The division of a number from a number line’s origin is referred to as a number’s absolute value. It expresses the magnitude of any integer ‘a’ and is indicated as |a|. Any integer, regardless of sign, has an absolute value that is equal to the real numbers, whether it is positive or negative. It is denoted by the modulus of a, which is two vertical lines |a|.

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## What is Absolute Values?

**The absolute value (or modulus) | x | of a real number x is the non-negative value of x without regard to its sign.**

For example, the absolute value of 5 is 5, and the absolute value of −5 is also 5. The absolute value of a number may be thought of as its distance from zero along real number line. Furthermore, the absolute value of the difference of two real numbers is the distance between them.

## Properties of Absolute Values

If x and y are real numbers, then the following properties are satisfied by the absolute values:

Property | Expression |

Non-negativity | | x | ≥ 0 |

Positive-definiteness | | x | = 0 ↔ a = 0 |

Multiplicativeness | | x × y| = |x| × |y| |

Subadditivity | | x + y| ≤ | x | + | y | |

Symmetry | |-x| = |x| |

Identity of indiscernible (equivalent to positive definiteness) | | x – y | = 0 ↔ a = b |

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

Preservation of division (equivalent to multiplicativeness) | | x / y| = | x | / | y | |

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

## Real Numbers in Absolute Values

The absolute value will meet the following requirements if x is a real number.

- If x ≥ 0, then | x | = x.
- If x < 0, then | x | = – x.

Let’s examine the absolute value of 2 in the following numerical line. Here, the distance between 2 and 0 is shown as |2|. Thus, the distance of 2 from the origin is equal to both +2 and -2. However, since distance is never measured in negative values, it would be regarded as 2.

## Complex Number of Absolute Value

Real and imaginary numbers make up complex numbers. As a result, it is challenging to determine their absolute value, in contrast to integers. Assume that the supplied complex number is x+iy.

- 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.

## Number Line in Absolute Value

The absolute value graph is the graph that displays absolute values. Since any real number has a positive absolute value, any number or function graph will only have an absolute value that is positive.

**Example:**Graph the absolute value of the number -9.**Solution:**Absolute value of |-9| is +9.

## Question to Practice

**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.

**Question 3:**

- Find the absolute value of a number -12/5

**Solution:**

- To find: |-12/5|
- |-12/5| = 12/5
- Hence, the absolute value of a number -12/5 is 12/5

**Question**** 4:**

- Find the absolute value for the following numbers:
- (a) |-1/2|
- (b) |72|
- (c) |3/4|

**Solution:**

- The absolute value of the numbers are:
- (a) |-1/2| = 1/2
- (b) |72| = 72
- (c) |3/4| = 3/4

### Practices This

- What is the absolute value of -13?
- Find the absolute value of 100.
- Arrange the following in ascending order.

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

- Plot the absolute value of -7 on the number line.

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## Conclusion

It is important to understand absolute values in mathematics and all of its applications. Absolute values include characteristics that make them helpful in describing magnitudes, solving equations, and determining how far a number is from zero on the number line. Calculus, geometry, complex numbers, optimization, and other areas find use for them, making them an important idea with a wide range of applications in both mathematics and real-world problem solving.