b) |72| = 72

### practical questions

1. What is the absolute value of -13?

2. Find the absolute value of 100.

3. Arrange the following in ascending order.

|3|, |-9|, |1|, |-2|, |-5|

4. Draw the absolute value of -7 on the number line.

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## Frequently Asked Questions About Absolute Values

### What is an absolute value of a number?

The absolute value of a number represents the number's distance from zero on a number line.

### What is the absolute value of 4?

The absolute value of 4 is 4. |4| = 4

### What is the absolute value of -12?

The absolute value of -12 is 12. Because the distance between -12 and 0 on a number line is 12.

### What is the symbol for absolute value?

The symbol used to represent the absolute value is |x|, where x is an integer.

### What is the absolute value of -7?

The absolute value of -7 is 7. It is represented by |-7| shown = 7.

### Is the absolute value negative?

The absolute value of an integer is always positive and never negative.