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College Math Resources - Topics in Precalculus

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The Real Number System

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What are the Integers?

The Integers are often described as just the set of all positive and negative
whole numbers.
But what does that mean unless we already have an
idea of "negative" numbers?

In fact, for most of us (as for humankind in general) the first concept
of number was just the positive counting numbers (often called the Natural
Numbers if we include zero), with operations of addition corresponding
to combination of sets and multiplication for repeated addition. In this
system, the problem of a+ ? =b has an answer only if
b is greater than a.

But with the advent of trade (which often required the borrowing of
substantial sums to support a venture long before there was any return)
it became necessary to account for the fact that a person could have both
credits and debts (cheques written to you and cheques written by you).

Often these were recorded in ledgers in Black and Red ink respectively
(so "in the black" is good and "in the red" is bad). It soon became obvious
that these two kinds of numbers could be combined. For example if you have
written a friend a cheque for $100 and she has written you one for $50,
then you would both be equally well off if you got together and tore up
the two cheques and replaced them with just one from you to her for $50.
In terms of the coloured ledger this could be expressed as

100 + 50
= 50
or in more "modern" notation
(-100)+50=(-50)
All of the "rules" of integer arithmetic make perfectly good sense in
terms of combinations of credits and debts. For example, what happens to
your net credit if you "take away" or have torn up 5 cheques that you wrote
for $20 each? Surely you are now $100 better off - which corresponds to
the arithmetic statement

-5(-20)=+100
If you have never done so, it is worthwhile to verify that all of the
rules for signed arithmetic make sense in a similar way. If you do that
you will probably never again forget them (or even think of them as something
that has to be remembered!). This is just one example of the fact that
in Mathematics (and probably in most other areas as well) a little understanding
is worth more than a lot of memorizing. People who are successful in Math
usually actually remember very little. They have better things to do with
their brain cells!

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.