The axioms used by mathematicians to define the real numbers are just a way to make precise some intuitions that we and our students have developed from an early age. We will base our definition of numbers on this “ingrained knowledge” instead.

The basic knowledge and skills necessary for the definition of number:

  1. Counting 1, 2, 3, 4, … 
  2. Ideas about physical space, like:
    • Two objects can’t occupy the same place at the same time.
    • If I move an object from one place to another, its size does not change.
  3. Ideas about points and lines. We still use our basic ideas about space to reason about them, like:
    • If two points occupy the same place, they are the same point.
    • A line segment is described by two points on a line (the left endpoint and the right endpoint).
    • I can move a line segment without changing its length (sometimes, it helps to think of this as “making a copy of the line segment”).
    • I can divide a line segment up into any number of equal pieces.
    • A line doesn’t have any holes in it.

From these, we can define:

Definition. A number line, or real line, or x-axis, is a line with two distinct labelled points, 0 and 1.

Definition. A number is a point on a number line.