## absolute value

### The absolute value of a number is the distance between that number and zero on a number line

The absolute value of a number is the distance between that number and zero on a number line. It is represented by two vertical bars surrounding the number. The absolute value of a number is always positive or zero.

Mathematically, if we have a real number “a”, then the absolute value of “a” is denoted as |a|. The absolute value function is defined as follows:

|a| = a, if a is greater than or equal to zero

|a| = -a, if a is less than zero

For example, let’s consider the number -5. The absolute value of -5 is 5 because it is the distance between -5 and 0 on a number line. Symbolically, | -5 | = 5.

Similarly, the absolute value of 3 is 3, because it is the distance between 3 and 0. Symbolically, | 3 | = 3.

Some key properties of absolute value include:

1. Non-negativity: The absolute value of any real number is always non-negative or zero. This means that |a| ≥ 0, for all real numbers “a”.

2. Symmetry: The absolute value function is symmetric around the origin. This means that |a| = |-a|, for all real numbers “a”.

3. Triangle inequality: For any two real numbers “a” and “b”, the absolute value of their sum is less than or equal to the sum of their absolute values. Mathematically, |a + b| ≤ |a| + |b|.

The concept of absolute value is important in various areas of mathematics, including algebra, calculus, and number theory. It is useful in solving equations involving inequalities, finding the distance between two points, and determining the magnitude of quantities.

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