The absolute value of the number line distance is simply the magnitude or size of that distance, always expressed as a non-negative number. It represents how far a number is from zero, irrespective of direction.
Understanding Absolute Value
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Definition: The absolute value of a number is its distance from zero on the number line. This distance is always a positive number or zero.
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Notation: We denote the absolute value of a number x as |x|.
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Example: 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.
Absolute Value and Distance
The question directly asks about the absolute value of the number line distance. This is subtly different from the general definition of absolute value. Consider two points, A and B, on a number line:
- The distance between A and B can be found by subtracting their coordinates (e.g., B - A). This could result in a positive or negative number, depending on which order you subtract.
- The absolute value of the distance between A and B is |B - A| or |A - B|. This will always be a non-negative value, representing the length of the segment connecting A and B.
Example:
Let point A be located at -2 and point B be located at 3 on the number line.
- The distance from A to B (B - A) is 3 - (-2) = 5
- The distance from B to A (A - B) is -2 - 3 = -5
- The absolute value of the distance from A to B is |5| = 5
- The absolute value of the distance from B to A is |-5| = 5
Regardless of the direction, the absolute value of the distance between A and B is 5.
Key Takeaways
- Absolute value expresses magnitude without regard to sign.
- Distance on a number line can be positive or negative depending on the direction considered.
- The absolute value of a number line distance is always non-negative and represents the length of the segment.
