Modulus Function

The modulus function, also known as the absolute value function, is a fundamental concept in mathematics that assigns to each real number its non-negative value. The modulus function is denoted by |x|, where x is a real number. The primary purpose of the modulus function is to measure the distance of a number from zero on the real number line, regardless of the direction.

Definition

The modulus function is defined as:

$$ |x| = \begin{cases} x & \text{if } x \geq 0 \ -x & \text{if } x < 0 \end{cases} $$

This definition can be understood as: if x is a non-negative number, then the modulus of x is x itself; if x is a negative number, then the modulus of x is the opposite of x (i.e., -x), which is a positive number.

Properties of the Modulus Function

The modulus function has several important properties:

1. Non-negativity: For any real number x, |x| is always non-negative.
2. Identity: For any non-negative real number x, |x| = x.
3. Opposite Numbers: For any real number x, |-x| = |x|.
4. Multiplicative: For any real numbers x and y, |xy| = |x||y|.
5. Triangle Inequality: For any real numbers x and y, |x + y| ≤ |x| + |y|.

Graph of the Modulus Function

The graph of the modulus function is a V-shaped curve that intersects the x-axis at the origin (0,0). For positive values of x, the graph coincides with the line y = x, and for negative values of x, the graph coincides with the line y = -x.

Examples

Here are some examples to illustrate the modulus function:

1. |3| = 3 because 3 is non-negative.
2. |-5| = 5 because the opposite of -5 is 5, which is non-negative.
3. |0| = 0 because 0 is neither negative nor positive.
4. |-2.5| = 2.5 because the opposite of -2.5 is 2.5.

Table of Differences and Important Points

Property	Description	Example
Non-negativity	The output is always non-negative.	`
Identity	If the input is non-negative, the output is the same as the input.	`
Opposite Numbers	The modulus of an opposite number is the same.	`
Multiplicative	The modulus of a product is the product of the moduli.	`
Triangle Inequality	The modulus of a sum is less than or equal to the sum of the moduli.	`

Applications

The modulus function is used in various fields such as engineering, physics, and computer science. It is particularly useful in situations where the magnitude of a quantity is important, but its direction is not. For example, in physics, the modulus function can represent the magnitude of a vector, and in computer science, it can be used to ensure that a number stays within a certain range (e.g., array indexing).

Conclusion

The modulus function is a simple yet powerful tool in mathematics. Its ability to provide the absolute value of a number makes it indispensable in both theoretical and applied mathematics. Understanding the properties and applications of the modulus function is essential for students and professionals alike.