How far apart?
Distance between two rational numbers
The gap
Two friends stand on a number line, one at −2 and one at 3. How far apart are they?
What is the distance between −2 and 3?
What this lesson covers
The idea
The distance between two rational numbers a and b on the number line is |a – b|.
The gap
Two friends stand on a number line, one at −2 and one at 3. How far apart are they?
What is the distance between −2 and 3?
- 1
- 5
- 6
Measure the gap
Move A and B. The green band is the distance between them. Look at a − b and at |a − b|.
|a − b|
The distance between two rational numbers a and b on the number line is |a − b|.
It does not matter which number you subtract from which: |a − b| = |b − a|. The distance is never negative.
Notes
The distance between two rational numbers a and b on the number line is |a − b|.
Check yourself
The distance between −2 and 3 is |−2 − 3| = |−5| = ?
Answer: 5
|−5| = 5. The two friends are 5 units apart.
How far apart are −1/2 and 3/2? Find |−1/2 − 3/2|.
Answer: 2
−1/2 − 3/2 = −4/2 = −2, and |−2| = 2.
Which expression gives the distance between a and b?
Is |a − b| always the same as |b − a|?
- a − b. a − b can be negative, but a distance cannot be.
- |a + b|. a + b has nothing to do with the gap between a and b.
- |a − b| — correct. Yes. The absolute value makes sure the distance is not negative.
- Yes. They are both the distance between a and b — correct. Yes. a − b and b − a differ only in sign, and the absolute value removes the sign.
- No. One of them is always negative. Absolute values are never negative.
- Only when a is greater than b. It holds for any a and b.