Question
Explain why the absolute value of a number is never a negative number.
Answers
GPT-4o mini
The absolute value of a number is defined as the distance of that number from zero on the number line, without considering the direction (positive or negative). This means that:
1. **Definition**: For any real number \( x \), the absolute value is represented as \( |x| \). It is defined as:
- \( |x| = x \) if \( x \geq 0 \)
- \( |x| = -x \) if \( x < 0 \)
2. **Always Non-negative**: In both cases of the definition:
- If \( x \) is non-negative (0 or positive), \( |x| \) is equal to \( x \), which is non-negative.
- If \( x \) is negative, \( |x| \) is calculated as \( -x \). Because negating a negative number gives us a positive result, \( -x \) is always positive.
3. **Distance Interpretation**: The concept of absolute value can also be understood geometrically. The distance between any point on the number line and the point zero is always a non-negative quantity, because distance cannot be negative.
Thus, by definition and interpretation, the absolute value of any real number is always greater than or equal to zero, and therefore it can never be a negative number.
1. **Definition**: For any real number \( x \), the absolute value is represented as \( |x| \). It is defined as:
- \( |x| = x \) if \( x \geq 0 \)
- \( |x| = -x \) if \( x < 0 \)
2. **Always Non-negative**: In both cases of the definition:
- If \( x \) is non-negative (0 or positive), \( |x| \) is equal to \( x \), which is non-negative.
- If \( x \) is negative, \( |x| \) is calculated as \( -x \). Because negating a negative number gives us a positive result, \( -x \) is always positive.
3. **Distance Interpretation**: The concept of absolute value can also be understood geometrically. The distance between any point on the number line and the point zero is always a non-negative quantity, because distance cannot be negative.
Thus, by definition and interpretation, the absolute value of any real number is always greater than or equal to zero, and therefore it can never be a negative number.