In fact relative to any choice of a fixed starting point (often called the Origin), with a unit of measurement, and a specified direction, each point on the line can be located by specifying exactly one real number (with magnitude equal to its number of units from the origin and sign being positive in the given direction and negative in the opposite direction). The key points are the Origin (which corresponds to the number 0), and the point one unit away in the "positive" direction (which corresponds to the number 1).
Once 0 and 1 have been placed on the line, this forces everything else:
First the positive whole numbers correspond to repeated steps the same
as from 0 to 1
Then the rest of the integers going the other way
Then fractional steps for the rationals
But that's not all!
There are in fact positions on the line which do not correspond to any exact fractional number of steps no matter how tiny the fractions we use. The numbers corresponding to these positions are called "irrational" (because they are not ratios of integers).
The University of Saskatchewan's 'Exercises in Math Readiness' site includes a section on Absolute Value and Distance . Their section on Decimal Expansions also is somewhat related to this.
At SFU's Centre for Experimental and Constructive Mathematics, their RevEng pages include java applets that will "reverse engineer" any decimal you type in, and try to find simple expressions whose values agree with your number.
If you have come across any good web-based illustrations of these
and related concepts,
please do let us know and we will add them here.
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