# Mathematics-Absolute Value

The Absolute Value ( or modulus ) of a number is the distance of a number from its origin. In other ways absolute value of a real number ‘a’ is numerical value of ‘a’ without regarding its sign.

The absolute value of 3 is 3

.

The absolute value of -2 is 2.

The notataion |a| was introduced by Karl Wiererstrass in 1841. Other Names of Absolute value include “Numerical Value” and “Magnitude”. For any real number ‘a’ absolute value or modulus of a number is denoted by |a| and is defined as

| a | = a if a = 0

| a | = -a if a < 0

The absolute value has following four fundamental properties::

Non-negativity | a | = 0

Positive –definiteness | a | = 0 <=> a = 0

Multiplicativeness | ab | = | a | | b |

Subadditivity | a + b | = | a | + | b |

Other Properties of Absolute value::

Self-composition | ( | a | ) | = | a |

Symmetry | – a | = | a |

Identity of indiscernibles ( equivalent to positive definiteness )

| a - b | = 0 <=> a = b

Triangle inequivality (=Subadditivity ) | a - b | = | a - c | + | c - b |

Preservation of division | a / b |= | a | / | b | ( if b not equal to zero )

Equivalent to subadditivity | a - b | = | | a | - | b | |

Examples::

| x - 6 | >= 12 <=> -12 >= x - 6 >= 12 <=> -6 >= x >= 18

| x - 4 | <= 5 <=> -5 <= x - 4 <= 5 <=> -1 <= x <= 9

The Absolute value of a number is used to define the Absolute difference of real numbers.

For any complex number, Z the absolute value is defined as follows:-

Z = x + iy

X = real part and Y = imaginery part

1. | Z | = v( ( x^{2} ) + y^{2 })

2. | z |^{2 }= x^{2 }+ y^{2}