## absolute value

### The absolute value is a mathematical function that gives the distance of a number from zero on the number line

The absolute value is a mathematical function that gives the distance of a number from zero on the number line. It is represented by the symbol “|” surrounding the number. For example, the absolute value of 5 is written as |5|, and its value is 5 since 5 is 5 units away from zero.

The absolute value function is defined as follows:

For any number x, if x is positive or zero, then the absolute value of x is x itself.

If x is negative, then the absolute value of x is -x. In other words, the negative sign is removed to make it positive.

For instance, the absolute value of -3 is |-3|, which equals 3 because it is 3 units away from zero. Similarly, the absolute value of 0 is |0|, which is also 0.

The absolute value function is often used to ensure that a result is always positive, no matter the input. It has numerous applications in mathematics and other fields, such as solving equations, finding distances, and working with inequalities.

Here are a few properties of absolute value:

1. Magnitude: The absolute value of a number is always greater than or equal to zero.

|x| ≥ 0

2. Identity: The absolute value of zero is zero.

|0| = 0

3. Symmetry: The absolute value of a number is the same as the absolute value of its negative counterpart.

|-x| = |x|

4. Triangle Inequality: For any two numbers a and b, the absolute value of their sum is less than or equal to the sum of their absolute values.

|a + b| ≤ |a| + |b|

Understanding the concept of absolute value is crucial in various mathematical topics, such as algebra, calculus, and number theory. It allows us to manipulate and solve equations and inequalities involving real numbers effectively.

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