Absolute Value: Definition, Equation and Properties with Examples

The separation of a number from a number line's origin is referred to as a number's absolute value. It is denoted by the symbol |a|, which specifies the size of any integer 'a'. Any integer's absolute value, regardless of its sign or whether it is positive or negative, will be a non-negative real number. To write the modulus of a, two vertical lines |a| are used to symbolize it.

JEE Main 2025: Sample Papers | Mock Tests | PYQs | Study Plan 100 Days

JEE Main 2025: Maths Formulas  | Study Materials

JEE Main 2025: Syllabus | Preparation Guide | High Scoring Topics 

This Story also Contains

1. What Is A Number's Absolute Value?
2. Symbol For Absolute Value
3. Absolute Value in Equations or Inequalities.
4. Examples For Absolute Value Of A Number
5. Absolute Value Function
6. Properties

What Is A Number's Absolute Value?

The absolute value of an integer or number is how far it is from zero on a number line. Because of this, the absolute value is always positive or zero and never negative. The following definition applies to the absolute values:

|a| = ( a if a ≥ 0 )

( -a if a < 0 )

Only positive numbers will be generated if the number is positive. Additionally, the number's modulus will also be negative if it is negative. It is represented by the symbol |n|, where n is an integer.

Symbol For Absolute Value

The absolute value sign is represented by the modulus symbol "| |" and the numbers inside the symbol. For example |9| is used to represent the absolute value of nine.

Any number's absolute value is determined by how far it is from the line's origin. Additionally, it indicates whether a number is positive or negative and its polarity. It can never be negative since it represents distance, which cannot be negative. It is therefore always positive.

‘| |’ – Absolute value symbols, ‘| | pipes’ are used to represent the absolute values.

The number "a'' is the one whose absolute value needs to be determined, for example: "|a|".

Absolute Value in Equations or Inequalities.

The symbol |a| is used to signify a number's absolute value. On a number line, this value or number signifies the separation (absolute difference) between a and 0. An absolute value equation is one that contains an absolute value expression. Absolute value inequalities come in two different varieties. One with |a| < b or less, and the other with |a|> b or more.

Examples For Absolute Value Of A Number

Let's look at a few illustrations of the absolute value of numbers.

|-1| = 1

|-14| = 14

|1| = 1

|0| = 0

|7| = 7

Absolute Value Function

If f(x) = |x|, the absolute value function is defined as follows:

f(x) = +x if x > 0,

And, f(x) = -x if x < 0

Properties

If x and y are real numbers, and the absolute values meet the criteria listed below: