MathematicsChapter 7: Coordinate Geometry
Distance On A Number Line
On a number line every point has one coordinate.
Learning objectives
- Explain distance on a number line in clear mathematical language
- Apply the ideas in absolute value
- Solve problems involving symmetric gaps
Definition
On a number line every point has one coordinate. The distance between points with coordinates a and b is |b-a|. Absolute value is never negative, so distance is a length, not a signed walk. Distance is also symmetric: |b-a| = |a-b|, so the gap from 2 to 7 is the same as the gap from 7 to 2. This one-dimensional idea is the seed of the later plane formula.
Where / symbols
- |b-a|
- The distance between points with coordinates a and b is |b-a|.
- never negative
- Absolute value is never negative, so distance is a length, not a signed walk.
- symmetric
- Distance is also symmetric: |b-a| = |a-b|, so the gap from 2 to 7 is the same as the gap from 7 to 2.
- coincide
- If a=b the distance is 0 and the points coincide.
Overview
Length has no direction on the line:
Absolute Value
Write |b-a| and drop any minus sign that appears after you subtract. From 3 to -5 the difference is -8, but |-8|=8, so the distance is 8. Distance is never negative: a length of -4 units does not make sense. If a=b the distance is 0 and the points coincide.
Examples |7-2|=5 · |-5-3|=8 · |4-4|=0
Symmetric Gaps
Symmetric means the order of the two points does not matter: |b-a| = |a-b|. You can start at either end. On a map, the road length between two milestones is the same whichever way you drive. Later, the plane distance formula will also be symmetric in the two points.
Examples |7-2| equals |2-7| · |0-(-3)| equals 3 · A to B same as B to A
What it means
On a number line every point has one coordinate.
Main ideas
1. ABSOLUTE VALUE: Write |b-a| and drop any minus sign that appears after you subtract. 2. SYMMETRIC GAPS: Symmetric means the order of the two points does not matter: |b-a| = |a-b|.
Why it matters
Think of |b-a| as how many unit steps lie between a and b. Directed travel can be negative (leftward), but distance reports only how far. Do not leave a minus sign in a distance answer.
How to answer in an exam
Distance on a line is |b-a|, never b-a with a possible minus. If you forget the bars you may write a negative length, which is marked wrong. Symmetric: swapping the two points does not change the answer.
Explanation
Think of |b-a| as how many unit steps lie between a and b. Directed travel can be negative (leftward), but distance reports only how far. Do not leave a minus sign in a distance answer.
In simple words
Distance on a line is |b-a|, never b-a with a possible minus. If you forget the bars you may write a negative length, which is marked wrong. Symmetric: swapping the two points does not change the answer.
Key points
- |b-a|
- never negative
- symmetric
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Distance On A Number Line
Sticky Notes · Quick Revision
Core Idea
- On a number line every point has one coordinate.
Key Terms
- |b-a|: The distance between points with coordinates a and b is |b-a|.
- never negative: Absolute value is never negative, so distance is a length, not a signed walk.
Main Pattern
- 1. ABSOLUTE VALUE: Write |b-a| and drop any minus sign that appears after you subtract.
- 2. SYMMETRIC GAPS: Symmetric means the order of the two points does not matter: |b-a| = |a-b|.
Worked Reminder
- |7-2|=5
- |-5-3|=8
- |4-4|=0
Common Mistake
- Distance on a line is |b-a|, never b-a with a possible minus. If you forget the bars you may write a negative length, which is marked wrong. Symmetric: swapping the two points does not change the answer.
Exam Tip
- State the rule for distance on a number line, show each algebraic step, and check the final answer against the question.
60-second revision
- On a number line every point has one coordinate.
- Write |b-a| and drop any minus sign that appears after you subtract.
- Symmetric means the order of the two points does not matter: |b-a| = |a-b|.
- Think of |b-a| as how many unit steps lie between a and b.
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