Show why the distance between two points on a number line (the x-axis) is | a – b |

Question

Use the Distance Formula and the x-axis of the coordinate plane. Show why the distance between two points on a number line (the x-axis) is | a – b |, where a and b are the x-coordinates of the points.

Answer 1

The distance formula for two points P1: (x₁,y₁) and P2:  (x₂,y₂) is:

      d = √( (x₂-x₁)² + (y₂-y₁)² )

 

On the number line a and b are the x-coordinates and the y-coordinates are both 0.  The points are  (a,0) and (b,0).

 

In the Distance Formula, this makes the distance:

    d = √( (x₂-x₁)² )

 

Now, since the square root may have either both positive and negative roots, we write the value as:

      d = ± (x₂-x₁)

 

     This is either (x₂-x₁)  or (-(x₂ - x₁))

When considering distance (a measurement without a sign attached), we write the positive distance:  |x₂ - x₁|.

     

Answer 2

To determine distance the starting point is subtracted from the ending point.

iE.g., if you go from 2 to 8 the distance is 8 - 2 or 6. If you go from 8 to 2 the distance is still 6 although 2 - 8 = -6.