Langara College - Department of Mathematics and Statistics

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Internet Resources for the Calculus Student - Topics in Precalculus

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The Real Number System

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The Number Line

Since the Real Numbers were invented to account for all possible measurements
along a line, it is not surprising that we can understand a lot about them
in terms of a picture of such a line.

<click on the picture to go to an interactive
version>
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).

Related Links:
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.