Question
Answer and Explanation
The absolute value of a number is its distance from zero on the number line, regardless of direction. It is always a non-negative value.
Essentially, the absolute value of a number is its magnitude or size, without considering whether the number is positive or negative.
Here's a breakdown:
Formal Definition:
For any real number 'x', the absolute value of 'x', denoted as |x|, is defined as:
|x| = x if x ≥ 0 (if 'x' is positive or zero)
|x| = -x if x < 0 (if 'x' is negative)
Key Characteristics:
- Always Non-negative: The absolute value of any number is always greater than or equal to zero. It can never be negative.
- Distance from Zero: It represents the distance of the number from zero on the number line. Distance is always measured as a positive quantity.
Examples:
- The absolute value of 5, written as |5|, is 5, because 5 is 5 units away from zero.
- The absolute value of -5, written as |-5|, is also 5, because -5 is also 5 units away from zero.
- The absolute value of 0, written as |0|, is 0, because 0 is 0 units away from zero.
Use Cases:
The concept of absolute value is used in a variety of mathematical and real-world scenarios:
- Distance Calculations: Used in geometry to find distances between points.
- Error and Tolerances: In engineering and measurement, it represents acceptable deviation.
- Programming: Used in algorithms when dealing with the magnitude of quantities.
In summary, understanding the absolute value is key for working with numbers, especially when you need to consider the magnitude or distance from zero irrespective of the sign.