In absolute values, the value is referring to the distance of a number or point from the origin or zero location of a number line. Geometrically the absolute values of |x| are a real number of x, and it is a geometric value without regard the sign. For example, a number 3 is the absolute values of both 3 and -3. The absolute values are represented by two vertical lines, such as | x | (read by modulus of x).
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Let us take, a and b are real numbers and then the absolute values are satisfying the following properties,
| a | ≥ 0
| a | = 0 ↔ a = 0
| a × b| = |a | × |b|
| a + b| ≤ | a | + | b |
5. Symmetry :
| - a | = | a |
6.Identity of indiscernible (equivalent to positive-definiteness) :
| a - b | = 0 ↔ a = b
7. Triangle inequality Triangle inequality (equivalent to subadditivity ) :
| a - b | ≤ | a - c | + | c - a |
8.Preservation of division (equivalent to multiplicativeness) :
| a / b| = | a | / | b |
9. Equivalent to subadditivity :
| a - b | ≥ | | a | - | b | |
In definition of absolute values, for all real number ‘x’ the absolute values, satisfy the following conditions
| x | = x , if x ≥ 0
| x | = - x , if x < 0.
In number line, the representations of the absolute values of a real number are the relative number's distance from zero or origin. For example, |2| is the distance of 2 from 0(zero).
Here, both 2 and -2 are single distance of 2 units from zero (0). |2| = |-2| = 2. In mathematics, the measurement of any distance is not negative values.
The absolute values are also called the positive square roots; it is represented by the square-root symbol without sign. Such as,
| x | = √( x2)
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