Final answer:
The distance formula on a number line is the absolute value of the difference between the coordinates of two points. It is expressed as d = |a - b|. In two and three dimensions, the distance formula expands to include the Pythagorean theorem, taking into account the coordinates along each axis.
Step-by-step explanation:
The distance formula on a number line is used to find the absolute distance between any two points. For a number line (one-dimensional space), the distance formula is quite simple: if you have two points, say A and B, with coordinates a and b respectively, the distance (d) between them is found by taking the absolute value of the difference of their coordinates. This can be expressed as d = |a - b|.
In a two-dimensional space, you would often use the Pythagorean theorem to find the distance between two points, which are defined by their (x, y) coordinates. In this case, if we have two points (x1, y1) and (x2, y2), the distance (d) between these two points is d = √((x2 - x1)² + (y2 - y1)²), which is derived from the theorem a² + b² = c² where a and b are the legs of a right triangle and c is the hypotenuse. This hypotenuse is the straight-line distance between the points.
When discussing three-dimensional space, points are specified by their (x, y, z) coordinates and the distance formula extends to d = √(x2 - x1)² + (y2 - y1)² + (z2 - z1)², which is again based on the Pythagorean theorem but applied in three dimensions.